ÿWPCL ûÿ2BJ|xÕ!Ð x ÐÐÐüð ä ØÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÐЊ‚ÐÈÐÁ`ÁSOURCES © page !ÕÐ °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐЊ‚ÐÈÐ Ã Ã1.ÁÁBIOGRAPHICAL MATERIALÄ Ä ©© in chronological order ÁÁà ÃALCUINÄ Ä (c735-804) Phillip Drennon Thomas. Alcuin of York. DSB I, 104-105. Robert Adamson. Alcuin, or Albinus. DNB, (I, 239-240), 20. Andrew Fleming West. Alcuin and the Rise of the Christian Schools. (The Great Educators ©© III.) Heinemann, 1893. The only book on Alcuin that I found which deals with the Propositiones. Stephen Allott. Alcuin of York c. A.D. 732 to 804 ©© his life and letters. William Sessions, York, 1974. ÁÁà ÃFIBONACCI [LEONARDO PISANO]Ä Ä (c1170©>1240) ÁÁSee also the entries for Fibonacci in Common References. Fibonacci. (1202 ©© first paragraph); 1228 ©© second paragraph, on p. 1. In this paragraph he narrates almost everything we know about him. [In the second ed., he inserted a dedication as the first paragraph.] ÁÁÁÁThe paragraph ends with the notable sentence which I have used as a motto for this work. "Si quid forte minus aut plus iusto vel necessario intermisi, mihi deprecor indulgeatur, cum nemo sit qui vitio careat et in omnibus undique sit circumspectus." (If I have perchance omitted anything more or less proper or necessary, I beg indulgence, since there is no one who is blameless and utterly provident in all things. [Grimm's translation.]) Richard E. Grimm. The autobiography of Leonardo Pisano. Fibonacci Quarterly 11:1 (Feb 1973) 99©104. He has collated six MSS of the autobiographical paragraph and presents his critical version of it, with English translation and notes. Sigler, below, gives another translation. I give Grimm's translation, omitting his notes. ÁÁÁÁAfter my father's appointment by his homeland as state official in the customs house of Bugia for the Pisan merchants who thronged to it, he took charge; and, in view of its future usefulness and convenience, had me in my boyhood come to him and there wanted me to devote myself to and be instructed in the study of calculation for some days. There, following my introduction, as a consequence of marvelous instruction in the art, to the nine digits of the Hindus, the knowledge of the art very much appealed to me before all others, and for it I realized that all its aspects were studied in Egypt, Syria, Greece, Sicily, and Provence, with their varying methods; and at these places thereafter, while on business, I pursued my study in depth and learned the give©and©take of disputation. But all this even, and the algorism, as well as the art of Pythagoras I considered as almost a mistake in respect to the method of the Hindus. Therefore, embracing more stringently that method of the Hindus, and taking stricter pains in its study, while adding certain things from my own understanding and inserting also certain things from the niceties of Euclid's geometric art, I have striven to compose this book in its entirety as understandably as I could, dividing it into fifteen chapters. Almost everything which I have introduced I have displayed with exact proof, in order that those further seeking this knowledge, with its pre©eminent method, might be instructed, and further, in order that the Latin people might not be discovered to be without it, as they have been up to now. If I have perchance omitted anything more or less proper or necessary, I beg indulgence, since there is no one who is blameless and utterly provident in all things. F. Bonaini. Memoria unica sincrona di Leonardo Fibonacci novamente scoperta. Giornale Storico degli Archivi Toscani 1:4 (Oct©Dec 1857) 239©246. This reports the discovery of a 1241 memorial of the Comune of Pisa, which I reproduce as it is not well known. This grants Leonardo an annual honorarium of 20 pounds. In 1867, a plaque bearing this inscription and an appropriate heading was placed in the atrium of the Archivio di Stato in Pisa. ÁÁÁÁ"Considerantes nostre civitatis et civium honorem atque profectum, qui eis, tam per doctrinam quam per sedula obsequia discreti et sapientis viri magistri Leonardi Bigolli, in abbacandis estimationibus et rationibus civitatis eiusque officialium et aliis quoties expedit, conferunter; ut eidem Leonardo, merito dilectionis et gratie, atque scientie sue prerogativa, in recompensationem laboris sui quem substinet in audiendis et consolidandis estimationibus et rationibus supradictis, a Comuni et camerariis publicis, de Comuni et pro Comuni, mercede sive salario suo, annis singulis, libre xx denariorum et amisceria consueta dari debeant (ipseque pisano Comuni et eius officialibus in abbacatione de cetero more solito serviat), presenti constitutione firmamus." ÁÁÁÁA translation follows, but it can probably be improved. My thanks to Steph Maury Gannon for many improvements over my initial version. ÁÁÁÁConsidering the honour and progress of our city and its citizens that is brought to them through both the knowledge and the diligent application of the discreet and wise Maestro Leonardo Bigallo in the art of calculation for valuations and accounts for the city and its officials and others, as often as necessary; we declare by this present decree that there shall be given to the same Leonardo, from the Comune and on behalf of the Comune, by reason of affection and gratitude, and for his excellence in science, in recompense for the labour which he has done in auditing and consolidating the above mentioned valuations and accounts for the Comune and the public bodies, as his wages or salary, 20 pounds in money each year and his usual fees (the same Pisano shall continue to render his usual services to the Comune and its officials in the art of calculation etc.). ÁÁÁÁBonaini also quotes a 1506 reference to Lionardo Fibonacci. Mario Lazzarini. Leonardo Fibonacci Le sue Opere e la sua Famiglia. Bolletino di Bibliografia e Storia delle Scienze Matematiche 6 (1903) 98-102 & 7 (1904) 1©7. Traces the family to late 11C, saying Leonardo's father was Guglielmo and his grandfather was probably Bonaccio. He estimates the birth date as c1170. He describes a contract of 28 Aug 1226 in which Leonardo Bigollo, his father, Guglielmo, and his brother, Bonaccingo, buy a piece of land from a relative. This land included a tower and other buildings, outside the city, near S. Pietro in Vincoli. [G. Milanesi; Documento inedito intorno a Leonardo Fibonacci; Rome, 1867 ©© ??NYS]. Says nothing is known of the 1202 ed of Liber Abbaci. Quotes the above memorial. R. B. McClenon. Leonardo of Pisa and his liber quadratorum. AMM 26:1 (Jan 1919) 1©8. Gino Loria. Leonardo Fibonacci. Gli Scienziati Italiana dall'inizio del medio evo ai nostri giorni. Ed. by Aldo Mieli. (Dott. Attilio Nardecchia Editore, Rome, 1921;) Casa Editrice Leonardo da Vinci, Rome, 1923. Vol. 1, pp. 4©12. This reproduces much of the material in Lazzarini and the opening biographical paragraph of Liber Abaci. Ettore Bortolotti. Article on Fibonacci in: Enciclopedia Italiana. G. Treccani, Rome, 1949 (reprint of 1932 ed.). Charles King. Leonardo Fibonacci. Fibonacci Quarterly 1:4 (Dec 1963) 15©19. Gino Arrighi, ed. Leonardo Fibonacci: La Practica di Geometria ©© Volgarizzata da Cristofano di Gherardo di Dino, cittadino pisano. Dal Codice 2186 della Biblioteca Riccardiana di Firenze. Domus Galilaeana, Pisa, 1966. The Frontispiece is the mythical portrait of Fibonacci, taken from I Benefattori dell'UmanitÀ!À, vol. VI; Ducci, Florence, 1850. (Smith, History II 214 says it is a "Modern engraving. The portrait is not based on authentic sources".) P. 15 shows the plaque erected in the Archivio di Stato di Pisa in 1855 which reproduces the above memorial with an appropriate heading, but Arrighi has no discussion of it. P. 19 is a photo of the statue in Pisa and p. 16 describes its commissioning in 1859. Joseph and Francis Gies. Leonard of Pisa and the New Mathematics of the Middle Ages. Crowell, NY, 1969. This is a book for school students and contains a number of dubious statements and several false statements. Kurt Vogel. Fibonacci, Leonardo, or Leonardo of Pisa. DSB IV, 604©613. A. F. Horadam. Eight hundred years young. Australian Mathematics Teacher 31 (1975) 123-134. Good survey of Fibonacci's life & work. Gives English of a few problems. This is available on Kimberling's website © see below. Ettore Picutti. Leonardo Pisano. Le Scienze 164 (Apr 1982) ??NYS. = Le Scienze, Quaderni; 1984, pp. 30©39. (Le Scienze is a magazine; the Quaderni are collections of articles into books.) Mostly concerned with the Liber Quadratorum, but surveys Fibonacci's life and work. Says he was born around 1170. Includes photo of the plaque in the Archivo di Stato di Pisa. Leonardo Pisano Fibonacci. Liber quadratorum, 1225. Translated and edited by L. E. Sigler as: The Book of Squares; Academic Press, NY, 1987. Introduction: A brief biography of Leonardo Pisano (Fibonacci) [1170 © post 1240], pp. xv©xx. This is the best recent biography, summarizing Picutti's article. Says he was born in 1170 and his father's name was Guilielmo ©© cf Loria above. Gives another translation of the biographical paragraph of the Liber Abbaci. A. F. Horadam & J. Lahr. Letter to the Editor. Fibonacci Quarterly 28:1 (Feb 1990) 90. The authors volunteer to act as coordinators for work on the life and work of Fibonacci. Addresses: A. F. Horadam, Mathematics etc., Univ. of New England, Armidale, New South Wales, 2351, Australia; J. Lahr, 14 rue des Sept Arpents, L-1139 Luxembourg, Luxembourg. Thomas Koshy. Fibonacci and Lucas Numbers with Applications. Wiley©Interscience, Wiley, 2001. Claims to be 'the first attempt to compile a definitive history and authoritative analysis' of the Fibonacci numbers, but the history is generally second©hand and marred with a substantial number of errors, The mathematical work is extensive, covering many topics not organised before, and is better done, but there are more errors than one would like. Laurence E. Sigler. Translation of Liber Abaci as: Fibonacci's Liber Abaci A Translation into Modern English of Leonardo Pisano's Book of Calculation. Springer, 2002. Clark Kimberling's site web includes biographical material on Fibonacci and other similar number theorists. http://cedar.evansville.edu/~ck6/bstud/fibo.html . Ron Knott has a huge website on Fibonacci numbers and their applications, with material on many related topics, e.g. continued fractions, À!À, etc. with some history. www.ee.surrey.ac.uk/personal/r.knott/fibonacci/fibnat.html . ÁÁLuca à ÃPACIOLIÄ Ä (c1445©1517) S. A. Jayawardene. Luca Pacioli. BDM 4, 1897©1900. Bernardino Baldi (Catagallina) (1553©1617). Vita di Pacioli. (1589, first published in his Cronica de Mathematici of 1707.) Reprinted in: Bollettino di bibliografia e di storia delle scienze matematiche e fisiche 12 (1879) 421©427. ??NYS ©© cited by Taylor, p. 338. Enrico Narducci. Intorno a due edizioni della "Summa de arithmetica" di Fra Luca Pacioli. Rome, 1863. ??NYS ©© cited by Riccardi [Biblioteca Matematica Italiana, 1952] D. Ivano Ricci. Luca Pacioli, l'uomo e lo scienziato. San Sepolcro, 1940. ??NYS ©© cited in BDM. R. Emmett Taylor. No Royal Road Luca Pacioli and His Times. Univ. of North Carolina Press, Chapel Hill, 1942. BDM describes this as lively but unreliable. Ettore Bortolotti. La Storia della Matematica nella UniversitÀ!À di Bologna. Nicola Zanichelli Editore, Bologna, 1947. Chap. I, ÀÀ V, pp. 27©33: Luca Pacioli. Margaret Daly Davis. Piero della Francesca's Mathematical Treatises The "Trattato d'abaco" and "Libellus de quinque corporibus regularibus". Longo Editore, Ravenna, 1977. This discusses Piero's reuse of his own material and Pacioli's reuse of Piero's material. Fenella K. C. Rankin. The Arithmetic and Algebra of Luca Pacioli. PhD thesis, Univ. of London, 1992 (copy at the Warburg Institute), ??NYR. Enrico Giusti, ed. Descriptive booklet accompanying the 1994 facsimile of the Summa ©© qv in Common References. Edward A. Fennell. Figures in Proportion: Art, Science and the Business Renaissance. The contribution of Luca Pacioli to culture and commerce in the High Renaissance. Catalogue for the exhibition, The Institute of Chartered Accountants in England and Wales, London, 1994. ÁÁClaude©Gaspar à ÃBACHETÄ Ä de MÀ)Àziriac (1581-1638) C.-G. Collet & J. Itard. Un mathÀ)Àmaticien humaniste ©© Claude-Gaspar Bachet de MÀ)Àziriac (1581-1638). Revue d'Histoire des Sciences et leurs Applications 1 (1947) 26-50. J. Itard. Avant©propos. IN: Bachet; Problemes; 1959 reprint, pp. v-viii. Based on the previous article. There is a Frontispiece portrait in the reprint. Underwood Dudley. The first recreational mathematics book. JRM 3 (1970) 164-169. On Bachet's Problemes. William Schaaf. Bachet de MÀ)Àziriac, Claude-Gaspar. DSB I, 367-368. ÁÁJean à ÃLEURECHONÄ Ä (c1591-1670) and Henrik à ÃVAN ETTENÄ Ä A. Deblaye. À(Àtude sur la rÀ)ÀcrÀ)Àation mathÀ)Àmatique du P. Jean Leurechon, JÀ)Àsuite. MÀ)Àmoires de la SociÀ)ÀtÀ)À Philotechnique de Pont©À!À©Mousson 1 (1874) 171©183. [MUS #314. Schaaf. Hall, OCB, pp. 86, 88 & 114, says the only known copy of this journal is at Harvard, which has kindly supplied me with a photocopy of this article. Hall indicates the article is in vol. II and says it is 12 pages, but only cites pp. 171 & 174.] This simply assumes Leurechon is the author and gives a summary of his life. The essential content is described by Hall. G. EnestrÀ?Àm. Girard Desargues und D.A.L.G. Biblioteca Mathematica (3) 14 (1914) 253-258. D.A.L.G. was an annotator of van Etten's book in c1630. Although D.A.L.G. was used by Mydorge on one of his other books, it had been conjectured that this stood for Des Argues Lyonnais Girard (or GÀ)ÀomÀ/Àtre). EnestrÀ?Àm can find no real evidence for this and feels that Mydorge is the most likely person. Trevor H. Hall. Mathematicall Recreations. An Exercise in Seventeenth Century Bibliography. Leeds Studies in Bibliography and Textual Criticism, No. 1. The Bibliography Room, School of English, University of Leeds, 1969, 38pp. Pp. 18-38 discuss the question of authorship and Hall feels that van Etten probably was the author and that there is very little evidence for Leurechon being the author. Much of the mathematical content is in Bachet's Problemes and may have been copied from it or some common source. [This booklet is reproduced as pp. 83©119 of Hall, OCB, with the title page of the 1633 first English edition reproduced as plate 5, opp. p. 112. Some changes have been made in the form of references since OCB is a big book, but the only other substantial change is that he spells the name of the dedicatee of the book as Verreyken rather than Verreycken.] William Schaaf. Leurechon, Jean. DSB VIII, 271-272. Jacques Voignier. Who was the author of "Recreation Mathematique" (1624)? The Perennial Mystics #9 (1991) 5©48 (& 1©2 which are the cover and its reverse). [This journal is edited and published by James Hagy, 2373 Arbeleda Lane, Northbrook, Illinois, 60062, USA.] Presents some indirect evidence for Leurechon's authorship. ÁÁJacques à ÃOZANAMÄ Ä (1640-1717) On the flyleaf of J. E. Hofmann's copy of the 1696 edition of Ozanam's Recreations is a pencil portrait labelled Ozanam ©© the only one I know of. This copy is at the Institut fÀGÀr Geschichte der Naturwissenschaft in Munich. Hofmann published the picture ©© see below. Charles Hutton. A Mathematical and Philosophical Dictionary. 1795©1796. Vol. II, pp. 184ª185. ??NYS [Hall, OCB, p. 166.] Charles Hutton. On the life and writings of Ozanam, the first author of these Mathematical Recreations. Ozanam©Hutton. Vol. I. 1803: xiii©xv; 1814: ix©xi. William L. Schaaf. Jacques Ozanam on mathematics .... MTr 50 (1957) 385©389. Mostly based on Hutton. Includes a sketchy bibliography of Ozanam's works, generally ignoring the Recreations. Joseph Ehrenfried Hofmann. Leibniz und Ozanams Problem, drei Zahlen so zu bestimmen, dass ihre Summe eine Quadratzahl und ihre Quadratsumme eine Biquadratzahl ergibt. Studia Leibnitiana 1:2 (1969) 103©126. Outlines Ozanam's life, gives a bibliography of his works and reproduces the above©mentioned drawing as a plate opp. p. 124. (My thanks to Menso Folkerts for this information and a copy of Hofmann's article.) William L. Schaaf. Ozanam, Jacques. DSB X, 263-265. ÁÁJean À(Àtienne à ÃMONTUCLAÄ Ä (1725©1799) Charles Hutton. Some account of the life and writings of Montucla. Ozanam-Hutton. Vol. I. 1803: viii©xii; 1814: iv©viii. Charles Hutton. A Philosophical and Mathematical Dictionary. 2nd ed. of the Dictionary cited under Ozanam, 1815, Vol. II, pp. 63©64. ??NYS. According to Hall, OCB, p. 167, this is not in the 1795©1796 ed. and is a reworking of the previous item. ÁÁLewis à ÃCARROLLÄ Ä (1832©1898) Ðа¤˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐÁÁPseudonym of Charles Lutwidge Dodgson. There is so much written on Carroll that I will only give references to his specifically recreational work and some basic references. ÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐ The Diaries of Lewis Carroll. Edited by Roger Lancelyn Green. (OUP, 1954); 2 vols, Greenwood Publishers, Westport, Connecticut, 1971, HB. Lewis Carroll's Diaries The private journals of Charles Lutwidge Dodgson (Lewis Carroll) The first complete version of the nine surviving volumes with notes and annotations by Edward Wakeling. Introduction by Roger Lancelyn©Green. The Lewis Carroll Society, Publications Unit, Luton, Bedfordshire. [There were 13 journals, but 4 are lost.] ÁÁÁÁVol. 1. Journal 2, Jan©Sep 1855. 1993, 158pp. ÁÁÁÁVol. 2. Journal 4, Jan©Dec 1856. 1994, 158pp. ÁÁÁÁVol. 3. Journal 5, Jan 1857 © Apr 1858. 1995, 199pp. ÁÁÁÁVol. 4. Journal 8, May 1862 © Sep 1864 and a reconstruction of the four missing ÁÁÁÁÁÁyears, 1858©1862. 1997, 399pp. ÁÁÁÁVol. 5. Journal 9, Sep 1864 © Jan 1868, including the Russian Journal. ÁÁÁÁÁÁ1999, 416pp. ÁÁÁÁVol. 6. Journal 10, Apr 1868 © Dec 1876. 2001, 552pp. ÁÁÁÁVol. 7. Journal 11, Jan 1877 © Jun 1883. 2003, 606pp. The Letters of Lewis Carroll. Edited by Morton N. Cohen with the assistance of Roger Lancelyn Green. Volume One ca.1837 © 1885; Volume Two 1886 © 1898. Macmillan London, 1979. Stuart Dodgson Collingwood. The Life and Letters of Lewis Carroll. T. Fisher Unwin, London, 1898. Stuart Dodgson Collingwood, ed. The Lewis Carroll Picture Book. T. Fisher Unwin, London, 1899. = Diversions and Digressions of Lewis Carroll, Dover, 1961. = The Unknown Lewis Carroll, Dover, 1961(?). Reprint, in reduced format, Collins, c1910. The pagination of the main text is the same in the 1899 and in both Dover reprints, but is quite different than the Collins. Cited as: Carroll©Collingwood, qv in Common References. R. B. Braithwaite. Lewis Carroll as logician. MG 16 (No. 219) (Jul 1932) 174©178. He notes that Carroll assumed that a universal statement implied the existence of an object satisfying the antecedent, e.g. 'all unicorns are blue' would imply the existence of unicorns, contrary to modern convention. Derek Hudson. Lewis Carroll ©© An Illustrated Biography. Constable, 1954; new illustrated ed., 1976. Warren Weaver. Lewis Carroll: Mathematician. SA 194:4 (Apr 1956) 116-128. + Letters and response. SA 194:6 (Jun 1956) 19©22. Martin Gardner. The Annotated Alice. C. N. Potter, NY, 1960. Penguin, 1965; 2nd ed., 1971. Revised as: More Annotated Alice, 1990, qv. Martin Gardner. The Annotated Snark. Bramhall House, 1962. Penguin, 1967; revised, 1973 & 1974. John Fisher. The Magic of Lewis Carroll. Nelson, 1973. Penguin, 1975. Morton N. Cohen, ed. The Selected Letters of Lewis Carroll. Papermac (Macmillan), 1982. Martin Gardner. More Annotated Alice. [Extension of The Annotated Alice.] Random House, 1990. Edward Wakeling. Lewis Carroll's Games and Puzzles. Dover and the Lewis Carroll Birthplace Trust, 1992. Cited as Carroll©Wakeling, qv in Common References. Francine F. Abeles, ed. The Pamphlets of Lewis Carroll ©© Vol. 2: The Mathematical Pamphlets of Charles Lutwidge Dodgson and Related Pieces. Lewis Carroll Society of North America, distributed by University Press of Virginia, Charlottesville, 1994. Edward Wakeling. Rediscovered Lewis Carroll Puzzles. Dover, 1995. Cited as Carroll-Wakeling II, qv in Common References. Martin Gardner. The Universe in a Handkerchief. Lewis Carroll's Mathematical Recreations, Games, Puzzles and Word Plays. Copernicus (Springer, NY), 1996. Cited as Carroll-Gardner, qv in Common References. Martin Gardner. The Annotated Alice: The Definitive Edition. 1999. [A combined version of The Annotated Alice and More Annotated Alice.] ÁÁProfessor Louis à ÃHOFFMANNÄ Ä (1839-1919) ÁÁPseudonym of Angelo John Lewis. Joseph Foster. Men©at©the©Bar: A biographical Hand©List of the Members of the Various Inns of Court, including Her Majesty's Judges, etc. 2nd ed, the author, 1885. P. 277 is the entry for Lewis. Born in London, eldest son of John Lewis. Graduated from Wadham College, Oxford. Entered Lincoln's Inn as a student in 1858, called to the bar there in 1861. Married Mary Ann Avery in 1864. Author of à ÃManual of Indian Penal CodeÄ Ä and à ÃManual of Indian Civil ProcedureÄ Ä. Address: 12 Crescent Place, Mornington Crescent, London, NW. (My thanks to the Library of Lincoln's Inn for this information.) Anonymous. Professor Hoffmann. Mahatma 4:1 (Jul 1900) 377©378. A brief note, with photograph, stating that he is Mr. Angelo Lewis, M.A. and Barrister©at©Law. Will Goldston. Will Goldston's Who's Who in Magic. My version is included in a compendium called: Tricks that Mystify; Will Goldston, London, nd [1934©NUC]. Pp. 106©107. Says he was a barrister, retired to Hastings about 1903 and died in 1917. Who Was Who, 1916©1928, p. 627. This says he attended North London Collegiate School and that he only practised law until 1876. He was on the staff of the Saturday Review and a contributor to many journals. Won the À À100 prize offered by Youth's Companion (Boston) for best short story for boys. Lists 36 books by him and 9 card games he invented. Address: Manningford, Upper Bolebrooke Road, Bexhill©on©Sea. (My thanks to the Library of Lincoln's Inn for this information.) J. B. Findlay & Thomas A. Sawyer. Professor Hoffmann: A Study. Published by Thomas A. Sawyer, Tustin, California, 1977. A short book, 12 + 67 pp, with two portraits (one from Mahatma) and 27pp of bibliography. He was born at 3 Crescent Place, Mornington Crescent, London. He was a barrister and wrote two books on Indian law. Charles Reynolds. Introduction ©© to the reprint of Hoffmann's Modern Magic, Dover, 1978, pp. v-xiv. This says Lewis was a barrister, which is mentioned in another reprint of a Hoffmann book and in S. H. Sharpe's translation of Ponsin on Conjuring. Edward Hordern. Foreword to this edition. In: Hoffmann's Puzzles Old and New (see under Common References), 1988 reprint, pp. v-vi. This says he was the Reverend Lewis, but this is corrected in Hoffmann©Hordern to saying he was a barrister. Hoffmann©Hordern, p. viii, is a version of the photograph in Mahatma. Hall, OCB, p. 189, gives Hoffmann's address as Ireton Lodge, Cromwell Ave., N. ©ª presumably the Cromwell Ave. in Highgate. Toole Stott 386 gives a little information about Hoffmann and Modern Magic, including an address in Mornington Crescent in 1877. No DNB or DSB entry ©© I have suggested a DNB entry. ÁÁSam à ÃLOYDÄ Ä (1841-1911) and Sam à ÃLOYD JR.Ä Ä (1873-1934) [W. R. Henry.] Samuel Loyd. [Biography.] Dubuque Chess Journal, No. 66 (Aug©Sep 1875) 361©365. ??NX ©© o/o (11 Jul 91). Loyd. US Design 4793 ©© Design for Puzzle©Blocks. 11 April 1871. These are solid pieces, but unfortunately the drawing did not come with this, so I am not clear what they are. ??Need drawing ©© o/o (11 Jul 91). Anonymous & Sam Loyd. Loyd's puzzles (Introductory column). Brooklyn Daily Eagle (22 Mar 1896) 23. Says he lives at 153 Halsey St., Brooklyn. L. D. Broughton Jr. Samuel Loyd. [A Biography.] Lasker's Chess Magazine 1:2 (Dec 1904) 83©85. About his chess problems with a mention of some of his puzzles. G. G. Bain. The prince of puzzle-makers. An interview with Sam Loyd. Strand Magazine 34 (No. 204) (Dec 1907) 771-777. Solutions of Sam Loyd's puzzles. Ibid. 35 (No. 205) (Jan 1908) 110. Walter Prichard Eaton. My fifty years in puzzleland ©© Sam Loyd and his ten thousand brain-teasers. The Delineator (New York) (April 1911) 274 & 328. Drawn portrait of Loyd, age 69. Anon. Puzzle inventor dead. New©York Daily Tribune (12 Apr 1911) 7. Says he died at his house, 153 Halsey St. "He declared no one had ever succeeded in solving [the "Disappearing Chinaman"]." Says he is survived by a son and two daughters (!! ©© has anyone ever tracked the daughters and their descendents??). Anon. Sam Loyd, puzzle man, dies. New York Times (12 Apr 1911) 13. Says he was for some time editor of The Sanitary Engineer and a shrewd operator on Wall Street. Anon. Sam Loyd. SA (22 Apr 1911) 40©41?? Says he was for some years chess editor of SA and was puzzle editor of Woman's Home Companion when he died. W. P. Eaton. Sam Loyd. The American Magazine 72 (May 1911) 50, 51, 53. Abridged version of Eaton's earlier article. Photo of Loyd on p. 50. P. J. Doyle. Letter to the Chess column. The Sunday Call [Newark, NJ] (21 May 1911), section III, p. 10. A. C. White. Sam Loyd and His Chess Problems. Whitehead and Miller, Leeds, UK, 1913; corrected, Dover, 1962. Alain C. White. Supplement to Sam Loyd and His Chess Problems. Good Companion Chess Problem Club, Philadelphia, vol. I, nos. 11©12 (Aug 1914), 12pp. This is mostly corrections of the chess problems, but adds a few family details with a picture of the Loyd Homestead and Grist Mill in Moylan, Pennsylvania. Alain C. White. Reminiscences of Sam Loyd's family. The Problem [Pittsburgh] (28 Mar 1914) 2, 3, 6, 7. Louis C. Karpinski. Loyd, Samuel. Dictionary of American Biography, Scribner's, NY, vol. XI, 1933, pp. 479-480. Loyd Jr. SLAHP. 1928. Preface gives some details of his life, making little mention of his father, "who was a famous mathematician and chess player". He claims to have created over 10,000 puzzles. There are some vague biographical details on pp. 1-22, e.g. 'Father conducted a printing establishment.' 'My "Missing Chinaman Puzzle"'. (It may have been some such assertion that led me to estimate his birthdate as 1865, but I now see it is well known to be 1873.) Anonymous. Sam Loyd dead; puzzle creator. New York Times (25 Feb 1934). Obituary of Sam Loyd Jr. Says he resided at 153 Halsey St., Brooklyn ©© the same address as his father ©© see the Brooklyn Daily Eagle article of 1896, above. He worked from a studio at 246 Fulton St., Brooklyn. It says Jr. invented 'How Old is Ann?'. Clark Kinnaird. Encyclopedia of Puzzles and Pastimes. Grosset & Dunlap, NY, 1946. Pp. 263-267: Sam Loyd. Asserts that Loyd Jr. invented 'How Old is Ann?' Gardner. Sam Loyd: America's greatest puzzlist. SA (Aug 1957) c= First Book, Chap. 9. Gardner. Advertising premiums. SA (Nov 1971) c= Wheels, chap. 12. Will Shortz is working on a biography. No DSB entry. ÁÁFranÀ'Àois Anatole À(Àdouard à ÃLUCASÄ Ä (1842-1891) Jeux Scientifiques de Ed. Lucas. Advertisement by Chambon & Baye (14 rue Etienne©Marcel, Paris) for the 1ÃÃreÄÄ Serie of six games. Cosmos. Revue des Sciences et Leurs Applications 39 (NS No. 254) (7 Dec 1889) no page number on my photocopy. B. Bailly [name not given, but supplied by Hinz]. Article on Lucas's puzzles. Cosmos. Revue des Sciences et Leurs Applications. NS, 39 (No. 259) (11 Jan 1890) 156©159. NEED 156-157. NÀ)Àcrologie: À(Àdouard Lucas. La Nature 19 (1891) II, 302. Obituary notice: "ÃÃLa NatureÄÄ announces the death of Prof. Edouard Lucas ...." Nature 44 (15 Oct 1891) 574©575. Duncan Harkin. On the mathematical work of FranÀ'Àois-À(Àdouard-Anatole Lucas. L'Enseignement Math. (2) 3 (1957) 276-288. Pp. 282-288 is a bibliography of 184 items. I have found many Lucas publication not listed here and have started a new Bibliography ©© see below. P. J. Campbell. Lucas' solution to the non-attacking rooks problem. JRM 9 (1976/77) 195-200. Gives life of Lucas. A photo of Lucas is available from BibliothÀ/Àque Nationale, Service Photographique, 58 rue Richelieu, F-75084 Paris Cedex 02, France. Quote Cote du Document LnÃÃ27ÄÄ . 43345 and Cote du Cliche 83 A 51772. (??*) I have obtained a copy, about 55 x 85 mm, with the photo in an oval surround. It looks like a carte©de©visite, but has À(Àdouard LUCAS (1842©1891). ©© Phot. Zagel. underneath. (Thanks to H. W. Lenstra for the information.) Norman T. Gridgeman. Lucas, FranÀ'Àois-À(Àdouard-Anatole. DSB VIII, 531-532. Susanna S. Epp. Discrete Mathematics with Applications. Wadsworth, Belmont, Calif., 1990, p. 477 gives a small photo of Lucas which looks nothing like the photo from the BN. I have since received a note from Epp via Paul Campbell that a wrong photo was used in the first edition, but this was corrected in later editions. Alain Zalmanski. Edouard Lucas Quand l'arithmÀ)Àtique devient amusante. Jouer Jeux MathÀ)Àmatiques 3 (Jul/Sep 1991) 5. Brief notice of his life and work. Andreas M. Hinz. Pascal's triangle and the Tower of Hanoi. AMM 99 (1992) 538©544. Sketches Lucas' life and work, giving details that are not in the above items. David Singmaster. The publications of À(Àdouard Lucas. Draft version, 14pp, 1998. I discovered many items in Dickson's History of the Theory of Numbers and elsewhere which are not given by Harkin (cf above). This has 248 items, though many of these are multiple items so the actual count is perhaps 275. However, Dickson does not give article titles, and may not give the pages of the entire article, so the same article may be cited more than once, at different pages. I hope to fill in the missing information at some time. ÁÁHermann CÀÀsar Hannibal à ÃSCHUBERTÄ Ä (1848©1911) Acta Mathematica 1882©1912. Table GÀ)ÀnÀ)Àrale des Tomes 1©35. 1913. P. 169. Portrait of Schubert. Werner Burau. Schubert, Hermann CÀÀsar Hannibal. DSB XII, 227-229. ÁÁWalter William Rouse à ÃBALLÄ Ä (1850-1925) Anon. Obituary: Mr. Rouse Ball. The Times (6 Apr 1925) 16. Anon. Funeral notice: Mr. W. W. R. Ball. The Times (9 Apr 1925) 13. (Lord) Phillimore. Letter: Mr. Rouse Ball. The Times (9 Apr 1925) 15. "An old pupil". The late Mr. Rouse Ball. The Times (13 Apr 1925) 12. J. J. Thomson. W. W. Rouse Ball. The Cambridge Review (24 Apr 1925) 341©342. Anon. Obituary of W. W. Rouse Ball. Nature 115 (23 May 1925) 808-809. Anon. The late Mr. W. W. Rouse Ball. The Trinity Magazine (Jun 1925) 53©54. Anon. Entry in Who's Who, 1925, p. 127. Anon. Wills and bequests: Mr. Walter William Rouse Ball. The Times (7 Sep 1925) 15. E. T. Whittaker. Obituary. W. W. Rouse Ball. Math. Gaz. 12 (No. 178) (Oct 1925) 449©454, with photo opp. p. 449. F. Cajori. Walter William Rouse Ball. Isis 8 (1926) 321-324. Photo on plate 15, opp. p. 321. Copy of Ball's 1924 Xmas card on p. 324. J. A. Venn. Alumni Cantabrigienses. Part II: From 1752 to 1900. Vol. I, p. 136. CUP, 1940. David Singmaster. Walter William Rouse Ball (1850©1925). 6pp handout for 1st UK Meeting on the History of Recreational Mathematics, 24 Oct 1992. Plus extended biographical (6pp) and bibliographical (8pp) notes which repeat some of the material in the handout. No DNB or DSB entry ©© however I have offered to write a DNB entry. I have since seen the proposed list of names for the next edition and Ball is already on it. ÁÁHenry Ernest à ÃDUDENEYÄ Ä (1857-1930) Anon. & Dudeney. A chat with the puzzle king. The Captain 2 (Dec? 1899) 314-320, with photo. Partly an interview. Includes photos of Littlewick Meadow. Anon. Solutions to "Sphinx's puzzles". The Captain 2:6 (Mar 1900) 598-599 & 3:1 (Apr 1900) 89. Anon. Master of the breakfast table problem. Daily Mail (1 Feb 1905) 7. An interview with Dudeney in which he gives the better version of his spider and fly problem. Fenn Sherie. The Puzzle King: An Interview with Henry E. Dudeney. Strand Magazine 71 (Apr 1926) 398-4O4. Alice Dudeney. Preface to PCP, dated Dec 1931, pp. vii-x. The date of his death is erroneously given as 1931. Gardner. Henry Ernest Dudeney: England's greatest puzzlist. SA (Jun 1958) c= Second Book, chap. 3. Angela Newing. The Life and Work of H. E. Dudeney. MS 21 (1988/89) 37-44. Angela Newing is working on a biography. No DNB or DSB entry. I have suggested a DNB entry. ÁÁWilhelm Ernst Martin Georg à ÃAHRENSÄ Ä (1872-1927) Wilhelm Lorey. Wilhelm Ahrens zum GedÀÀchtnis. Archiv fÀGÀr Geschichte der Mathematik, der Naturwissenschaften und der Technik 10 (1927/28) 328-333. Photo on p. 328. O. Staude. Dem Andenken an Dr. Wilhelm Ahrens. Jahresbericht DMV 37 (1928) 286©287. No DSB entry. ÁÁYakov Isidorovich à ÃPERELMANÄ Ä [À@ À. À À. À ÀÀ ÀÀ# ÀÀ ÀÀ ÀÀ ÀÀ ÀÀ À] (1882©1942) Perelman. FMP. 1984. P. 2 (opp. TP) is a sketch of his life and the history of the book. There is a small drawing of Perelman at the top of the page. Patricio Barros. Website ©© Yakov I. Perelman [in Spanish]: www.geocities.com/yakov_perelman/index.html. This includes a four page biography, in collaboration with Antonio Bravo, and two photos. ÁÁHubert à ÃPHILLIPSÄ Ä (1891©1964) Hubert Phillips. Journey to Nowhere. A Discursive Autobiography. Macgibbon & Kee, London, 1960. ??NYR No DNB entry ©© I have suggested one. à Ã2. ÁÁGENERAL PUZZLE COLLECTIONS AND SURVEYSÄ Ä H. E. Dudeney. Great puzzle crazes. London Magazine 13?? (Nov 1904) 478-482. Fifteen Puzzle. Pigs in Clover, Answers, Pick©me©up (spiral ramp) and other dexterity puzzles. Get Off the Earth. Conjurer's Medal (ring maze). Chinese Rings. Chinese Cross (six piece burr). Puzzle rings. Solitaire. The Mathematician's Puzzle (square, circle, triangle). Imperial Scale. Heart and Balls. H. E. Dudeney. Puzzles from games. Strand Magazine 35 (No. 207) (Mar 1908) 339-344. Solutions. Ibid. 35 (No. 208) (Apr 1908) 455-458. H. E. Dudeney. Some much-discussed puzzles. Strand Magazine 35 (No. 209) (May 1908) 580-584. Solutions. Ibid. 35 (No. 210) (Jun 1908) 696. H. E. Dudeney. The world's best puzzles. Strand Magazine 36 (No. 216) (Dec 1908) 779-787. Solutions. Ibid. 37 (No. 217) (Jan 1909) 113-116. H. E. Dudeney. The psychology of puzzle crazes. The Nineteenth Century 100:6 (Dec 1926) 868-879. Repeats much of his 1904 article. Sam Loyd Jr. Are you good at solving puzzles? The American Magazine (Sep 1931) 61-63, 133-137. Orville A. Sullivan. Problems involving unusual situations. SM 9 (1943) 114-118 & 13 (1947) 102-104. ÙÙ Ã Ã3.ÁÁGENERAL HISTORICAL AND BIBLIOGRAPHICAL MATERIALÄ Ä Ðа¤˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐÁÁI have tried to divide this material into historical and bibliographical parts, but the two overlap considerably. ÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐ Ã ÃÁÁ3.A.ÁÁGENERAL HISTORICAL MATERIALÄ Ä Raffaella Franci. Giochi matematici in trattati d'abaco del medioevo e del rinascimento. Atti del Convegno Nazionale sui Giochi Creative, Siena, 11©14 Jun 1981. Tipografia Senese for GIOCREA (SocietÀ!À Italiana Giochi Creativi), 1981. Pp. 18©43. Describes and quotes many typical problems. 17 references, several previously unknown to me. Heinrich Hermelink. Arabische Unterhaltungsmathematik als Spiegel Jahrtausendealter Kulturbeziehungen zwischen Ost und West. Janus 65 (1978) 105©117, with English summary. An English translation appeared as: Arabic recreational mathematics as a mirror of age©old cultural relations between Eastern and Western civilizations; in: Ahmad Y. Al©Hassan, Ghada Karmi & Nizar Namnum, eds.; Proceedings of the First International Symposium for the History of Arabic Science, April 1976 ©© Vol. Two: Papers in European Languages; Institute for the History of Arabic Science, Aleppo, 1978, pp. 44©52. (There are a few translation and typographical errors, which make it clear that the English version is a translation of the German.) D. E. Smith. On the origin of certain typical problems. AMM 24 (1917) 64-71. (This is mostly contained in his History, vol. II, pp. 536-548.) à ÃÁÁ3.B.ÁÁBIBLIOGRAPHICAL MATERIALÄ Ä Ðа¤˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐÁÁMany of the items cited in the Common References have extensive bibliographies. In particular: BLC; BMC; BNC; DNB; DSB; Halwas; NUC; Schaaf; Smith & De Morgan: Rara; Suter are basic bibliographical sources. Datta & Singh; Dickson; Heath: HGM; Murray; Sanford: H&S & Short History; Smith: History & Source Book; Struik; Tropfke are histories with extensive bibliographical references. AR; BR are editions of early texts with substantial bibliographical material. Ahrens: MUS; Ball: MRE; Berlekamp, Conway & Guy: Winning Ways; Gardner; Lucas: RM are recreational books with some useful bibliographical material. Of these, the material in Ahrens is by far the most useful. The magic bibliographies of Christopher, Clarke & Blind, Hall, Heyl, Price (see HPL), Toole Stott and Volkmann & Tummers have considerable overlap with the present material, particularly for older books, though Hall, Heyl and Toole Stott restrict themselves to English material, while Volkmann & Tummers only considers German. Santi is also very useful. Below I give some additional bibliographical material which may be useful, arranged in author order. ÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐAnonymous. Mathematical bibliography. SSM 48 (1948) 757-760. Covers recreations. Wilhelm Ahrens. Mathematische Spiele. Section I G 1 of Encyklopadie der Math. Wiss., Vol. I, part 2, Teubner, Leipzig, 1900-1904, pp. 1080-1093. Raymond Clare Archibald. Notes on some minor English mathematical serials. MG 14 (1928©29) 379©400. Elliott M. Avedon & Brian Sutton-Smith. The Study of Games. (Wiley, NY, 1971); Krieger, Huntington, NY, 1979. Anthony S. M. Dickins. A Catalogue of Fairy Chess Books and Opuscules Donated to Cambridge University Library, 1972-1973, by Anthony Dickins M.A. Third ed., Q Press, Kew Gardens, UK, 1983. Underwood Dudley. An annotated list of recreational mathematics books. JRM 2:1 (Jan 1969) 13©20. 61 titles, in English and in print at the time. Aviezri S. Fraenkel. Selected Bibliography on Combinatorial Games and Some Related Material. There have been several versions with slightly varying titles. The most recent printed version is: 400 items, 28 pp., including 4 pp of text, Sep 1990. Technical Report CS90-23, Weizmann Institute of Science, Rehovot, Israel. = Proc. Symp. Appl. Math. 43 (1991) 191©226. Fraenkel has since produced Update 1 to this which lists 430 items on 31pp, Aug 1992; and Update 2, 480 items on 33pp, with 5 pp of text, accidentally dated Aug 1992 at the top but produced in Feb 1994. On 22 Nov 1994, it became a dynamic survey on the Electronic J. Combinatorics and can be accessed from: ÁÁÁÁhttp://ejc.math.gatech.edu:8080/journal/surveys/index.html. ÁÁIt can also be accessed via anonymous ftp from ftp.wisdom.weizmann.ac.il. After logging in, do cd pub/fraenkel and then get one of the following three compressed files: games.tex.z; games.dvi.z; games.ps.z. Martin P. Gaffney & Lynn Arthur Steen. Annotated Bibliography of Expository Writing in the Mathematical Sciences. MAA, 1976. JoAnne S. Growney. Mathematics and the arts ©© A bibliography. Humanistic Mathematics Network Journal 8 (1993) 22©36. General references. Aesthetic standards for mathematics and other arts. Biographies/autobiographies of mathematicians. Mathematics and display of information (including mapmaking). Mathematics and humor. Mathematics and literature (fiction and fantasy). Mathematics and music. Mathematics and poetry. Mathematics and the visual arts. JoAnne S. Growney. Mathematics in Literature and Poetry. Humanistic Mathematics Network Journal 10 (Aug 1994) 25©30. Short survey. 3 pages of annotated references to 29 authors, some of several books. R. C. Gupta. A bibliography of selected book [sic] on history of mathematics. The Mathematics Education 23 (1989) 21©29. Trevor H. Hall. Mathematicall Recreations. Op. cit. in 1. This is primarily concerned with the history of the book by van Etten. [This booklet is revised as pp. 83©119 of Hall, OCB ©© see Section 1.] Catherine Perry Hargrave. A History of Playing Cards and a Bibliography of Cards and Gaming. (Houghton Mifflin, Boston, 1930); Dover, 1966. Susan Hill. Catalogue of the Turner Collection of the History of Mathematics Held in the Library of the University of Keele. University Library, Keele, 1982. (Sadly this collection was secretly sold by Keele University in 1998 and has now been dispersed.) Honeyman Collection ©© see: Sotheby's. Horblit Collection ©© see: Sotheby's and H. P. Kraus. Else HÀQÀyrup. Books about Mathematics. Roskilde Univ. Center, PO Box 260, DK-4000, Roskilde, Denmark, 1979. D. O. Koehler. Mathematics and literature. MM 55 (1982) 81©95. 64 references. See Utz for some further material. H. P. Kraus (16 East 46th Street, New York, 10017). The History of Science including Navigation. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐCatalogue 168. A First Selection of Books from the Library of Harrison D. Horblit. Nd [c1976]. Catalogue 169. A Further Selection of Books, 1641©1700 (Wing Period) from the Library of Harrison D. Horblit. Nd [c1976]. Catalogue 171. Another Selection of Books from the Library of Harrison D. Horblit. Nd [c1976]. ÁÁThese are the continuations of the catalogues issued by Sotheby's, qv. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐJohn S. Lew. Mathematical references in literature. Humanistic Mathematics Network Journal 7 (1992) 26©47. Antonius van der Linde. Das erst Jartausend [sic] der Schachlitteratur ©© (850-1880). (1880); Facsimile reprint by Caissa Limited Editions, Yorklyn, Delaware, 1979, HB. Andy Liu. Appendix III: A selected bibliography on popular mathematics. Delta©k 27:3 (Apr 1989) ©© Special issue: Mathematics for Gifted Students, 55©83. À(Àdouard Lucas. RÀ)ÀcrÀ)Àations mathÀ)Àmatiques, vol 1 (i.e. RM1), pp. 237©248 is an Index Bibliographique. Felix MÀGÀller. FÀGÀhrer durch die mathematische Literature mit besonderer BerÀGÀcksichtigung der historisch wichtigen Schriften. Abhandlungen zur Geschichte der Mathematik 27 (1903). Charles W. Newhall. "Recreations" in secondary mathematics. SSM 15 (1915) 277-293. Mathematical Association. 259 London Road, Leicester, LE2 3BE. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐCatalogue of Books and Pamphlets in the Library. No details, [c1912], 19pp, bound in at end of Mathematical Gazette, vol. 6 (1911-1912). A First List of Books & Pamphlets in the Library of the Mathematical Association ©© Books and Pamphlets acquired before 1924. Bell, London, 1926. A Second List of Books & Pamphlets in the Library of the Mathematical Association ©© Books and Pamphlets acquired during 1924 and 1925. Bell, London, 1929. A Third List of Books & Pamphlets in the Library of the Mathematical Association ©© Books and Pamphlets added from 1926 to 1929. Bell, London, 1930. A Fourth List of Books & Pamphlets in the Library of the Mathematical Association ©© Books and Pamphlets added from 1930 to 1935. Bell, London, 1936. ÁÁÁÁLists 1-4 edited by E. H. Neville. Books and Periodicals in the Library of the Mathematical Association. Ed. by R. L. Goodstein. MA, 1962. Includes the four previous lists and additions through 1961. SEE ALSO: Riley; Rollett; F. R. Watson. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐStanley Rabinowitz. Index to Mathematical Problems 1980©1984. MathPro Press, Westford, Massachusetts, 1992. Cecil B. Read & James K. Bidwell. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐSelected articles dealing with the history of elementary Mathematics. SSM 76 (1976) 477©483. Periodical articles dealing with the history of advanced mathematics ©© Parts I & II. SSM 76 (1976) 581©598 & 687©703. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐRudolf H. Rheinhardt. Bibliography on Whist and Playing Cards. From: Whist Scores and Card©table Talk, Chicago, 1887. Reprinted by L. & P. Parris, Llandrindod Wells, nd [1980s]. Pietro Riccardi. Biblioteca Matematica Italiana dalla Origine della Stampa ai Primi Anni del Secolo XIX. G. G. GÀ?Àrlich, Milan, 1952, 2 vols. This work appeared in several parts and supplements in the late 19C and early 20C, mostly published by the SocietÀ!À Tipografica Modense, Modena, 1878©1893. Because it appeared in parts, the contents of early copies are variable and even the reprints may vary. The contents of this set are as follows. ÁÁà ÃI.Ä Ä ÁÁ20pp prelims + Col. 1 © 656 (Abaco © Kirchoffer). [= original Vol. I.] ÁÁÁÁCol. 1 © 676 (La Cometa © Zuzzeri) + 2pp correzioni. [= original Vol. II.] ÁÁà ÃII.Ä ÄÁÁ4pp titles and reverses. Correzioni ed Aggiunte. [= original Appendice.] ÁÁÁÁSerie I.ÃÃaÄÄ Col. 1 © 78 + 1ÀÀpp Continuazione delle Correzioni (note that these ÁÁÁÁÁÁhave Pag. when they mean Col.). ÁÁÁÁSerie II.ÃÃaÄÄ. Col. 81 © 156. ÁÁÁÁSerie III.ÃÃaÄÄ. Col. 157 © 192 + Aggiunte al Catalogo delle Opere di sovente citate, ÁÁÁÁÁÁcol. 193©194 + 1p Continuazione delle Correzioni (note that these have ÁÁÁÁÁÁPag. when they mean Col.). ÁÁÁÁSerie IV.ÃÃaÄÄ. Col. 197 © 208 + Seconda Aggiunta al Catalogo delle Opere piÀIÀ di ÁÁÁÁÁÁsovente citate, col. 209 © 212 + Continuazione delle Correzioni in ÁÁÁÁÁÁcol. 211©212. ÁÁÁÁSerie V.ÃÃaÄÄ. Col. 1 © 180. ÁÁÁÁSerie VI.ÃÃaÄÄ. Col. 179 © 200. ÁÁÁÁÁÁSerie V & VI must have been published as one volume as Serie V ends ÁÁÁÁÁÁÁÁhalfway down a page and then Serie VI begins on the same page. ÁÁÁÁSerie VII.ÃÃaÄÄ. 2pp introductory note by Ettore Bortolotti in 1928 saying that this ÁÁÁÁÁÁmaterial was left as a manuscript by Riccardi and never previously ÁÁÁÁÁÁpublished + Col. 1 © 106. ÁÁÁÁIndice Alfabetico, of authors, covering the original material and all seven Series ÁÁÁÁÁÁof Correzioni ed Aggiunte, in 34 unnumbered columns. ÁÁÁÁParte Seconda. Classificazione per materie delle opere nella Parte I. 18pp ÁÁÁÁÁÁ(including a chronological table) + subject index, pp. 1 © 294. ÁÁÁÁCatalogo Delle opere piÀIÀ di sovente citate, col. 1 © 54. ÁÁÁÁ[I have seen an early version which had the following parts: Vol. I, 1893, col. 1-656; Vol. II, 1873, col. 1©676; Appendice, 1878©1880©1893, col. 1©228. Appendice, nd, col. 1©212. Serie V, col. 1©228. Parte 2, Vol. 1, 1880, pp. 1©294. Renner Katalog 87 describes it as 5 in 2 vols.] A. W. Riley. School Library Mathematics List ©© Supplement No. 1. MA, 1973. ÁÁSEE ALSO: Rollett. Tom Rodgers. Catalog of his collection of books on recreational mathematics, etc. The author, Atlanta, May 1991, 40pp. Leo F. Rogers. Finding Out in the History of Mathematics. Produced by the author, London, c1985, 52pp. A. P. Rollett. School Library Mathematics List. Bell, London, for MA, 1966. ÁÁSEE ALSO: Riley. Charles L. Rulfs. Origins of some conjuring works. Magicol 24 (May 1971) 3©5. JosÀ)À A. SÀÀnchez PÀ)Àrez. Las Matematicas en la Biblioteca del Escorial. Imprenta de Estanislao Maestre, Madrid, 1929. William L. Schaaf. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐList of works on recreational mathematics. SM 10 (1944) 193©200. ÁÁÁÁPLUS: A. Gloden; Additions to Schaaf's "List of works on mathematical recreations"; SM 13 (1947) 127. A Bibliography of Recreational Mathematics. Op. cit. in Common References, 4 vols., 1955©1978. In these volumes he gives several lists of relevant books. ÁÁÁÁBooks for the periods 1900©1925 and 1925©c1956 are given as Sections 1.1 (pp. 2©3) and 1.2 (pp. 4©12) in Vol. 1. ÁÁÁÁChapter 9, pp. 144©148, of Vol. 1, is a Supplement, generally covering c1954©c1962, but with some older items. ÁÁÁÁIn Vol. 2, 1970, the Appendix, pp. 181©191, extends to c1969, including some older items and repeating a few from the Supplement of Vol. 1. ÁÁÁÁAppendix A of Vol. 3, 1973, pp. 111©113, adds some more items up through 1972. ÁÁÁÁAppendix A, pp. 134©137, of Vol. 4, 1978, extends up through 1977. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐ The following VESTPOCKET BIBLIOGRAPHIES are extensions of the material ÁÁÁÁin his Bibliographies. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐNo. 1:ÁÁPythagoras and rational triangles; Geoboards and lattices. JRM 16:2 (1983©84) 81©88. No. 2:ÁÁCombinatorics; Gambling and sports. JRM 16:3 (1983©84) 170©181. No. 3:ÁÁTessellations and polyominoes; Art and music. JRM 16:4 (1983©84) 268-280. No. 4:ÁÁRecreational miscellany. JRM 17:1 (1984©85) 22©31. No. 5:ÁÁPolyhedra; Topology; Map coloring. JRM 17:2 (1984©85) 95©105. No. 6:ÁÁSundry algebraic notes. JRM 17:3 (1984©85) 195©203. No. 7:ÁÁSundry geometric notes. JRM 18:1 (1985©86) 36©44. No. 8:ÁÁProbability; Gambling. JRM 18:2 (1985©86) 101©109. No. 9:ÁÁGames and puzzles. JRM 18:3 (1985©86) 161©167. No. 10:ÁÁRecreational mathematics; Logical puzzles; Expository mathematics. JRM 18:4 (1985©86) 241©246. No. 11:ÁÁLogic, Artificial intelligence, and Mathematical foundations. JRM 19:1 (1987) 3©9. No. 12:ÁÁMagic squares and cubes; Latin squares; Mystic arrays and Number patterns. JRM 19:2 (1987) 81©86. The High School Mathematics Library. NCTM, (1960, 1963, 1967, 1970, 1973); 6th ed., 1976; 7th ed., 1982; 8th ed., 1987. ÁÁSEE ALSO: Wheeler; Wheeler & Hardgrove. Early Books on Magic Squares. JRM 16:1 (1983©84) 1©6. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐWilliam L. Schaaf & David Singmaster. Books on Recreational Mathematics. A Supplement to the Lists in William L. Schaaf's A Bibliography of Recreational Mathematics. Collected by William L. Schaaf; typed and annotated by David Singmaster. School of Computing, Information Systems and Mathematics, South Bank University, London, SE1 0AA. 18pp, Dec 1992 and revised several times afterwards. Peter Schreiber. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐMathematik und belletristik [1.] & 2. Teil. Mitteilungen der Mathematischen Gesellschaft der Deutschen Demokratischer Republik. (1986), no. 4, 57©71 & (1988), no. 1©2, 55©61. Good on German works relating mathematics and arts. Mathematiker als Memoirenschreiber. Alpha (Berlin) (1991), no. 4, no page numbers on copy received from author. Extends previous work. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐS. N. Sen. Scientific works in Sanskrit, translated into foreign languages and vice-versa in the 18th and 19th century A.D. Indian J. History of Science 7 (1972) 44-70. Will Shortz. Puzzleana [catalogue of his puzzle books]. Produced by the author. 14 editions have appeared. The latest is: May 1992, 88pp with 1175 entries in 26 categories, with indexes of authors and anonymous titles. Some entries cover multiple items. In Jan 1995, he produced a 19pp Supplement extending to a total of 1451 entries. David Singmaster. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐThe Bibliography of Some Recreational Mathematics Books. School of Computing, Information Systems ÁÁÁÁand Mathematics, South Bank Univ. ÁÁ13 Nov 1994, 39pp. Technical Report SBU©CISM©94©09. ÁÁ2nd ed., Aug 1995, 41pp. Technical Report SBU©CISM©95©08. ÁÁ3rd ed., Jun 1996, 42pp. Technical Report SBU©CISM©96©12. ÁÁ4th ed., Jun 1998, 44pp. Technical Report SBU©CISM©98©02. ÁÁÁÁ(Current version is 61pp.) Books on Recreational Mathematics. School of Computing, Information Systems and ÁÁÁÁMathematics, South Bank Univ., until 1996. ÁÁ21 Jan 1991. Approx. 2951 items on 120pp, ringbound. ÁÁ30 Jan 1992. Approx. 3314 items on 138pp, ringbound. ÁÁ10 Jan 1993. Approx. 3606 items on 95pp, ringbound. ÁÁ10 Dec 1994. Approx. 4303 items plus 67 Old Books on 110pp. Technical ÁÁÁÁReport SBU-CISM©94©11. ÁÁ10 Oct 1996. Approx. 4842 items plus 84 Old Books on 127pp. Technical ÁÁÁÁReport SBU©CISM©96©17. ÁÁ24 May 1999. Approx. 6015 items plus 133 Old Books on 166pp. Technical ÁÁÁÁReport SBU©CISM©99©14. ÁÁ26 Feb 2002. Approx. 7185 items plus 192 Old Books plus Supplement of ÁÁÁÁCalculating Devices, on 220pp. thermal bound. ÁÁ22 Nov 2003. Approx. 7811 items plus 202 Old Books plus Supplement of ÁÁÁÁCalculating Devices, on 244pp. thermal bound. Index to Martin Gardner's Columns and Cross Reference to His Books. (Oct 1993.) Slightly revised as: Technical Report SBU©CISM©95©09; School of Computing, Information Systems, and Mathematics; South Bank University, London, Aug 1995, 22pp. (Current version is 23pp and Don Knuth has sent 9pp of additional material and I will combine these at some time.) Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐHarold Adrian Smith. Dick and Fitzgerald Publishers. Books at Brown 34 (1987) 108©114. Sotheby's [Sotheby Parke Bernet]. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐCatalogue of the J. B. Findlay Collection Books and Periodicals on Conjuring and the Allied Arts. Part I: A©O 5©6 Jul 1979. Part II: P©Z plus: Mimeographed Books and Instructions; Flick Books Catalogues of Apparatus and Tricks Autograph Letters, Manuscripts, and Typescripts 4©5 Oct 1979. Part III: Posters and Playbills 3©4 Jul 1980. Each with estimates and results lists. The Celebrated Library of Harrison D. Horblit Esq. Early Science Navigation & Travel Including Americana with a few medical books. Part I A © C 10/11 Jun 1974. Part II D © G 11 Nov 1974. HB. The sale was then cancelled and the library was sold to E. P. Kraus, qv, who issued three further catalogues, c1976. The Honeyman Collection of Scientific Books and Manuscripts. Seven volumes, each ÁÁÁÁwith estimates and results booklets. ÁÁPart I: Printed Books A©B, 30©31 Oct 1978. ÁÁPart II: Printed Books C©E, 30 Apr © 1 May 1979. ÁÁPart III: Manuscripts and Autograph Letters of the 12th to the 20th Centuries. ÁÁPart IV: Printed Books F©J, 5©6 Nov 1979. ÁÁPart V: Printed Books K©M, 12©13 May 1980. ÁÁPart VI: Printed Books N©Sa, 10©11 Nov 1980. ÁÁPart VII: Printed Books Sc©Z and Addenda, 19©20 May 1981. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐLynn A. Steen, ed. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐLibrary Recommendations for Undergraduate Mathematics. MAA Reports No. 4, 1992. Two©Year College Mathematics Library Recommendations. MAA Reports No. 5, 1992. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐStrens/Guy Collection. Author/Title Listing. Univ. of Calgary. Preliminary Catalogue, 319 pp., July 1986. [The original has a lot of blank space. I have a computer version which is reduced to 67pp.] Eva Germaine Rimington Taylor. The Mathematical Practitioners of Tudor & Stuart England 1485©1714. CUP for the Institute of Navigation, 1970. Eva Germaine Rimington Taylor. The Mathematical Practitioners of Hanoverian England 1714-1840. CUP for the Institute of Navigation, 1966. ÁÁÁÁPLUS: Kate Bostock, Susan Hurt & Michael Hart; An Index to the Mathematical Practitioners of Hanoverian England 1714©1840; Harriet Wynter Ltd., London, 1980. W. R. Utz. Letter: Mathematics in literature. MM 55 (1982) 249-250. Utz has sent his 3pp original more detailed version along with 4pp of further citations. This extends Koehler's article. George Walker. The Art of Chess©Play: A New Treatise on the Game of Chess. 4th ed., Sherwood, Gilbert & Piper, London, 1846. Appendix: Bibliographical Catalogue of the chief printed books, writers, and miscellaneous articles on chess, up to the present time, pp. 339©375. Frank R. [Joe] Watson, ed. Booklists. MA. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐPuzzles, Problems, Games and Mathematical Recreations. 16pp, 1980. Selections from the Recommended Books. 18pp, 1980. Full List of Recommended Books. 105pp, 1984. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐMargariete Montague Wheeler. Mathematics Library ©© Elementary and Junior High School. 5th ed., NCTM, 1986. ÁÁSEE ALSO: Schaaf; Wheeler & Hardgrove. Margariete Montague Wheeler & Clarence Ethel Hardgrove. Mathematics Library ©© Elementary and Junior High School. NCTM, (1960; 1968; 1973); 4th ed., 1978. ÁÁSEE ALSO: Schaaf; Wheeler. Ernst WÀ?Àlffing. Mathematischer BÀGÀcherschatz. Systematisches Verzeichnis der wichtigsten deutschen und auslÀÀndischen LehrbÀGÀcher und Monographien des 19. Jahrhunderts auf dem Gebiete der mathematischen Wissenschaften. I: Reine Mathematik; (II: Angewandte Mathematik never appeared). AGM 16, part I (1903). à Ã4. ÁÁMATHEMATICAL GAMESÄ Ä Aviezri S. Fraenkel. Selected Bibliography on Combinatorial Games and Some Related Material. Op. cit. in 3.B. ÁÁà Ã4.A.ÁÁGENERAL THEORY AND NIM-LIKE GAMESÄ Ä Ðа¤˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐ ÁÁConway's extension of this theory is well described in Winning Ways and later work is listed in Fraenkel's Bibliography ©© see section 3.B & 4 ©© so I will not cover such material here. ÁÁà Ã4.A.1.ÁÁONE PILE GAMEÄ Ä ÁÁSee MUS I 145©147. ÁÁ(a, b) denotes the game where one can take 1, 2, ..., or a away from one pile, starting with b in the pile, with the last player winning. The version (10, 100) is sometimes called Piquet des Cavaliers or Piquet À!À Cheval, a name which initially perplexed me. Piquet is one of the older card games, being well known to Rabelais (1534) and was known in the 16C as Cent (or Saunt or Saint) because of its goal of 100 points. See: David Parlett; (Oxford Guide to Card Games, 1990 =) A History of Card Games; OUP, 1991, pp. 24 & 175©181. The connection with horses undoubtedly indicates that (10, 100) was viewed as a game which could be played without cards, while riding ©© see Les Amusemens, Decremps. ÁÁÁÁINDEX ( 3, 13)ÁÁÁÁDudeney, Stong ( 3, 15)ÁÁÁÁMittenzwey, Hoffmann, Mr. X, Dudeney, Blyth, ( 3, 17)ÁÁÁÁFourrey, ( 3, 21)ÁÁÁÁBlyth, Hummerston, ( 4, 15)ÁÁÁÁMittenzwey, ( 6, 30)ÁÁÁÁPacioli, Leske, Mittenzwey, Ducret, ( 6, 31)ÁÁÁÁBaker, ( 6, 50)ÁÁÁÁBall©FitzPatrick, ( 6, 52)ÁÁÁÁRational Recreations ( 6, 57)ÁÁÁÁHummerston, ( 7, 40)ÁÁÁÁMittenzwey, ( 7, 41)ÁÁÁÁSprague, ( 7, 45)ÁÁÁÁMittenzwey, ( 7, 50)ÁÁÁÁDecremps, ( 7, 60)ÁÁÁÁFourrey, ( 8, 100)ÁÁÁÁBachet, Carroll, ( 9, 100)ÁÁÁÁBachet, Ozanam, Alberti (10, 100)ÁÁÁÁBachet, Henrion, Ozanam, Alberti, Les Amusemens, Hooper, Decremps, ÁÁÁÁÁÁÁÁBadcock, Jackson, Rational Recreations, Manuel des Sorciers, ÁÁÁÁÁÁÁÁBoy's Own Book, Nuts to Crack, Young Man's Book, Carroll, ÁÁÁÁÁÁÁÁMagician's Own Book, Book of 500 Puzzles, Secret Out, ÁÁÁÁÁÁÁÁBoy's Own Conjuring Book, Vinot, Riecke, Fourrey, Ducret, Devant, (10, 120)ÁÁÁÁBachet, (12, 134)ÁÁÁÁDecremps, ÁÁGeneral case: Bachet, Ozanam, Alberti, Decremps, Boy's Own Book, Young Man's Book, Vinot, Mittenzwey, (others ?? check) ÁÁVersions with limited numbers of each value or using a die ©© see 4.A.1.a. ÁÁVersion where an odd number in total has to be taken: Dudeney, Grossman & Kramer, Sprague. ÁÁVersions with last player losing: Mittenzwey, ÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐ Pacioli. De Viribus. c1500. Ff. 73v © 76v. XXXIIII effecto afinire qualunch' numero na'ze al compagno anon prendere piu de un termi(n)ato .n. (34th effect to finish whatever number is before the company, not taking more than a limiting number) = Peirani 109-112. Phrases it as an addition problem. Considers (6, 30) and the general problem. David Parlett. (Originally: The Oxford Guide to Card Games; OUP, 1990); reissued as: A History of Card Games. Penguin, 1991, pp. 174©175. "Early references to 'les luettes', said to have been played by Anne de Bretagne and Archduke Philip the Fair in 1503, and by Gargantua in 1534, seem to suggest a game of the Nim family (removing numbers of objects from rows and columns)." Cardan. Practica Arithmetice. 1539. Chap. 61, section 18, ff. T.iiii.v © T.v.r (p. 113). "Ludi mentales". One has 1, 3, 6 and the other has 2, 4, 5; or one has 1, 3, 5, 8, 9 and the other has 2, 4, 6, 7, 10; one one wants to make 100. "Sunt magnÀ%À inventionis, & ego inveni À%Àquitando & sine aliquo auxilio cum socio potes ludere & memorium exercere ...." Baker. Well Spring of Sciences. 1562? Prob. 5: To play at 31 with Numbers, 1670: pp. 353-354. ??NX. (6, 31). Bachet. Problemes. 1612. Prob. XIX: 1612, 99©103. Prob. XXII, 1624: 170©173; 1884: 115-117. Phrases it as an addition problem. First considers (10, 100), then (10, 120), (8, 100), (9, 100), and the general case. Labosne omits the demonstration. Dennis Henrion. Nottes to van Etten. 1630. Pp. 19©20. (10, 100) as an addition problem, citing Bachet. Ozanam. 1694. Prob. 21, 1696: 71©72; 1708: 63-64. Prob. 25, 1725: 182-184. Prob. 14, 1778: 162©164; 1803: 163©164; 1814: 143©145. Prob. 13, 1840: 73©74. Phrases it as an addition problem. Considers (10, 100) and (9, 100) and remarks on the general case. Alberti. 1747. Due persone essendo convenuto ..., pp. 105-108 (66-67). This is a slight recasting of Ozanam. Les Amusemens. 1749. Prob. 10, p. 130: Le Piquet des Cavaliers. (10, 100) in additive form. "Deux amis voyagent À!À cheval, l'un propose À!À l'autre un cent de Piquet sans carte." William Hooper. Rational Recreations, In which the Principles of Numbers and Natural Philosophy Are clearly and copiously elucidated, by a series of Easy, Entertaining, Interesting Experiments. Among which are All those commonly performed with the cards. [Taken from my 2nd ed.] 4 vols., L. Davis et al., London, 1774; 2nd ed., corrected, L. Davis et al., London, 1783©1782 (vol. 1 says 1783, the others say 1782; BMC gives 1783©82); 3rd ed., corrected, 1787; 4th ed., corrected, B. Law et al., London, 1794. [Hall, BCB 180©184 & Toole Stott 389©392. Hall says the first four eds. have identical pagination. I have not seen any difference in the first four editions, except as noted in Section 6.P.2. Hall, OCB, p. 155. Heyl 177 notes the different datings of the 2nd ed, Hall, BCB 184 and Toole Stott 393 is a 2 vol. 4th ed., corrected, London, 1802. Toole Stott 394 is a 2 vol. ed. from Perth, 1801. I have a note that there was an 1816 ed, but I have no details. Since all relevant material seems the same in all volumes, I will cite this as 1774.] Vol. 1, recreation VIII: The magical century. (10, 100) in additive form. Mentions other versions and the general rule. ÁÁÁÁI don't see any connection between this and Rational Recreations, 1824. Henri Decremps. Codicile de JÀ)ÀrÀ=Àme Sharp, Professeur de Physique amusante; OÀIÀ l'on trouve parmi plusieurs Tours dont il n'est point parlÀ)À dans son Testament, diverses rÀ)ÀcrÀ)Àations relatives aux Sciences & Beaux©Arts; Pour servir de troisiÀ/Àme suite À À La Magie Blanche DÀ)ÀvoilÀ)Àe. Lesclapart, Paris, 1788. Chap. XXVII, pp. 177©184: Principes mathÀ)Àmatiques sur le piquet À!À cheval, ou l'art de gagner son diner en se promenant. Does (10, 100) in additive form, then discusses the general method, illustrating with (7, 50) and (12, 134). Badcock. Philosophical Recreations, or, Winter Amusements. [1820]. Pp. 33©34, no. 48: A curious recreation with a hundred numbers, usually called the magical century. (10, 100) as an additive problem where each person starts with 50 counters. Discusses general case, but doesn't notice that the limitation to 50 counters each considerably changes the game! Jackson. Rational Amusement. 1821. Arithmetical Puzzles, no. 47, pp. 11 & 64. Additive form of (10, 100). Rational Recreations. 1824. Exercise 12(?), pp. 57©58. As in Badcock. Then says it can be generalised and gives (6, 52). Manuel des Sorciers. 1825. Pp. 57©58, art. 30: Le piquet sans cartes. ??NX (10, 100) done subtractively. The Boy's Own Book. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐThe certain game. 1828: 177; 1828©2: 236; 1829 (US): 104; 1855: 386-387; 1868: 427. The magical century. 1828: 180; 1828©2: 236-237; 1829 (US): 104©105; 1855: 391-392. ÁÁBoth are additive phrasings of (10, 100). The latter mentions using other numbers and how to win then. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐNuts to Crack V (1836), no. 70. An arithmetical problem. (10, 100). Young Man's Book. 1839. Pp. 294©295. A curious Recreation with a Hundred Numbers, usually called the Magical Century. Almost identical to Boy's Own Book. Lewis Carroll. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐDiary entry for 5 Feb 1856. In Carroll©Gardner, pp. 42©43. (10, 100). Wakeling's note in the Diaries indicates he is not familiar with this game. Diary entry for 24 Oct 1872. Says he has written out the rules for Arithmetical Croquet, a game he recently invented. Roger Lancelyn Green's abridged version of the Diaries, 1954, prints a MS version dated 22 Apr 1889. Carroll©Wakeling, prob. 38, pp. 52©53 and Carroll©Gardner, pp. 39 & 42 reprint this, but Gardner has a misprinted date of 1899. Basically (8, 100), but passing the values 10, 20, ..., requires special moves and one may have to go backward. Also, when a move is made, some moves are then barred for the next player. Overall, the rules are typically Carrollian©baroque. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐMagician's Own Book. 1857. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐThe certain game, p. 243. As in Boy's Own Book. The magical century, pp. 244©245. As in Boy's Own Book. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐBook of 500 Puzzles. 1859. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐThe certain game, p. 57. As in Boy's Own Book. The magical century, pp. 58©59. As in Boy's Own Book. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐThe Secret Out. 1859. Piquet on horseback, pp. 397©398 (UK: 130-131) ©© additive (10, 100) unclearly explained. Boy's Own Conjuring Book. 1860. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐThe certain game, pp. 213-214. As in Boy's Own Book. Magical century, pp. 215. As in Boy's Own Book. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐVinot. 1860. Art. XI: Un cent de piquet sans cartes, pp. 19©20. (10. 100). Says the idea can be generalised, giving (7, 52) as an example. Leske. Illustriertes Spielbuch fÀGÀr MÀÀdchen. 1864? Prob. 563©III, pp. 247: Wer von 30 Rechenpfennigen den letzen wegnimmt, hat gewonnen. (6, 30). F. J. P. Riecke. Mathematische Unterhaltungen. 3 vols., Karl Aue, Stuttgart, 1867, 1868 & 1873; reprint in one vol., SÀÀndig, Wiesbaden, 1973. Vol. 3, art 22.2, p. 44. Additive form of (10, 100). Mittenzwey. 1880. Probs. 286©287, pp. 52 & 101©102; 1895?: 315©317, pp. 56 & 103©104; 1917: 315©317, pp. 51 & 98. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐ(6, 30), last player wins. (4, 15), last player loses, the solution discusses other cases: (7, 40), (7, 45) and indicates the general solution. (added in 1895?) (3, 15), last player loses. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐHoffmann. 1893. Chap VII, no. 19: The fifteen matches puzzle, pp. 292 & 300-301 = Hoffmann©Hordern, p. 197. (3, 15). c= Benson, 1904, The fifteen match puzzle, pp. 241-242. Ball©FitzPatrick. 1st ed., 1898. DeuxiÀ/Àme exemple, pp. 29©30. (6, 50). E. Fourrey. RÀ)ÀcrÀ)Àations ArithmÀ)Àtiques. (Nony, Paris, 1899; 2nd ed., 1901); 3rd ed., Vuibert & Nony, Paris, 1904; (4th ed., 1907); 8th ed., Librairie Vuibert, Paris, 1947. [The 3rd and 8th eds are identical except for the title page, so presumably are identical to the 1st ed.] Sections 65-66: Le jeu du piquet À!À cheval, pp. 48-49. Additive forms of (10, 100) and (7, 60). Then gives subtractive form for a pile of matches for (3, 17). À(Àtienne Ducret. RÀ)ÀcrÀ)Àations MathÀ)Àmatiques. Garnier FrÀ/Àres, Paris, nd [not in BN, but a similar book, nouv. ed., is 1892]. Pp. 102-104: Le piquet À!À cheval. Additive version of (10, 100) with some explanation of the use of the term piquet. Discusses (6, 30). Mr. X [possibly J. K. Benson ©© see entry for Benson in Abbreviations]. His Pages. The Royal Magazine 9:3 (Jan 1903) 298©299. A good game for two. (3, 15) as a subtraction game. David Devant. Tricks for Everyone. Clever Conjuring with Everyday Objects. C. Arthur Pearson, London, 1910. A counting race, pp. 52©53. (10, 100). Dudeney. AM. 1917. Prob. 392: The pebble game, pp. 117 & 240. (3, 15) & (3, 13) with the object being to take an odd number in total. For 15, first player wins; for 13, second player wins. (Barnard (50 Telegraph ..., 1985) gives the case (3, 13).) Blyth. Match©Stick Magic. 1921. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐFifteen matchstick game, pp. 87©88. (3, 15). Majority matchstick game, p. 88. (3, 21). Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐHummerston. Fun, Mirth & Mystery. 1924. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐTwo second©sight tricks (no. 2), p. 84. (6, 57), last player losing. A match mystery, p. 99. (3, 21), last player losing. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐH. D. Grossman & David Kramer. A new match©game. AMM 52 (1945) 441-443. Cites Dudeney and says Games Digest (April 1938) also gave a version, but without solution. Gives a general solution whether one wants to take an odd total or an even total. C. L. Stong. The Amateur Scientist. Ill. by Roger Hayward. S&S, 1960. How to design a "Pircuit" or Puzzle circuit, pp. 388©394. On pp. 388©391, Harry Rudloe describes a relay circuit for playing the subtractive form of (3, 13), which he calls the "battle of numbers" game. Ronald Sprague. Unterhaltsame Mathematik. Vieweg, Braunschweig, 1961. Translated by T. H. O'Beirne as: Recreations in Mathematics, Blackie, London, 1963. Problem 24: "Ungerade" gewinnt, pp. 16 & 44-45. (= 'Odd' is the winner, pp. 18 & 53-55.) (7, 41) with the winner being the one who takes an odd number in total. Solves (7, b) and states the structure for (a, b). ÁÁÁÁI also have some other recent references to this problem. Lewis (1983) gives a general solution which seems to be wrong. à ÃÁÁ4.A.1.a. ÁÁTHE 31 GAMEÄ Ä Ðа¤˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐ ÁÁNumerical variations: Badcock, Gibson, McKay. ÁÁDie versions: Secret Out (UK), Loyd, Mott©Smith, Murphy. ÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐ Baker. Well Spring of Sciences. 1562? Prob. 5: To play at 31 with Numbers, 1670: pp. 353-354. ??NX. (6, 31). ??CHECK if this has the limited use of numbers. John Fisher. Never Give a Sucker an Even Break. (1976); Sphere Books, London, 1978. Thirty©one, pp. 102©104. (6, 31) additively, but played with just 4 of each value, the 24 cards of ranks 1 ©© 6, and the first to exceed 31 loses. He says it is played extensively in Australia and often referred to as "The Australian Gambling Game of 31". Cites the 19C gambling expert Jonathan Harrington Green who says it was invented by Charles James Fox (1749-1806). Gives some analysis. Badcock. Philosophical Recreations, or, Winter Amusements. [1820]. Pp. 33©34, no. 48: A curious recreation with a hundred numbers, usually called the magical century. (10, 100) as an additive problem where each person starts with 50 counters. Discusses general case, but doesn't notice that the limitation to 50 counters each considerably changes the game! Nuts to Crack V (1836), no. 71. (6, 31) additively, with four of each value. "Set down on a slate, four rows of figures, thus:©© ... You agree to rub out one figure alternately, to see who shall first make the number thirty©one." Magician's Own Book. 1857. Art. 31: The trick of thirty-one, pp. 70-71. (6, 31) additively, but played with just 4 of each value ©© e.g. the 24 cards of ranks 1 ©© 6. The author advises you not to play it for money with "sporting men" and says it it due to Mr. Fox. Cf Fisher. = Boy's Own Conjuring Book; 1860; Art. 29: The trick of thirty-one, pp. 78-79. = The Secret Out; 1859, pp. 65©66, which adds a footnote that the trick is taken from the book One Hundred Gambler Tricks with Cards by J. H. Green, reformed gambler, published by Dick & Fitzgerald. The Secret Out (UK), c1860. To throw thirty-one with a die before your antagonist, p. 7. This is incomprehensible, but is probably the version discussed by Mott©Smith. Edward S. Sackett. US Patent 275,526 ©© Game. Filed: 9 Dec 1882; patented: 10 Apr 1883. 1p + 1p diagrams. Frame of six rows holding four blocks which can be slid from one side to the other to play the 31 game, though other numbers of rows, blocks and goal may be used. Gives an example of a play, but doesn't go into the strategy at all. Larry Freeman. Yesterday's Games. Taken from "an 1880 text" of games. (American edition by H. Chadwick.) Century House, Watkins Glen, NY, 1970. P. 107: Thirty©one. (6, 31) with 4 of each value ©© as in Magician's Own Book. Algernon Bray. Letter: "31" game. Knowledge 3 (4 May 1883) 268, item 806. "... has lately made its appearance in New York, ...." Seems to have no idea as how to win. Loyd. Problem 38: The twenty-five up puzzle. Tit-Bits 32 (12 Jun & 3 Jul 1897) 193 & 258. = Cyclopedia. 1914. The dice game, pp. 243 & 372. = SLAHP: How games originate, pp. 73 & 114. The first play is arbitrary. The second play is by throwing a die. Further values are obtained by rolling the die by a quarter turn. Ball©FitzPatrick. 1st ed., 1898. GÀ)ÀnÀ)Àralization rÀ)Àcente de cette question, pp. 30©31. (6, 50) with each number usable at most 3 times. Some analysis. Ball. MRE, 4th ed., 1905, p. 20. Some analysis of (6, 50) where each player can play a value at most 3 times ©© as in Ball©FitzPatrick, but with the additional sentence: "I have never seen this extension described in print ...." He also mentions playing with values limited to two times. In the 5th ed., 1911, pp. 19©21, he elaborates his analysis. Dudeney. CP. 1907. Prob. 79: The thirty©one game, pp. 125©127 & 224. Says it used to be popular with card©sharpers at racecourses, etc. States the first player can win if he starts with 1, 2 or 5, but the analysis of cases 1 and 2 is complicated. This occurs as No. 459: The thirty©one puzzle, Weekly Dispatch (17 Aug 1902) 13 & (31 Aug 1902) 13, but he leaves the case of opening move 2 to the reader, but I don't see the answer given in the next few columns. Devant. Tricks for Everyone. Op. cit. in 4.A.1. 1910. The thirty©one trick, pp. 53©54. Says to get to 3, 10, 17, 24. Hummerston. Fun, Mirth & Mystery. 1924. Thirty©one ©© a game of skill, pp. 95©96. This uses a layout of four copies of the numbers 1, 2, 3, 4, 5, 6 with one copy of 20 in a 5 x 5 square with the 20 in the centre. Says to get to 3, 10, 17, 24, but that this will lose to an experienced player. Loyd Jr. SLAHP. 1928. The "31 Puzzle Game", pp. 3 & 87. Loyd Jr says that as a boy, he often had to play it against all comers with a $50 prize to anyone who could beat 'Loyd's boy'. This is the game that Loyd Sr called 'Blind Luck', but I haven't found it in the Cyclopedia. States the first player wins with 1, 2 or 5, but only sketches the case for opening with 5. I have seen an example of Blind Luck ©© it has four each of the numbers 1 © 6 arranged around a frame containing a horseshoe with 13 in it. McKay. Party Night. 1940. The 21 race, pp. 166. Using the numbers 1, 2, 3, 4, at most four times, achieve 21. Says to get 1, 6, 11, 16. He doesn't realise that the sucker can be mislead into playing first with a 1 and losing! Says that with 1, ..., 5 at most four times, one wants to achieve 26 and that with 1, ..., 6 at most four times, one wants to achieve 31. Gives just the key numbers each time. Geoffrey Mott©Smith. Mathematical Puzzles for Beginners and Enthusiasts. (Blakiston, 1946); revised 2nd ed., Dover, 1954. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐProb. 179: The thirty©one game, pp. 117©119 & 231©232. As in Dudeney. Prob. 180: Thirty©one with dice, p. 119 & 232©233. Throw a die, then make quarter turns to produce a total of 31. Analysis based on digital roots (i.e. remainders (mod 9)). First player wins if the die comes up 4, otherwise the second player can win. He doesn't treat any other totals. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐ"Willane". Willane's Wizardry. Academy of Recorded Crafts, Arts and Sciences, Croydon, 1947. "Trente et un", pp. 56©57. Says he doesn't know any name for this. Get 31 using 4 each of the cards A, 2, ..., 6. Says first player loses easily if he starts with 4, 5, 6 (not true according to Dudeney) and that gamblers dupe the sucker by starting with 3 and winning enough that the sucker thinks he can win by starting with 3. But if he starts with a 1 or 2, then the second player must play low and hope for a break. Walter B. Gibson. Fell's Guide to Papercraft Tricks, Games and Puzzles. Frederick Fell, NY, 1963. Pp. 54©55: First to fifty. First describes (50, 6), but then adds a version with slips of paper: eight marked 1 and seven marked with 2, 3, 4, 5, 6 and you secretly extract a 6 slip when the other player starts. Harold Newman. The 31 Game. JRM 23:3 (1991) 205©209. Extended analysis. Confirms Dudeney. Only cites Dudeney & Mott©Smith. Bernard Murphy. The rotating die game. Plus 27 (Summer 1994) 14©16. Analyses the die version as described by Mott©Smith and finds the set, S(n), of winning moves for achieving a count of n by the first player, is periodic with period 9 from n = 8, i.e. S(n+9) = S(n) for n ÀÀ 8. There is no first player winning move if and only if n is a multiple of 9. [I have confirmed this independently.] Ken de Courcy. The Australian Gambling Game of 31. Supreme Magic Publication, Bideford, Devon, nd [1980s?]. Brief description of the game and some indications of how to win. He then plays the game with face©down cards! However, he insures that the cards by him are one of of each rank and he knows where they are. à ÃÁÁ4.A.2. ÁÁSYMMETRY ARGUMENTSÄ Ä Loyd?? Problem 43: The daisy game. Tit-Bits 32 (17 Jul & 7 Aug 1897) 291 & 349. (= Cyclopedia. 1914. A daisy puzzle game, pp. 85 & 350. c= MPSL2, prob. 57, pp. 40-41 & 140. c= SLAHP: The daisy game, pp. 42 & 99.) Circular version of Kayles with 13 objects. Solution uses a symmetry argument ©© but the Tit-Bits solution was written by Dudeney. Dudeney. Problem 500: The cigar puzzle. Weekly Dispatch (7 Jun, 21 Jun, 5 Jul, 1903) all p. 16. (= AM, prob. 398, pp. 119, 242.) Symmetry in placement game, using cigars on a table. Loyd. Cyclopedia. 1914. The great Columbus problem, pp. 169 & 361. (= MPSL1, prob. 65, pp. 62 & 144. = SLAHP: When men laid eggs, pp. 75 & 115.) Placing eggs on a table. Maurice Kraitchik. La MathÀ)Àmatique des Jeux. Stevens, Bruxelles, 1930. Section XII, prob. 1, p. 296. (= Mathematical Recreations; Allen & Unwin, London, 1943; Problem 1, pp. 13-14.) Child plays black and white against two chess players and guarantees to win one game. [MJ cites L'Echiquier (1925) 84, 151.] ÁÁÁÁCAUTION. The 2nd edition of Math. des Jeux, 1953, is a translation of Mathematical Recreations and hence omits much of the earlier edition. Leopold. At Ease! 1943. Chess wizardry in two minutes, pp. 105-106. Same as Kraitchik. à ÃÁÁ4.A.3.ÁÁKAYLESÄ Ä Ðа¤˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐ ÁÁThis has objects in a line or a circle and one can remove one object or two adjacent objects (or more adjacent objects in a generalized version of the game). This derives from earlier games with an array of pins at which one throws a ball or stick. ÁÁMurray 442 cites Act 17 of Edward IV, c.3 (1477): "Diversez novelx ymagines jeuez appellez Cloishe Kayles ..." This outlawed such games. A 14C picture is given in [J. A. R. Pimlott; Recreations; Studio Vista, 1968, plate 9, from BM Royal MS 10 E IV f.99] showing a 3 x 3 array of pins. A version is shown in Pieter Bruegel's painting "Children's Games" of 1560 with balls being thrown at a row of pins by a wall, in the back right of the scene. Versions of the game are given in the works of Strutt and Gomme cited in 4.B.1. Gomme II 115-116 discusses it under Roly-poly, citing Strutt and some other sources. Strutt 270-271 (= Strutt©Cox 219©220) calls it "Kayles, written also cayles and keiles, derived from the French word quilles". He has redrawings of two 14C engravings (neither that in Pimlott) showing lines of pins at which one throws a stick (= plate opp. 220 in Strutt©Cox). He also says Closh or Cloish seems to be the same game and cites prohibitions of it in c1478 et seq. Loggats was analogous and was prohibited under Henry VIII and is mentioned in Hamlet. ÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐ 14C MS in the British Museum, Royal Library, No. 2, B. vii. Reproduced in Strutt, p. 271. Shows a monk(?) standing by a line of eight conical pins and another monk(?) throwing a stick at the pins. Anonymous. Games of the 16th Century. The Rockliff New Project Series. Devised by Arthur B. Allen. The Spacious Days of Queen Elizabeth. Background Book No. 5. Rockliff Publishing, London, ÀÀ1950, 4th ptg. The Background Books seem to be consecutively paginated as this booklet is paginated 129©152. Pp. 133©134 describes loggats, quoting Hamlet and an unknown poet of 1611. P. 137 is a photograph of the above 14C illustration. The caption is "Skittles, or "Kayals", and Throwing a Whirling Stick". van Etten. 1624. Prob. 72 (misnumbered 58) (65), pp 68-69 (97-98): Du jeu des quilles (Of the play at Keyles or Nine©Pins). Describes the game as a kind of ninepins. Loyd. Problem 43: The daisy game. Tit-Bits 32 (17 Jul & 7 Aug 1897) 291 & 349. (= Cyclopedia. 1914. A daisy puzzle game, pp. 85 & 350. c= MPSL2, prob. 57, pp. 40-41 & 140. c= SLAHP: The daisy game, pp. 42 & 99.) Circular version of Kayles with 13 objects. See also 4.A.2. Dudeney. Sharpshooters puzzle. Problem 430. Weekly Dispatch (26 Jan, 9 Feb, 1902) both p. 13. Simple version of Kayles. Ball. MRE, 4th ed., 1905, pp. 19©20. Cites Loyd in Tit-Bits. Gives the general version: place p counters in a circle and one can take not more than m adjacent ones. Dudeney. CP. 1907. Prob. 73: The game of Kayles, pp. 118-119 & 220. Kayles with 13 objects. Loyd. Cyclopedia. 1914. Rip van Winkle puzzle, pp. 232 & 369-370. (c= MPSL2, prob. 6, pp. 5 & 122.) Linear version with 13 pins and the second knocked down. Gardner asserts that Dudeney invented Kayles, but it seems to be an abstraction from the old form of the game. Rohrbough. Puzzle Craft, later version, 1940s?. Daisy Game, p. 22. Kayles with 13 petals of a daisy. Philip Kaplan. More Posers. (Harper & Row, 1964); Macfadden©Bartell Books, 1965. Prob. 45, pp. 48 & 95. Circular kayles with five objects. Doubleday © 2. 1971. Take your pick, pp. 63©65. This is Kayles with a row of 10, but he says the first player can only take one. à ÃÁÁ4.A.4.ÁÁNIMÄ Ä Ðа¤˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐ ÁÁNim is the game with a number of piles and a player can take any number from one of the piles. Normally the last one to play wins. ÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐ David Parlett. (Originally: The Oxford Guide to Card Games; OUP, 1990); reissued as: A History of Card Games. Penguin, 1991. Pp. 174©175. "Early references to 'les luettes', said to have been played by Anne de Bretagne and Archduke Philip the Fair in 1503, and by Gargantua in 1534, seem to suggest a game of the Nim family (removing numbers of objects from rows and columns)." Charles L. Bouton. Nim: a game with a complete mathematical theory. Annals of Math. (2) 3 (1901/02) 35-39. He says Nim is played at American colleges and "has been called Fan-Tan, but as it is not the Chinese game of that name, the name in the title is proposed for it." He says Paul E. More showed him the misÀ/Àre (= last player loses) version in 1899, so it seems that Bouton did not actually invent the game himself. Ahrens. "Nim", ein amerikanisches Spiel mit mathematischer Theorie. Naturwissenschaftliche Wochenschrift 17:22 (2 Mar 1902) 258-260. He says that Bouton has admitted that he had confused Nim and Fan-Tan. Fan-Tan is a Chinese game where you bet on the number of counters (mod 4) in someone's hand. Parker, Ancient Ceylon, op. cit. in 4.B.1, pp. 570©571, describes a similar game, based on odd and even, as popular in Ceylon and "certainly one of the earliest of all games". ÁÁÁÁFor more about Fan©Tan, see the following. Stewart Culin. Chess and playing cards. Catalogue of games and implements for divination exhibited by the United States National Museum in connection with the Department of ArchÀ%Àology and Paleontology of the University of Pennsylvania at the Cotton States and International Exposition, Atlanta, Georgia, 1895. IN: Report of the U. S. National Museum, year ending June 30, 1896. Government Printing Office, Washington, 1898, HB, pp. 665©942. [There is a reprint by Ayer Co., Salem, Mass., c1990.] Fan©Tan (= FÀÀn tÀÀÀÀn = repeatedly spreading out) is described on pp. 891 & 896, with discussion of related games on pp. 889©902. Alan S. C. Ross. Note 2334: The name of the game of Nim. MG 37 (No. 320) (May 1953) 119-120. Conjectures Bouton formed the word 'nim' from the German 'nimm'. Gives some discussion of Fan-Tan and quotes MUS I 72. J. L. Walsh. Letter: The name of the game of Nim. MG 37 (No. 322) (Dec 1953) 290. Relates that Bouton said that he had chosen the word from the German 'nimm' and dropped one 'm'. W. A. Wythoff. A modification of the game of Nim. Nieuw Archief voor Wiskunde (Groningen) (2) 7 (1907) 199-202. He considers a Nim game with two piles allows the extra move of taking the same amount from both piles. [Is there a version with more piles where one can take any number from one pile or equal amounts from two piles?? See Barnard, below for a three pile version.] Ahrens. MUS I. 1910. III.3.VII: Nim, pp. 72-88. Notes that Nim is not the same as Fan-Tan, has been known in Germany for decades and is played in China. Gives a thorough discussion of the theory of Nim and of an equivalent game and of Wythoff's game. E. H. Moore. A generalization of the game called Nim. Annals of Math. (2) 11 (1910) 93-94. He considers a Nim game with n piles and one is allowed to take any number from at most k piles. Ball. MRE, 5th ed., 1911, p. 21. Sketches the game of Nim and its theory. A. B. Nordmann. One Hundred More Parlour Tricks and Problems. Wells, Gardner, Darton & Co., London, nd [1927 ©© BMC]. No. 13: The last match, pp. 10©11. Thirty matches divided at random into three heaps. Last player loses. Explanation of how to win is rather cryptic: "you must try and take away ... sufficient ... to leave the matches in the two or three heaps remaining, paired in ones, twos, fours, etc., in respect of each other." Loyd Jr. SLAHP. 1928. A tricky game, pp. 47 & 102. Nim (3, 4, 8). Emanuel Lasker. Brettspiele der VÀ?Àlker. 1931. See comments in 4.A.5. JÀ?Àrg Bewersdorff [email of 6 Jun 1999] says that Lasker considered a three person Nim and found an equilibrium for it ©© see: JÀ?Àrg Bewersdorff; GlÀGÀck, Logik und Bluff Mathematik im Spiel ©© Methoden, Ergebnisse und Grenzen; Vieweg, 1998, Section 2.3 Ein Spiel zu dritt, pp. 110©115. Lynn Rohrbough, ed. Fun in Small Spaces. Handy Series, Kit Q, Cooperative Recreation Service, Delaware, Ohio, nd [c1935]. Take Last, p. 10. Last player loses Nim (3, 5, 7). Rohrbough. Puzzle Craft. 1932. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐJapanese Corn Game, p. 6 (= p. 6 of 1940s?). Last player loses Nim (1, 2, 3, 4, 5). Japanese Corn Game, p. 23. Last player loses Nim (3, 5, 7). Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐRenÀ)À de Possel. Sur la ThÀ)Àorie MathÀ)Àmatique des Jeux de Hasard et de RÀ)Àflexion. ActualitÀ)Às Scientifiques et Industrielles 436. Hermann, Paris, 1936. Gives the theory of Nim and also the misÀ/Àre version. Depew. Cokesbury Game Book. 1939. Make him take it, pp. 187©188. Nim (3, 4, 5), last player loses. Edward U. Condon, Gereld L. Tawney & Willard A. Derr. US Patent 2,215,544 ©© Machine to Play Game of Nim. Filed: 26 Apr 1940; patented: 24 Sep 1940. 10pp + 11pp diagrams. E. U. Condon. The Nimatron. AMM 49 (1942) 330-332. Has photo of the machine. Benedict Nixon & Len Johnson. Letters to the Notes & Queries Column. The Guardian (4 Dec 1989) 27. Reprinted in: Notes & Queries, Vol. 1; Fourth Estate, London, 1990, pp. 14©15. These describe the Ferranti Nimrod machine for playing Nim at the Festival of Britain, 1951. Johnson says it played Nim (3, 5, 6) with a maximum move of 3. The Catalogue of the Exhibition of Science shows this as taking place in the Science Museum. H. S. M. Coxeter. The golden section, phyllotaxis, and Wythoff's game. SM 19 (1953) 135-143. Sketches history and interconnections. H. S. M. Coxeter. Introduction to Geometry. Wiley, 1961. Chap. 11: The golden section and phyllotaxis, pp. 160©172. Extends his 1953 material. A. P. Domoryad. Mathematical Games and Pastimes. (Moscow, 1961). Translated by Halina Moss. Pergamon, Oxford, 1963. Chap. 10: Games with piles of objects, pp. 61-70. On p. 62, he asserts that Wythoff's game is 'the Chinese national game tsyanshidzi ("picking stones")'. However M.-K. Siu cannot recognise such a Chinese game, unless it refers to a form of jacks, which has no obvious connection with Wythoff's game or other Nim games. He says there is a Chinese character, 'nian', which is pronounced 'nim' in Cantonese and means to pick up or take things. N. L. Haddock. Note 2973: A note on the game of Nim. MG 45 (No. 353) (Oct 1961) 245-246. Wonders if the game of Nim is related to Mancala games. T. H. O'Beirne. Puzzles and Paradoxes. OUP, 1965. Section on misÀ/Àre version of Wythoff's game, p. 133. Richard Guy (letter of 27 Feb 1985) says this is one of O'Beirne's few mistakes ©© cf next entry. Winning Ways. 1982. P. 407 says Wythoff's game is also called Chinese Nim or Tsyan-shizi. No reference given. See comment under Domoryad above. This says many authors have done this incorrectly. D. St. P. Barnard. 50 Daily Telegraph Brain-Twisters. Javelin Books, Poole, Dorset, 1985. Prob. 30: All buttoned up, pp. 49-50, 91 & 115. He suggests three pile game where one can take any number from one pile or an equal number from any two or all three piles. [See my note to Wythoff, above.] Matthias Mala. Schnelle Spiele. Hugendubel, Munich, 1988. San Shan, p. 66. This describes a nim©like game named San Shan and says it was played in ancient China. Jagannath V. Badami. Musings on Arithmetical Numbers Plus Delightful Magic Squares. Published by the author, Bangalore, India, nd [Preface dated 9 Sep 1999]. Section 4.16: The game of Nim, pp. 124©125. This is a rather confused description of one pile games (21, 5) and (41, 5), but he refers to solving them by (mentally) dividing the pile into piles. This makes me think of combining the two games, i.e. playing Nim with several piles but with a limit on the number one can take in a move. à ÃÁÁ4.A.5.ÁÁGENERAL THEORYÄ Ä Charles Babbage. The Philosophy of Analysis ©© unpublished collection of MSS in the BM as Add. MS 37202, c1820. ??NX. Ff. 134©144 are: Essay 10 Part 5. See 4.B.1 for more details. At the top of f. 134.r, he has added a note: "This is probably my earliest Note on Games of Skill. I do not recollect the date. 3 March 1865". He then describes Tit Tat To and makes some simple analysis, but he never uses a name for it. Charles Babbage. Notebooks ©© unpublished collection of MSS in the BM as Add. MS 37205. ??NX. See 4.B.1 for more details. On f. 304, he starts on analysis of games. Ff. 310-383 are almost entirely devoted to Tit©Tat©To, with some general discussions. F. 321.r, 10 Sep 1860, is the beginning of a summary of his work on games of skill in general. F. 324©333, Oct 1844, studies "General laws for all games of Skill between two players" and draws flow charts showing the basic recursive analysis of a game tree (ff. 325.v & 325.r). On f. 332, he counts the number of positions in Tit Tat To as 9! + 8! + ... + 1! = 409,113. F. 333 has an idea of the tree structure of a game. John M. Dubbey. The Mathematical Work of Charles Babbage. CUP, 1978, pp. 96-97 & 125-130. See 4.B.1 for more details. He discusses the above Babbage material. On p. 127, Dubbey has: "The basic problem is one that appears not to have been previously considered in the history of mathematics." Dubbey, on p. 129, says: "This analysis ... must count as the first recorded stochastic process in the history of mathematics." However, it is really a deterministic two©person game. E. Zermelo. ÀFÀber eine Anwendung der Mengenlehre auf die Theorie des Schachspiels. Proc. 5th ICM (1912), CUP, 1913, vol. II, 501-504. Gives general idea of first and second person games. Ahrens. A&N. 1918. P. 154, note. Says that each particular Dots and Boxes board, with rational play, has a definite outcome. W. Rivier. Archives des Sciences Physiques et Naturelles (Nov/Dec 1921). ??NYS ©© cited by Rivier (1935) who says that the later article is a new and simpler version of this one. H. Steinhaus. Difinicje potrzebne do teorji gry i poÀ¯Àcigu (Definitions for a theory of games and pursuit). MyÀ¯Àl Akademicka (LwÀ;Àw) 1:1 (Dec 1925) 13-14 (in Polish). Translated, with an introduction by Kuhn and a letter from Steinhaus in: Naval Research Logistics Quarterly 7 (1960) 105-108. DÀ)ÀnÀ/Às KÀ?Ànig. ÀFÀber eine Schlussweise aus dem Endlichen ins Unendliche. Mitteilungen der UniversitÀÀ Szeged 3 (1927) 121©130. ??NYS ©© cited by Rivier (1935). KalmÀÀr cites it to the same Acta as his article. LÀÀszlÀ;À KalmÀÀr. Zur Theorie der abstracten Spiele. Acta Litt. Sci. Regia Univ. Hungaricae Francisco-Josephine (Szeged) 4 (1927) 62-85. Says there is a gap in Zermelo which has been mended by KÀ?Ànig. Lengthy approach, but clearly gets the idea of first and second person games. Max Euwe. Proc. Koninklijke Akadamie van Wetenschappen te Amsterdam 32:5 (1929). ??NYS ©© cited by Rivier (1935). Emanuel Lasker. Brettspiele der VÀ?Àlker. RÀÀtsel- und mathematische Spiele. A. Scherl, Berlin, 1931, pp. 170-203. Studies the one pile game (100, 5) and the sum of two one-pile games: (100, 5) + (50, 3). Discusses Nimm, "an old Chinese game according to Ahrens" and says the solver is unknown. Gives Lasker's Nim ©© one can take any amount from a pile or split it in two ©© and several other variants. Notes that 2nd person + 2nd person is 2nd person while 2nd person + 1st person is 1st person. Gives the idea of equivalent positions. Studies three (and more) person games, assuming the pay-offs are all different. Studies some probabilistic games. JÀ?Àrg Bewersdorff [email of 6 Jun 1999] observes that Lasker's analysis of his Nim got very close to the idea of the Sprague©Grundy number. See: JÀ?Àrg Bewersdorff; GlÀGÀck, Logik und Bluff Mathematik im Spiel ©© Methoden, Ergebnisse und Grenzen; Vieweg, 1998, Section 2.5 LaskerªNim: Gewinn auf verborgenem Weg, pp. 118©124. W. Rivier. Une theorie mathÀ)Àmatique des jeux de combinaisions. Comptes©Rendus du Premier CongrÀ/Às International de RÀ)ÀcrÀ)Àation MathÀ)Àmatique, Bruxelles, 1935. Sphinx, Bruxelles, 1935, pp. 106-113. A revised and simplified version of his 1921 article. He cites and briefly discusses Zermelo, KÀ?Ànig and Euwe. He seems to be classifying games as first player or second player. RenÀ)À de Possel. Sur la ThÀ)Àorie MathÀ)Àmatique des Jeux de Hasard et de RÀ)Àflexion. ActualitÀ)Às Scientifiques et Industrielles 436. Hermann, Paris, 1936. Gives the theory of Nim and also the misÀ/Àre version. Shows that any combinatorial game is a win, loss or draw and describes the nature of first and second person positions. He then goes on to consider games with chance and/or bluffing, based on von Neumann's 1927 paper. R. Sprague. ÀFÀber mathematische Kampfspiele. TÀ=Àhoku Math. J. 41 (1935/36) 438-444. P. M. Grundy. Mathematics and games. Eureka 2 (1939) 6-8. Reprinted, ibid. 27 (1964) 9-11. These two papers develop the Sprague©Grundy Number of a game. D. W. Davies. A theory of chess and noughts and crosses. Penguin Science News 16 (Jun 1950) 40©64. Sketches general ideas of tree structure, Sprague©Grundy number, rational play, etc. H. Steinhaus. Games, an informal talk. AMM 72 (1965) 457-468. Discusses Zermelo and says he wasn't aware of Zermelo in 1925. Gives Mycielski's formulation and proof via de Morgan's laws. Goes into pursuit and infinite games and their relation to the Axiom of Choice. H. Steinhaus. (Proof that a game without ties has a strategy.) In: M. Kac; Hugo Steinhaus ©© a reminiscence and a tribute; AMM 81 (1974) 572-581. Repeats idea of his 1965 talk. à ÃÁÁ4.B.ÁÁPARTICULAR GAMESÄ Ä Ðа¤˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐ ÁÁSee 5.M for Sim and 5.R.5 for Fox and Geese, etc. ÁÁMost of the board games described here are classic and have been extensively described and illustrated in the various standard books on board games, particularly the works of Robert C. Bell, especially his Board and Table Games from Many Civilizations; OUP, vol. I, 1960, vol. II, 1969; combined and revised ed., Dover, 1979 and the older work of Edward G. Falkener; Games Ancient and Oriental and How to Play Them; Longmans, Green, 1892; Dover, 1961. The works by Culin (see 4.A.4, 4.B.5 and 4.B.9) are often useful. Several general works on games are cited in 4.B.1 and 4.B.5 ©© I have read Murray's History of Board Games Other than Chess, but not yet entered the material. Note that many of these works are more concerned with the game than with its history and have a tendency to exaggerate the ages of games by assuming, e.g. that a 3 x 3 board must have been used for Tic©Tac©Toe. I will not try to duplicate the descriptions by Bell, Falkener and others, but will try to outline the earliest history, especially when it is at variance with common belief. The most detailed mathematical analyses are generally in Winning Ways. ÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐ Ã ÃÁÁ4.B.1.ÁÁTIC-TAC-TOE = NOUGHTS AND CROSSESÄ Ä Ðа¤˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐÁÁPopular belief is that the game is ancient and universal ©© e.g. see Brandreth, 1976. However the game appears to have evolved from earlier three-in-a-row games, e.g. Nine Holes or Three Men's Morris, in the early 19C. See also the historical material in 4.B.5. The game is not mentioned in Strutt nor most other 19C books on games, not even in Kate Greenaway's Book of Games (1889), nor in Halliwell's section on slate games (op. cit. in 7.L.1, 1849, pp. 103©104), but there may be an 1875 description in Strutt©Cox of 1903. Babbage refers to it in his unpublished MSS of c1820 as a children's game, but without giving it a name. In 1842, he calls it Tit Tat To and he uses slight variations on this name in his extended studies of the game ©© see below. The OED's earliest references are: 1849 for Tip-tap-toe; 1855 for Tit-tat-toe; 1861 for Oughts and Crosses. However, the first two entries may be referring to some other game ©© e.g. the entries for Tick-tack-toe for 1884 & 1899 are clearly to the game that Gomme calls Tit-tat-toe. Von der Lasa cites a 1838©39 Swedish book for Tripp, Trapp, Trull. Van der Linde (1874, op. cit. in 5.F.1) gives Tik, Tak, Tol as the Dutch name. Using the works of Strutt, Gomme, Strutt©Cox, Fiske, Murray, the OED and some personal communications, I have compiled a separate index of 121 variant names which refer to 5 basic games, with a few variants and a few unknown games. The Murray and Parker material is given first, as it deals generally with the ancient history. Then I list several standard sources and then summarize their content. Other material follows that. Fiske says that van der Linde and von der Lasa (see 5.F.1) mention early appearances of Morris games, but rather briefly and I don't always have that material. ÁÁThe usual # shape board will be so indicated. If one is setting down pieces, then the board is often drawn as a 'crossed square', i.e. a square with its horizontal and vertical midlines drawn, and one plays on the intersections. Fiske 127 says this form is common in Germany, but unknown in England and the US. In addition, the diagonals are often drawn, producing a 'doubly crossed square'. The squares are sometime drawn as circles giving a 'crossed circle' and a 'doubly crossed circle', though it is hard to identify the corners in a crossed circle. The 3 x 3 array of dots sometimes occurs. The standard # pattern is sometimes surrounded by a square producing a '3 x 3 chessboard'. ÁÁFiske 129 says the English play with O and +, while the Swedes play with O and 1. My experience is that English and Americans play with O and X. One English friend said that where she grew up, it was called 'Exeter's Nose' as a deliberate corruption of 'Xs and Os'. ÁÁThe first clear references to the standard game of Noughts and Crosses are Babbage (1820) and the items discussed under Tic©tac©toe below. Further clear references are: Cassell's, Berg, A wrangler ..., Dudeney, White and everything entered below after White. ÁÁMisÀ/Àre version: Gardner (1957); Scotts (1975); ÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐMurray mentions Morris, which he generally calls Merels, many times. Besides the many specific references mentioned below and in 4.B.5, he shows, on p. 614, under Nine Holes and Three Men's Morris, a number of 3 x 3 diagrams. ÁÁÁÁKurna, Egypt, (©14C) ©© a double crossed square and a double crossed circle ©© see Parker below. ÁÁÁÁPtolemaic Egypt (in the BM, no. 14315) ©© a square with # drawn inside. See below where I describe this, from a recent exhibition, as just a # board. ÁÁÁÁCeylon ©© a doubly crossed square ©© see Parker below. ÁÁÁÁRome and Pompeii ©© doubly crossed circles. ÁÁUnder Nine Holes, he says a piece can be moved to any vacant point; under Three Men's Morris, he says a man can only be moved along a marked line to an adjacent point, i.e. horizontally, vertically or along a main diagonal. ÁÁÁÁUnder Nine Holes, he shows the # board for English Noughts and Crosses. He specifically notes that the pieces do not move. His only other mention of this board is for a Swedish game called Tripp, Trapp, Trull, but he does not state that the pieces do not move. He gives no other examples of the # board nor of non-moving pieces. ÁÁÁÁHe also mentions Five (or Six) Men's Morris, of which little is known. On p. 133, he mentions a 3 x 3 "board of nine points used for a game essentially identical with the 'three men's merels', which has existed in China from at least the time of the Liang dynasty (A.D. 502-557). The 'Swei shu' (first half of the 7th c.) gives the names of twenty books on this game." H. Parker. Ancient Ceylon. ??, London, 1909; Asian Educational Services, New Delhi, 1981. Nerenchi keliya, pp. 577-580 & 644. There is a crossed square with small holes at the intersections at the Temple of Kurna, Upper Egypt, -14C. [Rohrbough, loc. cit. in 4.B.5, says this temple was started by Ramses I and completed by Seti in ©1336/©1333, citing J. Royal Asiatic Soc. (1783) 17.] On p. 644, he shows 34 mason's diagrams from Kurna, which include #, # in a circle, crossed square with small holes at the intersections, doubly crossed square, doubly crossed circle. He cites Bell, Arch. Survey of Ceylon, Third Progress Report, p. 5 note, for for a doubly crossed square in Ceylon, c1C, but Noughts and Crosses is not found in the interior of Ceylon. The doubly crossed square was used in 18C Ireland. On pp. 643©665, he discusses appearances of the crossed square and doubly crossed circle as designs or characters and claims they have mystic significance. On p. 662, he lists many early appearances of the # pattern. Murray 440, note 63, includes a reference to Soutendam; Keurboek van Delft; Delft, c1425, f. 78 (or p. 78?); who says games of subtlety are allowed, e.g. ... ticktacken. There is no indication if this may be our game and the OED indicates that such names were used for backgammon back to 1558. The OED doesn't cite: W. Shakespeare; Measure for Measure, c1604. Act I, scene ii, line 180 (or 196): "foolishly lost at a game of ticktack". Later it was more common as Tric©trac. Murray 746 notes a Welsh game Gwyddbwyll mentioned in the Mabinogion (14C). The name is cognate with the Irish Fidchell and may be a Three Men's Morris, but the game was already forgotten by the 15C. ÁÁÁÁSTANDARD SOURCES ON GAMES Joseph Strutt. The Sports and Pastimes of the People of England. (With title starting: Glig-Gamena Angel©ÀNÀeod., or the Sports ...; J. White, London, 1791, 1801, 1810). A new edition, with a copious index, by William Hone. Tegg, London, 1830, 1831, 1833, 1834, 1838, 1841, 1850, 1855, 1875, 1876, 1891. [The 1830 ed. has a preface, omitted in 1833, stating that the 1810 ed. is the same as the 1801 ed. and that Hone has only changed it by adding the Index and incorporating some footnotes into the text.] [Hall, BCB 263©266 are: 1801, 1810, 1830, 1831. Toole Stott 647©656 are: 1791; 1801; 1810; 1828©1830 in 10 monthly parts with Index by Hone; 1830; 1830; 1833; 1838; 1841; 1876, an expanded ed, ed by Hone. Heyl 300©302 gives 1830; 1838; 1850. Toole Stott 653 says the sheets were remaindered to Hone, who omitted the first 8pp and issued it in 1833, 1834, 1838, 1841. I have seen an 1855 ed. C&B list 1801, 1810, 1830, 1903. BMC has 1801, 1810, 1830, 1833, 1834, 1838, 1841, 1875, 1876, 1898.] ÁÁÁÁStrutt©Cox. The Sports and Pastimes of the People of England. By Joseph Strutt. 1801. A new edition, much enlarged and corrected by J. Charles Cox. Methuen, 1903. The Preface sketches Strutt's life and says this is based on the 'original' 1801 in quarto, with separate plates which were often hand coloured, but not consistently, while the 1810 reissue had them all done in a terra-cotta shade. Hone reissued it in octavo in 1830 with the plates replaced by woodcuts in the text and this was reissued in 1837, 1841 and 1875. (From above we see that there were other reissues.) "Mr. Strutt has been left for the most part to speak in his own characteristic fashion .... A few obvious mistakes and rash conclusions have been corrected, ... certain unimportant omissions have been made. ... Nearly a third of the book is new." Reprinted in 1969 and in the 1960s? J. T. Micklethwaite. On the indoor games of school boys in the middle ages. Archaeological Journal 49 (Dec 1892) 319©328. Describes various 3 x 3 boards and games on them, including Nine Holes and "ÃÃtick, tack, toeÄÄ; or ÃÃoughts and crossesÄÄ, which I suppose still survives wherever slate and pencil are used as implements of education", Three Men's Morris and also Nine Men's Morris, Fox and Geese, etc. Alice B. Gomme. The Traditional Games of England, Scotland, and Ireland. 2 vols., David Nutt, London, 1894 & 1898. Reprinted in one vol., Thames & Hudson, London, 1984. Willard Fiske. Chess in Iceland and in Icelandic Literature with Historical Notes on Other Table©Games. The Florentine Typographical Society, Florence, 1905. Esp. pp. 97©156 of the Stray Notes. P. 122 lists a number of works on ancient games. Ðа¤˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐÁÁThese and the OED have several entries on Noughts and Crosses and Tic-tac-toe and many on related games, which are summarised below. Gomme often cites or quotes Strutt. The OED often gives the same quotes as Gomme. Gomme's references are highly abbreviated but full details of the sources can usually be found in the OED. ÁÁà Ã(Nine Men's) MorrisÄ Ä, where Morris is spelled about 30 different ways, e.g. Marl, Merelles, Mill, Miracles, Morals, and Nine Men's may be given as, e.g. Nine-peg, Nine Penny, Nine Pin. Also known as Peg Morris and Shepherd's Mill. Gomme I 80 & 414-419 and Strutt 317-318 (c= Strutt©Cox 256©258 & plate opp. 246, which adds reference to Micklethwaite) are the main entries. See 4.B.5 for material more specifically on this game. ÁÁà ÃNine HolesÄ Ä, also known as Bubble-justice, Bumble-puppy, Crates, and possibly Troll-madam, Troule-in-Madame. Gomme I 413-414 and Strutt 274-275 & 384 (c= Strutt-Cox 222©223 & 304) are the main entries. Twelve Holes is similar [Gomme II 321 gives a quote from 1611]. There seem to be cases where Nine Men's Morris was used in referring to Nine Holes [Gomme I 414-419]. There are two forms of the game: one form has holes in an upright board that one must roll a ball or marble through; the other form has holes in the ground, usually in a 3  x  3 array, that one must roll balls into. Unfortunately, none of the references implies that one has to get three in a row ©© see Every Little Boys Book for a version where this is certainly not the case. There are references going back to 1572 for Crates (but mentioning eleven holes) [Gomme I 81 & II 309] and 1573 [OED] for Nine Holes. Botermans et al.; The World of Games; op. cit. in 4.B.5; 1989; p. 213, shows a 17C engraving by MÀ)Ànian showing Le Jeu de Troumadame as having a board with holes in it, held vertically on a table and one must roll marbles through the holes. They say it is nowadays known as 'bridge'. ÁÁà ÃThree Men's Morris.Ä Ä This is less common, but occurs in several variant spellings corresponding to the variants of Nine Men's Morris, including, e.g. Three-penny Morris, Tremerel. The game is played on a 3  x 3 board and each player has three men. After making three plays each, consisting of setting men on the cells, further play consists of picking up one of your own men and placing it on a vacant cell, with the object of getting three in a row. There are several versions of this game, depending on which cells one may play to, but the descriptions given rarely make this clear. [Gomme I 414-419] quotes from F. Douce; Illustrations of Shakespeare and of Ancient Manners; 1807, i.184. "In the French merelles each party had three counters only, which were to be placed in a line to win the game. It appears to have been the tremerel mentioned in an old fabliau. See Le Grand, Fabliaux et Contes, ii.208. Dr. Hyde thinks the morris, or merrils, was known during the time that the Normans continued in possession of England, and that the name was afterwards corrupted into three men's morals, or nine men's morals." [Hyde. Hist. Nederluddi [sic], p. 202.] In practice, the board is often or usually drawn as a crossed square. If one can move along all winning lines, then it would be natural to draw a doubly crossed square. See under Alfonso MS (1283) in 4.B.5 for versions called marro, tres en raya and riga di tre. Again, much of the material on this game is in 4.B.5. ÁÁà ÃFive-penny Morris.Ä Ä None of the references make it clear, but this seems to be (a form of) Three Men's Morris. Gomme I 122 and the OED [under Morrell] quote: W. Hawkins; Apollo Shroving (a play of 1627), act III, scene iv, pp. 48©49. ÁÁ"..., ÃÃOvidÄÄ hath honour'd my exercises. He describes in verse our boyes play. ÁÁTwise three stones, set in a crossed square where he wins the game ÁÁThat can set his three along in a row, ÁÁAnd that is fippeny morrell I trow." Most of the references (and myself) are perplexed by the reference to five, though the fact that one has at most five moves in Tic-tac-toe might have something to do with it?? Since Three Men's Morris is less well known, some writers have assumed Five-penny Morris was Nine Men's Morris and others have called all such games by the same name. A few lines later, Hawkins has: "I challenge him at all games from blowpoint upward to football, and so on to mumchance, and ticketacke. ... rather than sit out, I will give ÃÃApolloÄÄ three of the nine at Ticketacke, ..." ÁÁà ÃCorsicrownÄ Ä [Gomme I 80] seems to be a version of Three Men's Morris, but using seven of the nine cells, omitting two opposite side cells. Gomme quotes from J. Mactaggart; The Scottish Gallovidian Encyclopedia; (1871 or possibly 1824?): "each has three men .... there are seven points for these men to move about on, six on the edges of the square and one at the centre." ÁÁà ÃTic-tac-toe.Ä Ä The earliest clearly described versions are given in Babbage (with no name given), c1820, and Gomme I 311, under Kit-cat-cannio, where she quotes from: Edward Moor; Suffolk Words and Phrases; 1823 (This word does not occur in the OED). Gomme also gives entries for Noughts and Crosses [I 420-421] and Tip-tap-toe [II 295-296] with variants Tick-tack-toe and Tit-tat-toe. In 1842©1865, Babbage uses Tit Tat To and slight variants. Under Tip-tap-toe, Gomme says the players make squares and crosses and that a tie game is a score for Old Nick or Old Tom. (When I was young, we called it Cat's Game, and this is an old Scottish term [James T. R. Ritchie; à ÃThe Singing Street Scottish Children's Games, Rhymes and SayingsÄ Ä; (O&B, 1964); Mercat Press, Edinburgh, 2000, p. 61].) She quotes regional glossaries for Tip-tap-toe (1877), Tit-tat-toe (1866 & 1888), Tick-tack-toe (1892). The OED entry for Oughts and Crosses seems to be this game and gives an 1861 quote. Von der Lasa cites a 1838©39 Swedish book for Tripp, Trapp, Trull. Van der Linde (1874, op. cit. in 5.F.1) gives Tik, Tak, Tol as the Dutch name. ÁÁà ÃTit-tat-toeÄ Ä [Gomme II 296-298]. This is a game using a slate marked with a circle and numbered sectors. The player closes his eyes and taps three times with a pencil and tries to land on a good sector. Gomme gives the verse: ÁÁÁÁTit, tat, toe, my first go, ÁÁÁÁThree jolly butcher boys all in a row ÁÁÁÁStick one up, stick one down, ÁÁÁÁStick one in the old man's ground. But cf Games and Sports for Young Boys, 1859, below. ÁÁThe OED entries under Tick-tack, Tip-tap and Tit give a number of variant spellings and several quotations, which are often clearly to this game, but are sometimes unclear. Also some forms seem to refer to backgammon. ÁÁIn her 'Memoir on the study of children's games' [Gomme II 472-473], Gomme gives a somewhat Victorian explanation of the origin of Old Nick as the winner of a tie game as stemming from "the primitive custom of assigning a certain proportion of the crops or pieces of land to the devil, or other earth spirit." ÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐ Franco Agostini & Nicola Alberto De Carlo. Intelligence Games. (As: Giochi della Intelligenza; Mondadori, Milan, 1985.) Simon & Schuster, NY, 1987. P. 81 says examples of boards were discovered in the lowest level of Troy and in the Bronze Age tombs in Co. Wicklow, Ireland. Their description is a bit vague but indicates that the Italian version of Tic©tac©toe is actually Three Men's Morris. Anonymous. Play the game. Guardian Education section (21 Sep 1993) 18©19. Shows a stone board with the # incised on it 'from Bet Shamesh, Israel, 2000 BC'. This might be the same as the first board below?? A small exhibition of board games organized by Irving Finkel at the British Museum, 1991, displayed the following. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐStone slab with the usual # Tick©Tac©Toe board incised on it, but really a 4 x 3 board. With nine stone men. From Giza, >©850. BM items EA 14315 & 14309, donated by W. M. Flinders Petrie. Now on display in Room 63, Case C. Stone Nine Holes board from the Temple of Artemis, Ephesus, 2C©4C. Item BM GR 1873.5.5.150. This is a 3 x 3 array of depressions. Now on display in Room 69, Case 9. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐRobbie Bell & Michael Cornelius. Board Games Round the World. CUP, 1988. P. 6 states that the crossed square board has been found at Kurna (c©1400) and at the Ptolemaic temple at Komombo (c©300). They state that Three Men's Morris is the game mentioned by Ovid in Ars Amatoria. They say that it was known to the Chinese at the time of Confucius (c©500) under the name of Yih, but is now known as Luk tsut k'i. They also say the game is also known as Nine Holes ©© which seems wrong to me. The Spanish Treatise on Chess©Play written by order of King Alfonso the Sage in the year 1283. [= Libro de Acedrex, Dados e Tablas of Alfonso El Sabio, generally known as the Alfonso MS.] MS in Royal Library of the Escorial (j.T.6. fol). Complete reproduction in 194 Phototypic Plates. 2 vols., Karl W. Hirsemann, Leipzig, 1913. (There was also an edition by Arnald Steiger, Geneva, 1941.) See 4.B.5 for more details of this work. Vol. 2, f. 93v, p. CLXXXVI, shows a doubly crossed square board. ??NX ©© need to study text. Pieter Bruegel (the Elder). Children's Games. Painting dated 1560 at the Kunsthistorisches Museum, Vienna. In the right background, children are playing a game involving throwing balls into holes in the ground, but the holes appear to be in a straight line. Anonymous. Games of the 16th Century. 1950. Op. cit. in 4.A.3. P. 134 describes nineªholes, quoting an unknown poet of 1611: "To play at loggats, Nine©holes, or Tenªpinnes". The author doesn't specify what positions the balls are to be rolled into. P. 152 describes Troll©my©dames or Troule©in©madame: "they may have in the end of a bench eleven holes made, into which to troll pummets, or bowls of lead, ...." William Wordsworth. The Prelude, Book 1. Completed 1805, published 1850. Lines 509-513. ÁÁÁÁAt evening, when with pencil, and smooth slate ÁÁÁÁIn square divisions parcelled out and all ÁÁÁÁWith crosses and with cyphers scribbled o'er, ÁÁÁÁWe schemed and puzzled, head opposed to head ÁÁÁÁIn strife too humble to be named in verse. ÁÁIt is not clear if this is referring to Noughts and Crosses. Charles Babbage. The Philosophy of Analysis ©© unpublished collection of MSS in the BM as Add. MS 37202, c1820. ??NX. F. 4r is part of the Table of Contents. It shows Noughts and Crosses games played on the # board and on a 4 x 4 board adjacent to entry 4: The Mill. Ff. 124©146 are: Essay 10 ©© Of questions requiring the invention of new modes of analysis. On f. 128.r, he refers to a game in which "the relative positions of three of the marks is the object of inquiry." Though the reference is incomplete, a Noughts and Crosses game is drawn on the facing page, f. 127.v. Ff. 134©144 are: Essay 10 Part 5. At the top of f. 134.r, he has added a note: "This is probably my earliest Note on Games of Skill. I do not recollect the date. 3 March 1865". The Essay begins: "Amongst the simplest of those games requiring any degree of skill which amuse our early years is one which is played at in the following manner." He then describes the game in detail and makes some simple analysis, but he never uses a name for it. Charles Babbage. Notebooks ©© unpublished collection of MSS in the BM as Add. MS 37205. ??NX. On f. 304, he starts on analysis of games. Ff. 310©383 are almost entirely devoted to Tit©Tat©To, with some general discussions. Most of this material comprises a few sheets of working, carefully dated, sometimes amended and with the date of the amendment. A number of sheets describe parts of the automaton that he was planning to build which would play the game, but no such machine was built until 1949. The sheets are not always in strict chronological order. ÁÁÁÁF. 310.r is the first discussion of the game, called Tit Tat To, dated 17 Sep 1842. On F. 312.r, 20 Sep 1843, he says he has "Reduced the 3024 cases D to 199 which include many Duplicates by Symmetry." F. 321.r, 10 Sep 1860, is the beginning of a summary of his work on games of skill in general. He refers to Tit©tat©too. F. 322.r continues and he says: "I have found no game of skill more simple that that which children often play and which they call Tit-tat©to." F. 324©333, Oct 1844, studies "General laws for all games of Skill between two players" and draws flow charts showing the basic recursive analysis of a game tree (ff. 325.v & 325.r). On f. 332, he counts the number of positions as 9! + 8! + ... + 1!  = 409,113. F. 333 has an idea of the tree structure of a game. On ff. 337©338, 8 Sep 1848, he has Tit©tat too. On ff. 347.r©347.v, he suggests Nine Men's Morris boards in triangular and pentagonal shapes and does various counting on the different shapes. On ff. 348©349, 26 Oct 1859, he uses Tit©Tat©To. John M. Dubbey. The Mathematical Work of Charles Babbage. CUP, 1978, pp. 96-97 & 125-130. He discusses the above Babbage material. On p. 127, Dubbey has: "After a surprisingly lengthy explanation of the rules, he attempts a mathematical formulation. The basic problem is one that appears not to have been previously considered in the history of mathematics." Babbage represents the game using roots of unity. Dubbey, on p. 129, says: "This analysis ... must count as the first recorded stochastic process in the history of mathematics." However, it is really a deterministic two©person game. Games and Sports for Young Boys. Routledge, nd [1859 © BLC]. P. 70, under Rhymes and Calls: "In the game of Tit©tat©toe, which is played by very young boys with slate and pencil, this jingle is used:©© ÁÁÁÁTit, tat, toe, my first go: ÁÁÁÁThree jolly butcher boys all in a row; ÁÁÁÁStick one up, stick one down. ÁÁÁÁStick one on the old man's crown." Baron Tassilo von Heydebrand und von der Lasa. Ueber die griechischen und rÀ?Àmischen Spiele, welche einige ÀÀhnlichkeit mit dem Schach hatten. Deutsche Schachzeitung (1863) 162©172, 198©199, 225©234, 257©264. ??NYS ©© described on Fiske 121©122 & 137, who says van der Linde I 40©47 copies much of it. Von der Lasa asserts that the Parva Tabella of Ovid is Kleine MÀGÀhle (Three Men's Morris). He says the game is called Tripp, Trapp, Trull in the Swedish book Hand©Bibliothek fÀ?Àr SÀÀllkapsnÀ?Àjen, of 1839, vol. II, p. 65 (or 57) ©© ??NYS. Van der Linde says that the Dutch name is Tik, Tak, Tol. Fiske notes that both of these refer to Noughts and Crosses, but it is unclear if von der Lasa or van der Linde recognised the difference between Three Men's Morris and Noughts and Crosses. C. Babbage. Passages from the Life of a Philosopher. 1864. Chapter XXXIV ©© section on Games of Skill, pp. 465-471. (= pp. 152-156 in: Charles Babbage and His Calculating Engines, Dover, 1961.) Partial analysis. He calls it tit-tat-to. The Play Room: or, In©door Games for Boys and Girls. Dick & Fitzgerald(?), 1866. [Reprinted as: How to Amuse an Evening Party. Dick & Fitzgerald, NY, 1869.] ??NX ª© the 1869 was seen at Shortz's. P. 22: Tit©tat©to. Uses O and +. "This is a game that small boys enjoy, and some big ones who won't own it." Anonymous. Every Little Boy's Book A Complete CyclopÀ%Àdia of in and outdoor games with and without toys, domestic pets, conjuring, shows, riddles, etc. With two hundred and fifty illustrations. Routledge, London, nd. HPL gives c1850, but the text is clearly derived from Every Boy's Book, whose first edition was 1856. But the main part of the text considered here is not in the 1856 ed of Every Boy's Book (with J. G. Wood as unnamed editor), but is in the 8th ed of 1868 (published for Christmas 1867), which was the first seriously revised edition, with Edmund Routledge as editor. So this may be c1868. This is the first published use of the term Noughts and Crosses found so far ©© the OED's 1861 quote is to Oughts and Crosses.. ÁÁÁÁPp. 46©47: Slate games: Noughts and crosses. "This is a capital game, and one which every school©boy truly enjoys." Though the example shown is a draw, there is no mention of the fact that the game should always be a tie. ÁÁÁÁPp. 85©86: Nine©holes. This has nine holes in a row and each player has a hole. The ball is rolled to them and the person in whose hole it lands must run and pick up the ball and try to hit one of the others who are running away. So this has nothing to do with our games or other forms of Nine Holes. ÁÁÁÁP. 106: Nine©holes or Bridge©board. This has nine holes in an upright board and the object is get one's marbles through the holes. (This material is in the 1856 ed. of Every Boy's Book.) Correspondent to Notes and Queries (1875) ??NYS ©© quoted by Strutt©Cox 257. Describes a game called Three Mans' Marriage [sic] in Derbyshire which seems to be Noughts and Crosses played on a crossed square board. Pieces are not described as moving, but in the next description of a Nine Men's Morris, they are specifically described as moving. However, the use of a crossed square board may indicate that diagonals were not considered. Cassell's. 1881. Slate Games: Noughts and Crosses, or Tit-Tat-To, p. 84, with cross reference under Tit©Tat©To, p. 87. = Manson, 1911, pp. 202©203 & 208. Albert Norman. Ungdomens Bok [Book for Youth] (in Swedish). 2nd ed., Stockholm, 1883. Vol. I, p. 162++. ??NYS ©© quoted and described in Fiske 134©136. Description of Tripp, Trapp, Trull, with winning cry: "Tripp, trapp, trull, min qvarn ÀÀr full." (Qvarn = mill.) Lucas. RM2, 1883. Pp. 73©99. Analysis of Three Men's Morris, on a board with the main diagonals drawn, with moves of only one square along a winning line. He shows this is a first person game. If the first player is not permitted to play in the centre, then it is a tie game. No mention of Tic©Tac©Toe. Albert Ellery Berg, ed. The Universal Self-Instructor. Thomas Kelly, NY, 1883. Tit-tat-to, p. 379. Brief description. Mark Twain. The Adventures of Huckleberry Finn. 1884. Chap. XXXIV, about half©way through. "It's as simple as tit©tat©toe, three©in©a©row, ..., Huck Finn." "A wrangler and late master at Harrow school." The science of naughts and crosses. Boy's Own Paper 10: (No. 498) (28 Jul 1888) 702-703; (No. 499) (4 Aug 1888) 717; (No. 500) (11 Aug 1888) 735; (No. 501) (18 Aug 1888) 743. Exhaustive analysis, including odds of second player making a correct response to each opening. For first move in: middle, side, corner, the odds of a correct response are: 1/2, 1/2, 1/8. He implies that the analysis is not widely known. "Tom Wilson". Illustred Spelbok (in Swedish). Nd [late 1880s??]. ??NYS ©© described by Fiske 136©137. This gives Tripp, Trapp, Trull as a Three Men's Morris game on the crossed square, with moves "according to one way of playing, to whatever points they please, but according to another, only to the nearest point along the lines on which the pieces stand. This last method is always employed when the board has, in addition to the right lines, or lines joining the middles of the exterior lines, also diagonals connecting the angles". He then describes a drawn version using the # board and 0 and + (or 1 and 2 in the North) which seems to be genuinely Noughts and Crosses. Fiske says the book seems to be based on an early edition of the EncyclopÀ)Àdie des Jeux or a similar book, so it is uncertain how much the above represents the current Swedish game. Fiske was unable to determine the author's real name, though he was still living in Stockholm at the time. Il Libro del Giuochi. Florence, 1894. ??NYS ©© described in Fiske, pp. 109©110. Gives doubly crossed square board and mentions a Three Men's Morris game. T. de Moulidars. Grande EncyclopÀ)Àdie des Jeux. Montgredien or Librairie Illustree, Paris, nd. ??NYS ©© Fiske 115 (in 1905) says it appeared 'not very long ago' and that Gelli seems to be based on it. Fiske quotes the clear description of Three Men's Morris as Marelle Simple, using a doubly crossed square, saying that pieces move to adjoining cells, following a line, and that the first player should win if he plays in the centre. Fiske notes that Noughts and Crosses is not mentioned. J. Gelli. Come Posso Divertirmi? Milan, 1900. ??NYS ©© described in Fiske 107. Fiske quotes the description of Three Men's Morris as Mulinello Semplice, essentially a translation from Moulidars. Dudeney. CP. 1907. Prob. 109: Noughts and crosses, pp. 156 & 248. (c= MP, prob. 202: Noughts and crosses, pp. 89 & 175-176. = 536, prob. 471: Tic tac toe, pp. 185 & 390-392. Asserts the game is a tie, but gives only a sketchy analysis. MP gives a reasonably exhaustive analysis. Looks at Ovid's game. A. C. White. Tit-tat-toe. British Chess Magazine (Jul 1919) 217-220. Attempt at a complete analysis, but has a mistake. See Gardner, SA (Mar 1957) = 1st Book, chap. 4. D'Arcy Wentworth Thompson. Science and the Classics. OUP, 1940. Section V Games and Playthings, pp. 148©165. On p. 160, he quotes Ovid and says it is Noughts and Crosses, or in Ireland, Tip©top©castle. The Home Book of Quizzes, Games and Jokes. Blue Ribbon Books, NY, 1941. This is excerpted from several books ©© this material is most likely taken from: Clement Wood & Gloria Goddard; Complete Book of Games; same publisher, nd [late 1930s]. P. 150: Tit©tat©toe, noughts and crosses. Brief description of the game on the # board. "To win requires great ingenuity." G. E. Felton & R. H. Macmillan. Noughts and Crosses. Eureka 11 (Jan 1949) 5©9. Mentions Dudeney's work on the 3 x 3 board and asks for generalizations. Mentions pegotty = go©bang. Then studies the 4 x 4 x 4 game ©© see 4.B.1.a. Adds some remarks on pegotty, citing Falkener, Lucas and Tarry. Stanley Byard. Robots which play games. Penguin Science News 16 (Jun 1950) 65©77. On p. 73, he says D. W. Davies, of the National Physical Laboratory, has built, and exhibited to the Royal Society in May 1949, an electro©mechanical noughts and crosses machine. A photo of the machine is plate 16. He also mentions Babbage's interest in such a machine and an 1874 paper to the US National Academy by a Dr. Rogers ©© ??NYS. P. C. Parks. Building a noughts and crosses machine. Eureka 14 (Oct 1951) 15©17. Cites Babbage, Rogers, Davies, Byard. Parks built a simple machine with wire and tin cans in 1950 at a cost of about À À6. Says G. Eastell of Thetford, Norfolk, built a machine using sixty valves for the Festival of Britain. Gardner. Ticktacktoe. SA (Mar 1957) c= 1st Book, chap. 4. Quotes Wordsworth, discusses Three Men's Morris (citing Ovid) and its variants (including versions on 4 x 4 and 5 x 5 boards), the misÀ/Àre version (the person who makes three in a row loses), three and n dimensional forms (giving L. Moser's result on the number of winning lines on a kÃÃnÄÄ board), go©moku, Babbage's proposed machine, A. C. White's article. Addendum mentions the Opies' assertion that the name comes from the rhyme starting "Tit, tat, toe, My first go,". C. L. Stong. The Amateur Scientist. Ill. by Roger Hayward. S&S, 1960. A ticktacktoe machine, pp. 384©385. Noel Elliott gives a brief description of a relay logic machine to play the game. Donald Michie. Trial and error. Penguin Science Survey 2 (1961) 129©145. ??NYS. Describes his matchbox and bead learning machine, MENACE (Matchbox educable noughts and crosses engine), for the game. Gardner. A matchbox game©learning machine. SA (Mar 1962) c= Unexpected, chap. 8. Describes Michie's MENACE. Says it used 300 matchboxes. Gardner adapts it to Hexapawn, which is much simpler, requiring only 24 matchboxes. Discusses other games playable by 'computers'. Addendum discusses results sent in by readers including other games. Barnard. 50 Observer Brain©Twisters. 1962. Prob. 34: Noughts and crosses, pp. 39-40, 64 & 93-94. Asserts there are 1884 final winning positions. He doesn't consider equivalence by symmetry and he allows either player to start. Donald Michie & R. A. Chambers. Boxes: an experiment in adaptive control. Machine Intelligence 2 (1968) 136©152. Discusses MENACE, with photo of the pile of boxes. Says there are 288 boxes, but doesn't explain exactly how he found them. Chambers has implemented MENACE as a general game©learning computer program using adaptive control techniques designed by Michie. Results for various games are given. S. Sivasankaranarayana Pillai. A pastime common among South Indian school children. In: P. K. Srinivasan, ed.; Ramanujan Memorial Volumes: 1: Ramanujan ©© Letters and Reminiscences; 2: Ramanujan ©© An Inspiration; Muthialpet High School, Number Friends Society, Old Boys' Committee, Madras, 1968. Vol. 2, pp. 81©85. [Taken from Mathematics Student, but no date or details given ©© ??] Shows ordinary tic©tac©toe is a draw and considers trying to get t in a row on an n x n board. Shows that n = t ÀÀ 3 is a draw and that if t ÀÀ n + 1 © ÀÀ(n/6), then the game is a draw. L. A. Graham. The Surprise Attack in Mathematical Problems. Dover, 1968. Tic©tac©toe for gamblers, prob. 8, pp. 19©22. Proposed by F. E. Clark, solutions by Robert A. Harrington & Robert E. Corby. Find the probability of the first player winning if the game is played at random. Two detailed analyses shows that the probabilities for first player, second player, tie are (737, 363, 160)/1260. [Henry] Joseph & Lenore Scott. Quiz Bizz. Puzzles for Everyone ©© Vol. 6. Ace Books (Charter Communications), NY, 1975. P. 143: Ha©ho©ha. MisÀ/Àre version of noughts and crosses proposed. No discussion. Gyles Brandreth. Pencil and Paper Games and Puzzles. Carousel, 1976. Noughts and Crosses, pp. 11©12. Asserts "It's been played all around the world for hundreds, if not thousands, of years ...." I've included it as a typical example of popular belief about the game. = Pencil & Paper Puzzle Games; Watermill Press, Mahwah, New Jersey, 1989, Tic©Tac©Toe, pp. 11-12. Winning Ways. 1982. Pp. 667©680. Complete and careful analysis, including various uncommon traps. Several equivalent games. Discusses extensions of board size and dimension. Sheila Anne Barry. The World's Best Travel Games. Sterling, NY, 1987. Tic©tac©toe squared, pp. 88©89. Get 3 in a row on the 4 x 4 board. Also considers Tic©tac©toe©toe ©© get 4 in a row on 5 x 5 board. George Markovsky. Numerical tic©tac©toe ©© I. JRM 22:2 (1990) 114©123. Describes and studies two versions where the moves are numbered 1, 2, .... One is due to Ron Graham, the other to P. H. Nygaard and Markowsky sketches the histories. Ira Rosenholtz. Solving some variations on a variant of tic©tac©toe using invariant subsets. JRM 25:2 (1993) 128©135. The basic variant is to avoid making three in a row on a 4 x 4 board. By playing symmetrically, the second player avoids losing and 252 of the 256 centrally symmetric positions give a win for the second player. Extends analysis to 2n x 2n, 5 x 5, 4 x 4 x 4, etc. Bernhard Wiezorke. Sliding caution. CFF 32 (Aug 1993) 24©25 & 33 (Feb 1994) 32. This describes a sliding piece puzzle on the doubly crossed square board ©© see under 5.A. See: Yuri I. Averbakh; Board games and real events; 1995; in 5.R.5, for a possible connection. ÁÁà Ã4.B.1.aÁÁÁÁIN HIGHER DIMENSIONSÄ Ä C. Planck. Four-fold magics. Part 2 of chap. XIV, pp. 363-375, of W. S. Andrews, et al.; Magic Squares and Cubes; 2nd ed., Open Court, 1917; Dover, 1960. On p. 370, he notes that the number of m-dimensional directions through a cell of the n-dimensional board is the m-th term of the binomial expansion of ÀÀ(1+2)ÃÃnÄÄ. Maurice Wilkes says he played 3©D noughts and crosses at Cambridge in the late 1930s, but the game was to get the most lines on a 3 x 3 x 3 board. I recall seeing a commercial version, called Plato?, of this in 1970. Cedric Smith says he played 3©D and 4©D versions at Cambridge in the early 1940s. Arthur Stone (letter to me of 9 Aug 1985) says '3 and 4 dimensional forms of tic©tac©toe produced by Brooks, Smith, Tutte and myself', but it's not quite clear if they invented these. Tutte became expert on the 4ÃÃ3ÄÄ board and thought it was a first person game. They only played the 5ÃÃ4ÄÄ game once © it took a long time. Funkenbusch & Eagle, ÃÃNational Mathematics Mag.ÄÄ (1944) ??NYR. G. E. Felton & R. H. Macmillan. Noughts and crosses. Eureka 11 (1949) 5-9. They say they first met the 4 x 4 x 4 game at Cambridge in 1940 and they give some analysis of it, with tactics and problems. William Funkenbusch & Edwin Eagle. Hyper-spacial tit-tat-toe or tit-tat-toe in four dimensions. National Mathematics Magazine 19:3 (Dec 1944) 119-122. ??NYR A. L. Rubinoff, proposer; L. Moser, solver. Problem E773 ©© Noughts and crosses. AMM 54 (1947) 281 & 55 (1948) 99. Number of winning lines on a kÃÃnÄÄ board is {(k+2)ÃÃnÄÄ - kÃÃnÄÄ}/2. Putting k = 1 gives Planck's result. L. Buxton. Four dimensions for the fourth form. MG 26 (1964) 38-39. 3 x 3 x 3 and 3 x 3 x 3 x 3 games are obviously first person, but he proposes playing for most lines and with the centre blocked on the 3 x 3 x 3 x 3 board. Suggests 3ÃÃnÄÄ and 4 x 4 x 4 games. Anon. Puzzle page: Noughts and crosses. MTg 33 (1965) 35. Says practice shows that the 4 x 4 x 4 game is a draw. [I only ever had one drawn game!] Conjectures nÃÃnÄÄ is first player and (n+1)ÃÃnÄÄ is a draw. Roland Silver. The group of automorphisms of the game of 3-dimensional ticktacktoe. AMM 74 (1967) 247-254. Finds the group of permutations of cells that preserve winning lines is generated by the rigid motions of the cube and certain 'eviscerations'. [It is believed that this is true for the kÃÃnÄÄ board, but I don't know of a simple proof.] Ross Honsberger. Mathematical Morsels. MAA, 1978. Prob. 13: X's and O's, p. 26. Obtains L. Moser's result. Kathleen Ollerenshaw. Presidential Address: The magic of mathematics. Bull. Inst. Math. Appl. 15:1 (Jan 1979) 2©12. P. 6 discusses my rediscovery of L. Moser's 1948 result. Paul Taylor. Counting lines and planes in generalised noughts and crosses. MG 63 (No. 424) (Jun 1979) 77©82. Determines the number pÃÃrÄÄ(k) of r©sections of a kÃÃnÄÄ board by means of a recurrence pÃÃrÄÄ(k) = [pÃÃr©1ÄÄ(k+2) © pÃÃr©1ÄÄ(k)]/2r which generalises L. Moser's 1948 result. He then gets an explicit sum for it. Studies some other relationships. This work was done while he was a sixth form student. Oren Patashnik. Qubic: 4 x 4 x 4 tic-tac-toe. MM 53 (1980) 202-216. Computer assisted proof that 4 x 4 x 4 game is a first player win. Winning Ways. 1982. Pp. 673©679, esp. 678©679. Discusses getting k in a row on a n x n board. Discusses 4ÃÃ3ÄÄ game (Tic©Toc©Tac©Toe) and kÃÃnÄÄ game. Victor Serebriakoff. A Mensa Puzzle Book. Muller, London, 1982. (Later combined with A Second Mensa Puzzle Book, 1985, Muller, London, as: The Mensa Puzzle Book, Treasure Press, London, 1991.) Chapter 7: Conceptual conflict in multi©dimensional space, pp. 80©94 (1991: 98©112) & answers on pp. 99, 100, 106 & 131 (1991: 115, 116, 122 & 147). He considers various higher dimensional noughts and crosses on the 3ÃÃ3ÄÄ, 3ÃÃ4ÄÄ and 3ÃÃ5ÄÄ boards. He finds that there are 49 winning lines on the 3ÃÃ3ÄÄ and he finds how to determine the number of d©facets on an n©cube as the coefficients in the expansion of (2x + 1)ÃÃnÄÄ. He also considers games where one has to complete a 3 x 3 plane to win and gives a problem: OXO three hypercube planes, p. 91 (1991: 109) & Answer 29, p. 106 (1991: 122) which asks for the number of planes in the hypercube 3ÃÃ4ÄÄ. The answer says there are 123 of them, but in 1985 I found 154 and the general formula for the number of d©sections of a kÃÃnÄÄ board. When I wrote to Serebriakoff, he responded that he could not follow the mathematics and that "I arrived at the figures ... from a simple formula published in one of Art [sic] Gardner's books which checked out as far as I could take it. Several other mathematicians have looked through it and not disagreed." I wrote for a reference to Gardner but never had a response. I presented my work to the British Mathematical Colloquium at Cambridge on 2 Apr 1985 and discovered that the results were known ©© I had found the explicit sum given by Taylor above, but not the recurrence. à ÃÁÁ4.B.2.ÁÁHEXÄ Ä David Fielker sent some pages from a Danish book on games, but the TP is not present in his copies, so we don't have details. This says that Hein introduced the game in a lecture to students at the Institute for Theoretical Physics (now the Niels Bohr Institute) in Copenhagen in 1942. After its appearance in Politiken, specially printed pads for playing the game were sold, and a game board was marketed in the US as Hex in 1952. Piet Hein. Article or column in Politiken (Copenhagen) (26 Dec 1942). ??NYR, but the diagrams show a board of hexagons. Gardner (1957) and others have related that the game was independently invented by John Nash at Princeton in 1948©1949. Gardner had considerable correspondence after his article which I have examined. The key point is that one of Niels Bohr's sons, who had known the game in Copenhagen, was a visitor at the Institute for Advanced Study at the time and showed it to friends. I concluded that it was likely that some idea of the game had permeated to Nash who had forgotten this and later recalled and extensively developed the idea, thinking it was new to him. I met Harold Kuhn in 1998, who was a student with Nash at the time and he has no doubt that Nash invented the idea. In particular, Nash started with the triangular lattice, i.e. the dual of Hein's board, for some time before realising the convenience of the hexagonal lattice. Nash came to Princeton as a graduate student in autumn 1948 and had invented the game by the spring of 1949. Kuhn says he observed Nash developing the ideas and recognising the connections with the Jordan Curve Theorem, etc. Kuhn also says that there was not much connection between students at Princeton and at the Institute and relates that von Neumann saw the game at Princeton and asked what it was, indicating that it was not well known at the Institute. In view of this, it seems most likely that Nash's invention was independent, but I know from my own experience that it can be difficult to remember the sources of one's ideas ©© a casual remark about a hexagonal game could have re©emerged weeks or months later when Nash was studying games, as the idea of looking at hexagonal boards in some form, from which the game would be re©invented. Sylvester was notorious for publishing ideas which he had actually refereed or edited some years earlier, but had completely forgotten the earlier sources. In situations like Hex, we will never know exactly what happened ©© even if we were present at the time, it is difficult to know what is going on in the mind of the protagonist and the protagonist himself may not know what subconscious connections his mind is making. Even if we could discover that Nash had been told something about a hexagonal game, we cannot tell how his mind dealt with this information and we cannot assume this was what inspired his work. In other words, even a time machine will not settle such historical questions ©© we need something that displays the conscious and the unconscious workings of a person's mind. Parker Brothers. Literature on Hex, 1952. ??NYS or NYR. Claude E. Shannon. Computers and automata. Proc. Institute of Radio Engineers 41 (Oct 1953) 1234-1241. Describes his Hex machine on p. 1237. M. Gardner. The game of Hex. SA (Jul 1957) = 1st Book, chap. 8. Description of Shannon's 8 by 7 'Hoax' machine, pp. 81-82, and its second person strategy, p. 79. Anatole Beck, Michael N. Bleicher & Donald W. Crowe. Excursions into Mathematics. Worth Publishers, NY, 1969. Chap. 5: Games (by Beck), Section 3: The game of Hex, pp. 327©339 (with photo of Hein on p. 328). Says it has been attributed to Hein and Nash. At Yale in 1952, they played on a 14 x 14 board. Shows it is a first player win, invoking the Jordan Curve Theorem David Gale. The game of Hex and the Brouwer fixed©point theorem. AMM 86:10 (Dec 1979) 818©827. Shows that the non©existence of ties (Hex Theorem) is equivalent to the Brouwer Fixed©Point Theorem in two and in n dimensions. Says the use of the Jordan Curve Theorem is unnecessary. Winning Ways. 1982. Pp. 679©680 sketches the game and the strategy stealing argument which is attributed to Nash. C. E. Shannon. Photo of his Hoax machine sent to me in 1983. Cameron Browne. Hex Strategy: Making the Right Connections. A. K. Peters, Natick, Massachusetts, 2000. à ÃÁÁ4.B.3.ÁÁDOTS AND BOXESÄ Ä Lucas. Le jeu de l'À(Àcole Polytechnique. RM2, 1883, pp. 90-91. He gives a brief description, starting: "Depuis quelques annÀ)Àes, les À)ÀlÀ/Àves de l'À(Àcole Polytechnique ont imaginÀ)À un nouveaux jeu de combinaison assez original." He clearly describes drawing the edges of the game board and that the completer of a box gets to go again. He concludes: "Ce jeu nous a paru assez curieux pour en donner ici la description; mais, jusqu'a prÀ)Àsent, nous ne connaissons pas encore d'observations ni de remarques assez importantes pour en dire davantage." Lucas. Nouveaux jeux scientifiques de M. À(Àdouard Lucas. La Nature 17 (1889) 301-303. Clearly describes a game version of La Pipopipette on p. 302, picture on p. 301, "... un nouveau jeu ... dÀ)ÀdiÀ)À aux À)ÀlÀ/Àves de l'À)Àcole Polytechnique." This is dots and boxes with the outer edges already drawn in. Lucas. L'ArithmÀ)Àtique Amusante. 1895. Note III: Les jeux scientifiques de Lucas, pp. 203-209 ©© includes his booklet: La Pipopipette, Nouveau jeu de combinaisons, DÀ)ÀdiÀ)À aux À)ÀlÀ/Àves de l'À(Àcole Polytechnique, Par un Antique de la promotion de 1861, (1889), on pp. 204-208. On p. 207, he says the game was devised by several of his former pupils at the À(Àcole Polytechnique. On p. 37, he remarks that "ÃÃPipoÄÄ est la dÀ)Àsignation abrÀ)ÀgÀ)Àe de Polytechnique, par les À)ÀlÀ/Àves de l'X, ...." Robert Marquard & Georg Frieckert. German Patent 108,830 ©© Gesellschaftsspiel. Patented: 15 Jun 1899. 1p + 1p diagrams. 8 x 8 array of boxes on a board with slots for inserting edges. No indication that the player who completes a box gets to play again. They have some squares with values but also allow all squares to have equal value. C. Ganse. The dot game. Ladies' Home Journal (Jun 1903) 41. Describes the game and states that one who makes a box gets to go again. Loyd. The boxer's puzzle. Cyclopedia, 1914, pp. 104 & 352. = MPSL1, prob. 91, pp. 88-89 & 152-153. c= SLAHP: Oriental tit-tat-toe, pp. 28 & 92-93. Loyd doesn't start with the boundaries drawn. He asserts it is 'from the East'. Ahrens. A&N. 1918. Chap. XIV: Pipopipette, pp. 147-155, describes it in more detail than Lucas does. He says the game appeared recently. Blyth. Match©Stick Magic. 1921. Boxes, pp. 84©85. "The above game is familiar to most boys and girls ...." No indication that the completer of a box gets to play again. Heinrich Voggenreiter. Deutsches Spielbuch Sechster Teil: Heimspiele. Ludwig Voggenreiter, Potsdam, 1930. Pp. 84©85: Die KÀÀsekiste. Describes a version for two or more players. The first player must start at a corner and players must always connect to previously drawn lines. A player who completes a box gets to play again. Meyer. Big Fun Book. 1940. Boxes, p. 661. Brief description, somewhat vaguely stating that a player who completes a box can play again. The Home Book of Quizzes, Games and Jokes. Op. cit. in 4.B.1, 1941. P. 151: Dots and squares. Clearly says the completer gets to play again. "The game calls for great ingenuity." "Zodiastar". Fun with Matches and Match Boxes. (Cover says: Match Tricks From the 1880s to the 1940s.) Universal Publications, London, nd [late 1940s?]. The game of boxes, pp. 48©49. Starts by laying out four matches in a square and players put down matches which must touch the previous matches. Completing a box gives another play. No indication that matches must be on lattice lines, but perhaps this is intended. Readers' Research Department. RMM 2 (Apr 1961) 38-41, 3 (Jun 1961) 51-52, 4 (Aug 1961) 52-55. On pp. 40-41 of No. 2, it says that Martin Gardner suggests seeking the best strategy. Editor notes there are two versions of the rules ©© where the one who makes a box gets an extra turn, and where he doesn't ©© and that the game can be played on other arrays. On p. 51 of No. 3, there is a symmetry analysis of the no-extra-turn game on a board with an odd number of squares. On pp. 52-54 of No. 4, there is some analysis of the extra-turn case on a board with an odd number of boxes. Everett V. Jackson. Dots and cubes. JRM 6:4 (Fall 1973) 273-279. Studies 3-dimensional game where a play is a square in the cubical lattice. Gyles Brandreth. Pencil and Paper Games and Puzzles. Carousel, 1976. Worm, pp. 18©19. This is a sort of 'anti©boxes' ©© one draws segments on the lattice forming a path without any cycles ©© last player wins. = Pencil & Paper Puzzle Games; Watermill Press, Mahwah, New Jersey, 1989, pp. 18©19. Winning Ways. 1982. Chap. 16: Dots©and©Boxes, pp. 507©550 David B. Lewis. Eureka! Perigee (Putnam), NY, 1983. Pp. 44-45 suggests playing on the triangular lattice. Sheila Anne Barry. The World's Best Travel Games. Sterling, NY, 1987. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐEternal triangles, pp. 80©81. Gives the game on the triangular lattice. Snakes, pp. 81©82. Same as Brandreth's Worm. I think 'snake' would be a better title as only one path is drawn. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐÙÙ Ã ÃÁÁ4.B.4.ÁÁSPROUTSÄ Ä M. Gardner. SA (Jul 1967) = Carnival, chap. 1. Describes Michael Stewart Paterson and John Horton Conway's invention of the game on 21 Feb 1967 at tea time in the Department common room at Cambridge. The idea of adding a spot was due to Paterson and they agreed the credit for the game should be 60% Paterson to 40% Conway. Gyles Brandreth. Pencil and Paper Games and Puzzles. Carousel, 1976. Sprouts, p. 13. "... actually born in Cambridge about ten years ago." c= Pencil & Paper Puzzle Games; Watermill Press, Mahwah, New Jersey, 1989, p. 13: "... was invented about ten years ago." Winning Ways. 1982. Sprouts, pp. 564©570 & 573. Says the game was "introduced by M. S. Paterson and J. H. Conway some time ago". Also describes Brussels Sprouts and Starsªand©Stripes. An answer for Brussels Sprouts and some references are on p. 573. Sheila Anne Barry. The World's Best Travel Games. Sterling, NY, 1987. Sprouts, pp. 95©97. Karl©Heinz Koch. Pencil & Paper Games. (As: Spiele mit Papier und Bleistift, no details); translated by Elisabeth E. Reinersmann. Sterling, NY, 1992. Sprouts, pp. 36©37, says it was invented by J. H. Conway & M. S. Paterson on 21 Feb 1976 [sic ©© misprint of 1967] during their five o'clock tea hour. à ÃÁÁ4.B.5.ÁÁOVID'S GAME AND NINE MEN'S MORRISÄ Ä Ðа¤˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐÁÁSee also 4.B.1 for historical material. ÁÁThe classic Nine Men's Morris board consists of three concentric squares with their midpoints joined by four lines. The corners are sometimes also joined by another four diagonal lines, but this seems to be used with twelve men per side and is sometimes called Twelve Men's Morris ©© see 1891 below. Fiske 108 says this is common in America but infrequent in Europe, though on 127 he says both forms were known in England before 1600, and both were carried to the US, though the Nine form is probably older. ÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐ Murray 615 discusses Nine Men's Morris. He cites Kurna, Egypt (-14C), medieval Spain (Alquerque de Nueve), the Gokstad ship and the steps of the Acropolis of Athens. He says the board sometimes has diagonals added and then is played with 9, 11 or 12 pieces. Dudeney. AM. 1917. Introduction to Moving Counter Problems, pp. 58©59. This gives a brief survey, mentioning a number of details that I have not seen elsewhere, e.g. its occurrence in Poland and on the Amazon. Says the board was found on a Roman tile at Silchester and on the steps of the Acropolis in Athens among other sites. J. A. Cuddon. The Macmillan Dictionary of Sports and Games. Macmillan, London, 1980. Pp. 563-564. Discusses the history. Says there is a c-1400 board cut in stone at Kurna, Egypt and similar boards were made in years 9 to 21 at Mihintale, Ceylon. Says Ars Amatoria may be describing Three Men's Morris and Tristia may be describing a kind of Tic-tac-toe. Cites numerous medieval descriptions and variants. Claudia Zaslavsky. Tic Tac Toe and Other Three-in-a-Row Games from Ancient Egypt to the Modern Computer. Crowell, NY, 1982. This is really a book for children and there are no references for the historical statements. I have found most of them elsewhere, and the author has kindly send me a list of source books, but I have not yet tracked down the following items ©© ??. ÁÁÁÁThere is an English court record of 1699 of punishment for playing Nine Holes in church. ÁÁÁÁThere is a Nine Men's Morris board on a stone on the temple of Seti I (presumably this is at Kurna). There is a picture in the 13C Spanish 'Book of Games' (presumably the Alfonso MS ©© see below) of children playing Alquerque de Tres (c= Three Men's Morris). A 14C inventory of the Duc de Berry lists tables for MÀ)Àrelles (=? Nine Men's Morris) (see Fiske 113©115 below) and a book by Petrarch shows two apes playing the game. H. Parker. Ancient Ceylon. Loc. cit. in 4.B.1. Nine Men's Morris board in the Temple of Kurna, Egypt, -14C. [Rohrbough, below, says this temple was started by Ramses I and completed by Seti in ©1336/©1333, citing J. Royal Asiatic Soc. (1783) 17.] Two diagrams for Nine Men's Morris are cut into the great flight of steps at Mihintale, Ceylon and these are dated c1C. He cites Bell; Arch. Survey of Ceylon, Third Progress Report, p. 5 note, for another diagram of similar age. Jack Botermans, Tony Burrett, Pieter van Delft & Carla van Spluntern. The World of Games. (In Dutch, 1987); Facts on File, NY, 1989. ÁÁÁÁP. 35 describes Yih, a form of Three Men's Morris, played on a doubly crossed square with a man moving "one step along any line". A note adds that only the French have a rule forbidding the first player to play in the centre, which makes the game more challenging and is recommended. ÁÁÁÁPp. 103©107 is the beginning of a section: Games of alignment and configuration and discusses various games, but rather vaguely and without references. They mention Al©Qurna, Mihintale, Gokstad and some other early sites. They say Yih was described by Confucius, was played c©500 and is "the game, that we now know as tic©tac©toe, or three men's morris." They describe Noughts and Crosses in the usual way. They then distinguish Tic©Tac©Toe, saying "In Britain it is generally known as three men's morris ...." and say it is the same as Yih, "which was known in ancient Egypt". They say "Ovid mentions tic©tac©toe" in Ars Amatoria, that several Roman boards have survived and that it was very popular in 14C England with several boards for this and Nine Men's Morris cut into cloister seats. They then describe Three©in©a©Row, which allows pieces to move one step in any direction, as a game played in Egypt. They then describe Five or Six Men's Morris, Nine Men's Morris, Twelve Men's Morris and Nine Men's Morris with Dice, with nice 13C & 15C illustration of Nine Men's Morris. Bell & Cornelius. Board Games Round the World. Op. cit. in 4.B.1. 1988. Pp. 6©8. They discuss the crossed square board ©© see 4.B.1 ©© and describe Three Men's Morris with moves only along the lines to an adjacent vacant point. They then describe Achi, from Ghana, on the doubly crossed square with the same rules. They then describe Six Men's Morris which was apparently popular in medieval Europe but became obsolete by c1600. Ovid. Ars Amatoria. ©1. II, 203©208 & III, 353©366. Translated by J. H. Mozley; Loeb Classical Library, 1929, pp. 80©81 & 142©145. Translated by B. P. Moore, 1935, used in A. D. Melville; Ovid The Love Poems; OUP, 1990, pp. 113, 137, 229 & 241. ÁÁÁÁII, 203©208 are three couplets apparently referring to three games: two dice games and Ludus Latrunculorum. Mozley's prose translation is: ÂÂÂÂÁÁ"If she be gaming, and throwing with her hand the ivory dice, do you throw amiss and move your throws amiss; or if is the large dice you are throwing, let no forfeit follow if she lose; see that the ruinous dogs often fall to you; or if the piece be marching under the semblance of a robbers' band, let your warrior fall before his glassy foe." ÁÁ'Dogs' is the worst throw in Roman dice games. ÁÁÁÁMoore's verse translation of 207©208 is: ÂÂÂÂÁÁ"And when the raiding chessmen take the field, Your champion to his crystal foe must yield." ÁÁMelville's note says the original has 'bandits' and says the game is Ludus Latrunculorum. ÁÁÁÁIII, 357©360 is probably a reference to the same game since 'robbers' occurs again, though translated as brigands by Mozley, and again it immediately follows a reference to throwing dice. Mozley's translation of 353©366 is: ÂÂÂÂÁÁ"I am ashamed to advise in little things, that she should know the throws of the dice, and thy powers, O flung counter. Now let her throw three dice, and now reflect which side she may fitly join in her cunning, and which challenge, Let her cautiously and not foolishly play the battle of the brigands, when one piece falls before his double foe and the warrior caught without his mate fights on, and the enemy retraces many a time the path he has begun. And let smooth balls be flung into the open net, nor must any ball be moved save that which you will take out. There is a sort of game confined by subtle method into as many lines as the slippery year has months: a small board has three counters on either side, whereon to join your pieces together is to conquer." ÁÁMoore's translation of 357©360 is: ÂÂÂÂÁÁ"To guide with wary skill the chessmen's fight, When foemen twain o'erpower the single knight, And caught without his queen the king must face The foe and oft his eager steps retrace". ÁÁThis is clearly not a morris game ©© Mozley's note above and the next entry make it clear it is Ludus Latrunculorum, which had a number of forms. Mozley's note on pp. 142©143 refers to Tristia II, 478 and cites a number of other references for Ludus Latrunculorum. ÁÁÁÁMoore's translation of 363©366 is: ÂÂÂÂÁÁ"A game there is marked out in slender zones As many as the fleeting year has moons; A smaller board with three a side is manned, And victory's his who first aligns his band." ÁÁMozley's notes and Melville's notes say the first two lines refer to the Roman game of Ludus Duodecim Scriptorum ©© the Twelve Line Game ©© which is the ancestor of Backgammon. Mozley says the game in the latter two lines is mentioned in Tristia, "but we have no information about it." Melville says it is "a 'position' game, something like Nine Men's Morris" and cites R. C. Bell's article on 'Board and tile games' in the Encyclopaedia Britannica, 15th ed., Macropaedia ii.1152-1153, ??NYS. Ovid. Tristia. c10. II, 471-484. Translated by A. L. Wheeler. Loeb Classical Library, 1945, pp. 88-91. This mentions several games and the text parallels that of Ars Amatoria III. ÂÂÂÂÁÁ"Others have written of the arts of playing at dice ©© this was no light sin in the eyes of our ancestors ©© what is the value of the ÃÃtaliÄÄ, with what throw one can make the highest point, avoiding the ruinous dogs; how the ÃÃtesseraÄÄ is counted, and when the opponent is challenged, how it is fitting to throw, how to move according to the throws; how the variegated soldier steals to the attack along the straight path when the piece between two enemies is lost, and how he understands warfare by pursuit and how to recall the man before him and to retreat in safety not without escort; how a small board is provided with three men on a side and victory lies in keeping one's men abreast; and the other games ©© I will not describe them all ©© which are wont to waste that precious thing, our time." ÁÁA note says some see a reference to Ludus Duodecim Scriptorum at the beginning of this. The next note says the next text refers to Ludus Latrunculorum, a game on a squared board with 30 men on a side, with at least two kinds of men. The note for the last game says "This game seems to have resembled a game of draughts played with few men." and refers to Ars Amatoria and the German MÀGÀhlespiel, which he describes as 'a sort of draughts', but which is Nine Men's Morris. R. G. Austin. Roman board games ©© I & II. Greece and Rome 4 (No. 10) (Oct 1934) 24-34 & 4 (No. 11) (Feb 1935) 76©82. Claims the Ovid references are to Ludus Latrunculorum (a kind of Draughts?), Ludus Duodecim Scriptorum (later Tabula, an ancestor of Backgammon) and (Ars Amatoria.iii.365©366) a kind of Three Men's Morris. In the last, he shows a doubly crossed 3 x 3 board, but it is not clear which rule he adopts for the later movement of pieces, but he says: "the first player is always able to force a win if he places his first man on the centre point, and this suggests that the dice may have been used to determine priority of play, although there is no evidence of this." He says no Roman name for this game has survived. He discusses various known artifacts for all the game, citing several Roman 8 x 8 boards found in Britain. He gives an informal bibliography with comments as to the value of the works. D'Arcy Wentworth Thompson. Science and the Classics. OUP, 1940. Section V Games and Playthings, pp. 148©165. On p. 160, he quotes Ovid, Ars Amatoria.iii.365©366 and says it is Noughts and Crosses, or in Ireland, Tip©top©castle. The British Museum has a Nine Men's Morris board from the Temple of Artemis, Ephesus, 2C©4C. Item BM GR 1872,8©3,44. This was in a small exhibition of board games in 1990. I didn't see it on display in late 1996. Murray, p. 189. There was an Arabic game called Qirq, which Murray identifies with Morris. "Fourteen was a game played with small stones on a wooden board which had three rows of holes (al-QÀÀbÀEÀnÀ3À)." AbÀEÀ-HanÀ3Àfa, c750, held that Fourteen was illegal and Qirq was held illegal by writers soon afterward. On p. 194, Murray gives a 10C passage mentioning Qirq being played at Mecca. Fiske 255 cites the KitÀ]Àb al AghÀ]Àni, c960, for a reference to qirkat, i.e. morris boards. Paul B. Du Chaillu. The Viking Age. Two vols., John Murray, London, 1889. Vol. II, p.168, fig. 992 ©© Fragments of wood from Gokstad ship. Shows a partial board for Nine Men's Morris found in the Gokstad ship burial. There is no description of this illustration and there is only a vague indication that this is 10C, but other sources say it is c900. Gutorm Gjessing. The Viking Ship Finds. Revised ed., Universitets Oldsaksamling, Oslo, 1957. P. 8: "... there are two boards which were used for two kinds of games; on one side figures appear for use in a game which is frequently played even now (known as "MÀQÀlle")." Thorlief SjÀQÀvold. The Viking Ships in Oslo. Universitets Oldsaksamling, Oslo, 1979. P. 54: "... a gaming board with one antler gaming piece, ...." In medieval Europe, the game is called Ludus Marellorum or Merellorum or just Marelli or Merelli or Merels, meaning the game of counters. Murray 399 says the connection with Qirq is unclear. However, medieval Spain played various games called Alquerque, which is obviously derived from Qirq. Alquerque de Nueve seems to be Nine Men's Morris. However, in Italy and in medieval France, Marelle or Merels could mean Alquerque (de Doze), a draughts-like game with 12 men on a side played on a 5 x 5 board (Murray 615). Also Marro, Marella can refer to Draughts which seems to originate in Europe somewhat before 1400. Stewart Culin. Korean Games, with Notes on the Corresponding Games of China and Japan. University of Pennsylvania, Philadelphia, 1895. Reprinted as: Games of the Orient; Tuttle, Rutland, Vermont, 1958. Reprinted under the original title, Dover and The Brooklyn Museum, 1991. P. 102, section 80: Kon©tjil ©© merrells. This is the usual Nine Men's Morris. The Chinese name is SÀÀm©k'i (Three Chess). "I am told by a Chinese merchant that this game was invented by Chao Kw'ang©yin (917©975), founder of the Sung dynasty." This is the only indication of an oriental source that I have seen. Gerhard Leopold. Skulptierte WerkstÀGÀcke in der Krypta der Wipertikirche zu Quedlinburg. IN: Friedrich MÀ?Àbius & Ernst Schubert, eds.; Skulptur des Mittelalters; Hermann BÀ?Àhlaus Nachfolger, Weimar, 1987, pp. 27©43; esp. pp. 37 & 43. Describes and gives photos of several Nine©Men's©Morris boards carved on a pillar of the crypt of the Wipertikirche, Quedlinburg, Sachsen©Anhalt, probably from the 10/11 C. Richard de Fournivall. De Vetula. 13C. This describes various games, including Merels. Indeed the French title is: Ci parle du gieu des Merelles .... ??NYS ©© cited by Murray, pp. 439, 507, 520, 628. Murray 620 cites several MSS and publications of the text. "Bonus Socius" [Nicolas de NicolaÀ5À?]. This is a collection of chess problems, compiled c1275, which exists in many manuscript forms and languages. See 5.F.1 for more details of these MSS. See Murray 618-642. On pp. 619-624 & 627, he mentions several MSS which include 23, 24, 25 or 28 Merels problems. On p. 621, he cites "Merelles a Neuf" from 14C. Fiske 104 & 110©111 discusses some MSS of this collection. The Spanish Treatise on Chess©Play written by order of King Alfonso the Sage in the year 1283. [= Libro de Acedrex, Dados e Tablas of Alfonso El Sabio, generally known as the Alfonso MS.] MS in Royal Library of the Escorial (j.T.6. fol). Complete reproduction in 194 Phototropic Plates. 2 vols., Karl W. Horseman, Leipzig, 1913. (See in 4.A.1 for another ed.) This is a collection of chess problems produced for Alfonso X, the Wise, King of Castile (Castilla). Vol. 2, ff. 92v-93r, pp. CLXXXIV-CLXXXV, shows Nine Men's Morris boards. ??NX ©© need to study text. See: Murray 568-573; van der Linde I 137 & 279 ??NYS & Quellenstudien 73 & 277-278, ??NYS (both cited by Fiske 98); van der Lasa 116, ??NYS (cited by Fiske 99). ÁÁÁÁFiske 98©99 says that the MS also mentions Alquerque, Cercar de Liebre and Alquerque de Neuve (with 12 men against one). Fiske 253©255 gives a more detailed study of the MS based on a transcript. He also quotes a communication citing al Querque or al Kirk in Kazirmirski's Arabic dictionary and in the KitÀ]Àb al AghÀ]Àni, c960. ÁÁÁÁJosÀ)À Brunet y Bellet. El Ajedrez. Barcelona, 1890. ??NYS ©© described by Fiske 98. This has a chapter on the Alfonso MS and refers to Alquerque de Doce, saying that it is known as Tres en Raya in Castilian and Marro in Catalan (Fiske 102 says this word is no longer used in Spanish). Brunet notes that there are five miniatures pertaining to alquerque. Fiske says that all this information leaves us uncertain as to what the games were. Fiske says Brunet's chapter has an appendix dealing with Carrera's 1617 discussion of 'line games' and describing Riga di Tre as the same as Marro or Tres en Raya as a form of Three Men's Morris Murray gives many brief references to the game, which I will note here simply by his page number and the date of the item. ÁÁÁÁ438-439 (12C); 446 (14C); ÁÁÁÁ449 (c1400 ©© 'un marrelier', i.e. a Merels board); ÁÁÁÁ431 (c1430); 447 (1491); 446 (1538). Anon. Romance of Alexander. 1338. (Bodleian Library, Mss Bodl. 264). ??NYS. Nice illustration clearly showing Nine Men's Morris board. I. Disraeli (Amenities of Literature, vol. I, p. 86) also cites British Museum, Bib. Reg. 15, E.6 as a prose MS version with illustrations. Prof. D. J. A. Ross tells me there is nothing in the text corresponding to the illustrations and that the Bodleian text was edited by M. R. James, c1920, ??NYS. Illustration reproduced in: A. C. Horth; 101 Games to Make and Play; Batsford, London, (1943; 2nd ed., 1944); 3rd ed., 1946; plate VI facing p. 44, in B&W. Also in: Pia Hsiao et al.; Games You Make and Play; Macdonald and Jane's, London, 1975, p. 7, in colour. Fiske 113©115 gives a number of quotations from medieval French sources as far back as mid 14C, including an inventory of the Duc de Berry in 1416 listing two boards. Fiske notes that the game has given rise to several French phrases. He quotes a 1412 source calling it Ludus Sanct Mederici or Jeu Saint Marry and also mentions references in city statutes of 1404 and 1414. MS, Montpellier, Faculty of Medicine, H279 (Fonts de Boulier, E.93). 14C. This is a version of the Bonus Socius collection. Described in Murray 623©624, denoted M, and in van der Linde I 301, denoted K. Lucas, RM2, 1883, pp. 98©99 mentions it and RM4, 1894, QuatriÀ/Àme RÀ)ÀcrÀ)Àation: Le jeu des mÀ)Àrelles au XIIIÃÃeÄÄ siÀ/Àcle, pp. 67©85 discusses it extensively. This includes 28 Merels problems which are given and analysed by Lucas. Lucas dates the MS to the 13C. Household accounts of Edward IV, c1470. ??NYS ©© see Murray 617. Record of purchase of "two foxis and 46 hounds" to form two sets of "marelles". Civis Bononiae [Citizen of Bologna]. This is a collection of chess problems compiled c1475, which exists in several MSS. See Murray 643-703. It has 48 or 53 merels problems. On p. 644, 'merelleorum' is quoted. A Hundred Sons. Chinese scroll of Ming period (1368©1644). 18C copy in BM. ??NYS ©© extensively reproduced and described in: Marguerite Fawdry; Chinese Childhood; Pollock's Toy Theatres, London, 1977. On p. 12 of Fawdry is a scene, apparently from the scroll, in which some children appear to be playing on a Twelve Men's Morris board. Elaborate boards from Germany (c1530) and Venice (16C) survive in the National Museum, Munich and in South Kensington (Murray 757-758). Murray shows the first in B&W facing p. 757. William Shakespeare. A Midsummer Night's Dream. c1610. Act II, scene I, lines 98©100: "The nine men's morris is fill'd up with mud, And the quaint mazes in the wanton green For lack of tread are indistinguishable." Fiske 126 opines that the latter two lines may indicate that the board was made in the turf, though he admits that they may refer just to dancers' tracks, but to me it clearly refers to turf mazes. J. C. Bulenger. De Ludis Privatis ac Domesticus Veterum. Lyons, 1627. ??NYS Fiske 115 & 119 quote his description of and philological note on Madrellas (Three Men's Morris). Paul Fleming (1609©1640). In one of his lyrics, he has MÀGÀhlen. ??NYS ©© quoted by Fiske 132, who says this is the first German mention of Morris. Fiske 133 gives the earliest Russian reference to Morris as 1675. Thomas Hyde. Historia Nerdiludii, hoc est dicere, Trunculorum; .... (= Vol. 2 of De Ludis Orientalibus, see 7.B for vol. 1.) From the Sheldonian Theatre (i.e. OUP), Oxford, 1694. Historia Triodii, pp. 202©214, is on morris games. (Described in Fiske 118©124, who says there is further material in the Elenchus at the end of the volume ©© ??NYS) Hyde asserts that the game was well known to the Romans, though he cannot find a Roman name for it! He cites and discusses Bulenger, but disagrees with his philology. Gives lots of names for the game, ranging as far as Russian and Armenian. He gives both the Nine and Twelve Men's Morris boards on p. 210, but he has not found the Twelve board in Eastern works. On p. 211, he gives the doubly crossed square board with a title in Chinese characters, pronounced 'Che©lo', meaning 'six places', and having three white and three black men already placed along two sides. He says the Irish name is Cashlan Gherra (Short Castle) and that the name Copped Crown is common in Cumberland and Westmorland. He then describes playing the Twelve Man and Nine Man games, and then he considers the game on the doubly crossed square board. He seems to say there are different rules as to how one can move. ??need to study the Latin in detail. This is said to throw light on the Ovid passages. Hyde believes the game was well known to the Romans and hence must be much older. Fiske remarks that this is history by guesswork. Murray 383 describes Russian chess. He says Amelung identifies the Russian game "saki with HÀ?Àlzchenspiel (?merels)". Saki is mentioned on this page as being played at the Tsar's court, c1675. Archiv der Spiele. 3 volumes, Berlin, 1819©1821. Vol. 2 (1820) 21©27. ??NYS Described and quoted by Fiske 129©132. This only describes the crossed square and the Nine Men's Morris boards. It says that the Three Men's Morris on the crossed square board is a tie, i.e. continues without end, but it is not clear how the pieces are allowed to move. Fiske says this gives the most complete explanation he knows of the rules for Nine Men's Morris. Charles Babbage. Notebooks ©© unpublished collection of MSS in the BM as Add. MS 37205. ??NX. For more details, see 4.B.1. On ff. 347.r©347.v, 8 Sep 1848, he suggests Nine Men's Morris boards in triangular and pentagonal shapes and does various counting on the different shapes. The Family Friend (1856) 57. Puzzle 17. ©© Two and a Bushel. Shows the standard # board. "This very simple and amusing games, ©© which we do not remember to have seen described in any book of games, ©© is played, like draughts, by two persons with counters. Each player must have three, ... and the game is won when one of the players succeeds in placing his three men in a row; ...." There is no specification of how the men move. The word 'bushel' occurs in some old descriptions of Three Men's Morris and Nine Men's Morris as the name of the central area. The Sociable. 1858. Merelles: or, nine men's morris, pp. 279©280. Brief description, notable for the use of Merelles in an English book. Von der Lasa. Ueber die griechischen und rÀ?Àmischen Spiele, welche einige ÀÀhnlichkeit mit dem Schach hatten. Deutsche Schachzeitung (1863) 162©172, 198©199, 225©234, 257-264. ??NYS ©© described on Fiske 121©122 & 137, who says van der Linde I 40©47 copies much of it. He asserts that the Parva Tabella of Ovid is Kleine MÀGÀhle (Three Men's Morris). Von der Lasa says the game is called Tripp, Trapp, Trull in the Swedish book Hand©Bibliothek fÀ?Àr SÀÀllkapsnÀ?Àjen, of 1839, vol. II, p. 65 (or 57??). Van der Linde says that the Dutch name is Tik, Tak, Tol. Fiske notes that both of these refer to Noughts and Crosses, but it is unclear if von der Lasa or van der Linde recognised the difference between Three Men's Morris and Noughts and Crosses. Albert Norman. Ungdomens Bok [Book for Youth] (in Swedish). 2nd ed., Stockholm, 1883. Vol. I, p. 162++. ??NYS ©© quoted and described in Fiske 134©136. Plays Nine Men's Morris on a Twelve Men's Morris board. Webster's Dictionary. 1891. ??NYS ©© Fiske 118 quotes a definition (not clear which) which includes "twelve men's morris". Fiske says: "Here we have almost the only, and certainly the first mention of the game by its most common New England name, "twelve men's morris," and also the only hint we have found in print that the more complicated of the morris boards ©© with the diagonal lines ... ©© is used with twelve men, instead of nine, on each side." Fiske 127 says the name only appears in American dictionaries. Dudeney. CP. 1907. Prob. 110: Ovid's game, pp. 156-157 & 248. Says the game "is distinctly mentioned in the works of Ovid." He gives Three Men's Morris, with moves to adjacent cells horizontally or diagonally, and says it is a first player win. Blyth. Match©Stick Magic. 1921. Black versus white, pp. 79©80. 4 x 4 board with four men each. But the men must be initially placed WBWB in the first row and BWBW in the last row. They can move one square "in any direction" and the object is to get four in a row of your colour. Games and Tricks ©© to make the Party "Go". Supplement to "Pearson's Weekly", Nov. 7th, no year indicated [1930s??]. A matchstick game, p. 11. On a 4 x 4 board, place eight men, WBWB on the top row and BWBW on the bottom row. Players alternately move one of their men by one square in any direction ©© the object is to make four in a line. Lynn Rohrbough, ed. Ancient Games. Handy Series, Kit N, Cooperative Recreation Service, Delaware, Ohio, (1938), 1939. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐMorris was Player [sic] 3,300 Years Ago, p. 27. Says the temple of Kurna was started by Ramses I and completed by Seti in ©1336/©1333, citing J. Royal Asiatic Soc. (1783) 17. Three Men's Morris, p. 27. After placing their three men, players 'then move trying to get three men in a row.' Contributor says he played it in Cardiff more than 50 years ago. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐWinning Ways. 1982. Pp. 672©673. Says Ovid's Game is conjectured to be Three Men's Morris. The current version allows moves by one square orthogonally and is a first person win if the first person plays in the centre. If the first player cannot play in the centre, it is a draw. They use Three Men's Morris for the case with one step moves along winning lines, i.e. orthogonally or along main diagonals. An American Indian game, Hopscotch, permits one step moves orthogonally or diagonally (along any diagonal). A French game, Les Pendus, allows any move to a vacant cell. All of these are draws, even allowing the first player to play in the centre. They briefly describe Six and Nine Men's Morris. Ralph Gasser & J. Nievergelt. Es ist entscheiden: Das Muehle©Spiel ist unentscheiden. Informatik Spektrum 17 (1994) 314©317. ??NYS ©© cited by JÀ?Àrg Bewersdorff [email of 6 Jun 1999]. L. V. Allis. Beating the World Champion ©© The state of the art in computer game playing. IN: Alexander J. de Voogt, ed.; New Approaches to Board Games Research: Asian Origins and Future Perspectives; International Institute for Asian Studies, Leiden, 1995; pp. 155©175. On p. 163, he states that Ralph Gasser showed that Nine Men's Morris is a draw in Oct 1993, but the only reference is to a letter from Gasser. Ralph Gasser. Solving Nine Men's Morris. IN: Games of No Chance; ed. by Richard Nowakowski; CUP, 1996, pp. 101©113. ??NYS ©© cited by Bewersdorff [loc. cit.] and described in William Hartston; What mathematicians get up to; The Independent Long Weekend (29 Mar 1997) 2. Demonstrates that Nine Men's Morris is a draw. Gasser's abstract: "We describe the combination of two search methods used to solve Nine Men's Morris. An improved analysis algorithm computes endgame databases comprising about 10ÃÃ10ÄÄ states. An 18©ply alpha©beta search the used these databases to prove that the value of the initial position is a draw. Nine Men's Morris is the first non©trivial game to be solved that does not seem to benefit from knowledge©based methods." I'm not sure about the last statement ©© 4 x 4 x 4 noughts and crosses (see 4.B.1.a) and Connect©4 were solved in 1980 and 1988, though the first was a computer aided proof and the original brute force solution of Connect©4 by James Allen in Sep 1988 was improved to a knowledge©based approach by L. V. Allis by Aug 1989. The five©in©aªrow version of Connect©4 was shown to be a first person win in 1993. Bewersdorff [email of 11 Jun 1999] clarifies this by observing that draw here means a game that continues forever ©© one cannot come to a stalemate where neither side can move. à ÃÁÁ4.B.6.ÁÁPHUTBALLÄ Ä Winning Ways. 1982. Philosopher's football, pp. 688-691. In 1985, Guy said this was the only published description of the game. à ÃÁÁ4.B.7.ÁÁBRIDG-ITÄ Ä Ðа¤˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐÁÁThis is best viewed as played on a n x n array of squares. The n(n+1) vertical edges belong to one player, say red, while the n(n+1) horizontal edges belong to black. Players alternate marking a square with a line of their colour between edges of their colour. A square cannot be marked twice. The object is to complete a path across the board. In practice, the edges are replaced by coloured dots which are joined by lines. As with Hex, there can be no ties and there must be a first person strategy. ÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐ M. Gardner. SA (Oct 1958) c= 2nd Book, Chap. 7. Introduces David Gales's game, later called Bridg-it. Addendum in the book notes that it is identical to Shannon's 'Bird Cage' game of 1951 and that it was marketed as Bridg-it in 1960. M. Gardner. SA (Jul 1961) c= New MD, Chap. 18. Describes Oliver Gross's simple strategy for the first player to win. Addendum in the book gives references to other solutions and mentions. M. Gardner. SA (Jan 1973) c= Knotted, Chap. 9. Article says Bridg-it was still on the market. Winning Ways. 1982. Pp. 680©682. Covers Bridg©it and Shannon Switching Game. In Oct 2000, I bought a second©hand copy of a 5 x 5 version called Connections, attributed to Tom McNamara, but with no date. à ÃÁÁ4.B.8.ÁÁCHOMPÄ Ä Fred Schuh. Spel van delers (Game of divisors). Nieuw Tijdschrift vor Wiskunde 39 (1951-52) 299-304. ??NYS ©© cited by Gardner, below. M. Gardner. SA (Jan 1973) c= Knotted, Chap. 9. Gives David Gale's description of his game and results on it. Addendum in Knotted points out that it is equivalent to Schuh's game and gives other references. David Gale. A curious Nim©type game. AMM 81 (1974) 876©879. Describes the game and the basic results. Wonders if the winning move is unique. Considers three dimensional and infinite forms. A note added in proof refers to Gardner's article, says two programmers have consequently found that the 8 x 10 game has two winning first moves and mentions Schuh's game. Winning Ways. 1982. Pp. 598©600. Brief description with extensive table of good moves. Cites an earlier paper of Gale and Stewart which does not deal with this game. à ÃÁÁ4.B.9.ÁÁSNAKES AND LADDERSÄ Ä Ðа¤˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐÁÁI have included this because it has an interesting history and because I found a nice way to express it as a kind of Markov process or random walk, and this gives an expression for the average time the game lasts. I then found that the paper by Daykin et al. gives most of these ideas. ÁÁThe game has two or three rules for finishing. ÁÁÃÃA.ÄÄÁÁOne finishes by going exactly to the last square, or beyond it. ÁÁÃÃB.ÄÄÁÁOne finishes by going exactly to the last square. If one throws too much, then one stands still. ÁÁÃÃC.ÄÄÁÁOne finishes by going exactly to the last square. If one throws too much, one must count back from the last square. E.g., if there are 100 squares and one is at 98 and one throws 6, then one counts: 99, 100, 99, 98, 97, 96 and winds up on 96. (I learned this from a neighbour's child but have only seen it in one place ©© in the first Culin item below.) ÁÁGames of this generic form are often called promotion games. If one considers the game with no snakes or ladders, then it is a straightforward race game, and these date back to Egyptian and Babylonian times, if not earlier. ÁÁIn fact, the same theory applies to random walks of various sorts, e.g. random walks of pieces on a chessboard, where the ending is arrival exactly at the desired square. ÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐ In the British Museum, Room 52, Case 24 has a Babylonian ceramic board (WA 1991©7.20,I) for a kind of snakes and ladders from c©1000. The label says this game was popular during the second and first millennia BC. Sheng©kuan t'u [The game of promotion]. 7C. Chinese game. This is described in: Nagao Tatsuzo; Shina Minzoku©shi [Manners and Customs of the Chinese]; Tokyo, 1940©1942, perhaps vol. 2, p. 707, ??NYS This is cited in: Marguerite Fawdry; Chinese Childhood; Pollock's Toy Theatres, London, 1977, p. 183, where the game is described as "played on a board or plan representing an official career from the lowest to the highest grade, according to the imperial examination system. It is a kind of Snakes and Ladders, played with four dice; the object of each player being to secure promotion over the others." Thomas Hyde. Historia Nerdiludii, hoc est dicere, Trunculorum; .... (= Vol. 2 of De Ludis Orientalibus, see 7.B for vol. 1.) From the Sheldonian Theatre (i.e. OUP), Oxford, 1694. De ludo promotionis mandarinorum, pp. 70©101 ©© ??NX. This is a long description of Shing quon tu, a game on a board of 98 spaces, each of which has a specific description which Hyde gives. There is a folding plate showing the Chinese board, but the copy in the Graves collection is too fragile to photocopy. I did not see any date given for the game. Stewart Culin. Chinese Games with Dice and Dominoes. From the Report of the U. S. National Museum for 1893, pp. 489-537. Pp. 502©507 describes several versions of the Japanese Sugoroku (Double Sixes) which is a generic name for games using dice to determine moves, including backgammon and simple race games, as well as Snakes and Ladders games. One version has ending in the form C. Then says Shing KÀCÀn TÀÀÀAÀ (The Game of the Promotion of Officials) is described by Hyde as The Game of the Promotion of the Mandarins and gives an extended description of it. There is a legend that the game was invented when the Emperor Kienlung (1736©1796) heard a candidate playing dice and the candidate was summoned to explain. He made up a story about the game, saying that it was a way for him and his friends to learn the different ranks of the civil service. He was sent off to bring back the game and then made up a board overnight. However Hyde had described the game a century before this date. It seems that this is not really a Snakes and Ladders game as the moves are determined by the throw of the dice and the position ©© there are no interconnections between cells. But Culin notes that the game is complicated by being played for money or counters which permit bribery and rewards, etc. Culin. Chess and Playing Cards. Op. cit. in 4.A.4. 1898. ÁÁÁÁPp. 820©822 & plates 24 & 25 between 821 & 822. Says Shing KÀCÀn TÀÀÀAÀ (The Game of the Promotion of Officials) is described by Hyde as The Game of the Promotion of the Mandarins and refers to the above for an extended description. Describes the Korean version: Tjyong©Kyeng©To (The Game of Dignitaries) and several others from Korea and Tibet, with 108, 144, 169 and 64 squares. ÁÁÁÁPp. 840©842 & plate 28, opp. p. 841 describes Chong ÀGÀn ChÀÀau (Game of the Chief of the Literati) as 'in many respects analogous' to Shing KÀCÀn TÀÀÀAÀ and the Japanese game Sugoroku (Double Sixes) ©© in several versions. Then mentions modern western versions ©© Jeu de L'Oie, Giuoco dell'Oca, Juego de la Oca, Snake Game. Pp. 843©848 is a table listing 122 versions of the game in the University of Pennsylvania Museum of Archaeology and Paleontology. These are in 11 languages, varying from 22 to 409 squares. Bell & Cornelius. Board Games Round the World. Op. cit. in 4.B.1. 1988. Snakes and Ladders and the Chinese Promotion Game, pp. 65-67. They describe the Hindu version of Snakes and Ladders, called Moksha©patamu. Then they discuss Shing Kun t'o (Promotion of the Mandarins), which was played in the Ming (1368©1616) with four or more players racing on a board with 98 spaces and throwing 6 dice to see how many equal faces appeared. They describe numerous modern variants. Deepak Shimkhada. A preliminary study of the game of Karma in India, Nepal, and Tibet. Artibus Asiae 44 (1983) 4. ??NYS © cited in Belloli et al, p. 68. Andrew Topsfield. The Indian game of snakes and ladders. Artibus Asiae 46:3 (1985) 203-214 + 14 figures. Basically a catalogue of extant Indian boards. He says the game is called GyÀ]Àn caupad or GyÀ]Àn chaupar in Hindi. He states that Moksha©patamu sounds like it is Telugu and that this name appeared in Grunfield's Games of the World (1975) with no reference to a source and that Bell has repeated this. Game boards were drawn or painted on paper or cloth and hence were perishable. The oldest extant version is believed to be an 84 square board of 1735, in the Museum of Indology, Jaipur. There were Hindu, Jain, Muslim and Tibetan versions representing a kind of Pilgrim's Progress, finally arriving at God or Heaven or Nirvana. The number of squares varies from 72 to 360. ÁÁÁÁHe gives many references and further details. An Indian version of the game was described by F. E. Pargiter; An Indian game: Heaven or Hell; J. Royal Asiatic Soc. (1916) 539©542, ??NYS. He cites the version by Ayres (and Love's reproduction of it ©© see below) as the first English version. He cites several other late 19C versions. F. H. Ayres. [Snakes and ladders game.] No. 200682 Regd. Example in the Bethnal Green Museum, Misc. 8 © 1974. Reproduced in: Brian Love; Play The Game; Michael Joseph, London, 1978; Snakes & Ladders 1, pp. 22©23. This is the earliest known English version of the game, with 100 cells in a spiral and 5 snakes and 5 ladders. N. W. Bazely & P. J. Davis. Accuracy of Monte Carlo methods in computing finite Markov chains. J. of Res. of the Nat. Bureau of Standards ©© Mathematics and Mathematical Physics 64B:4 (Oct©Dec 1960) 211©215. ??NYS ©© cited by Davis & Chinn and Bewersdorff. Bewersdorff [email of 6 Jun 1999] brought these items to my attention and says it is an analysis based on absorbing Markov chains. D. E. Daykin, J. E. Jeacocke & D. G. Neal. Markov chains and snakes and ladders. MG 51 (No. 378) (Dec 1967) 313©317. Shows that the game can be modelled as a Markov process and works out the expected length of play for one player (47.98 moves) or two players (27.44 moves) on a particular board with finishing rule A. Philip J. Davis & William G. Chinn. 3.1416 and All That. S&S, 1969, ??NYS; 2nd ed, BirkhÀÀuser, 1985, chap. 23 (by Davis): "Mr. Milton, Mr. Bradley ©© meet Andrey Andreyevich Markov", pp. 164©171. Simply describes how to set up the Markov chain transition matrix for a game with 100 cells and ending B. Doesn't give any results. Lewis Carroll. Board game for one. In: Lewis Carroll's Bedside Book; ed. by Gyles Brandreth (under the pseud. Edgar Cuthwellis); Methuen, 1979, pp. 19©21. ??look for source; not in Carroll©Wakeling, Carroll©Wakeling II or Carroll©Gardner. Board of 27 cells with pictures in the odd cells. If you land on any odd cell, except the last one, you have to return to square 1. "Sleep is almost certain to have overwhelmed the player before he reaches the final square." Ending A is probably intended. (The average duration of this game should be computable.) S. C. Althoen, L. King & K. Schilling. How long is a game of snakes and ladders? MG 77 (No. 478) (Mar 1993) 71©76. Similar analysis to Daykin, Jeacocke & Neal, using finishing rule B, getting 39.2 moves. They also use a simulation to find the number of moves is about 39.1. David Singmaster. Letter [on Snakes and ladders]. MG 79 (No. 485) (Jul 1995) 396©397. In response to Althoen et al. Discusses history, other ending rules and wonders how the length depends on the number of snakes and ladders. Irving L. Finkel. Notes on two Tibetan dice games. IN: Alexander J. de Voogt, ed.; New Approaches to Board Games Research: Asian Origins and Future Perspectives; International Institute for Asian Studies, Leiden, 1995; pp. 24©47. Part II: The Tibetan 'Game of Liberation', pp. 34©47, discusses promotion games with many references to the literature and describes a particular game. JÀ?Àrg Bewersdorff. GlÀGÀck, Logik und Bluff Mathematik im Spiel ©© Methoden, Ergebnisse und Grenzen. Vieweg, 1998. Das Leiterspiel, pp. 67©68 & Das Leiterspiel als Markow-Kette. Discusses setting up the Markov chain, citing Bazley & Davis, with the same board as in Davis & Chinn, then states that the average duration is 39.224 moves. Jay Belloli, ed. The Universe A Convergence of Art, Music, and Science. [Catalogue for a group of exhibitions and concerts in Pasadena and San Marino, Sep 2000 © Jun 2001.] Armory Center for the Arts, Pasadena, 2001. P. 68 has a discussion of the Jain versions of the game, called 'gyanbazi', with a colour plate of a 19C example with a 9 x 9 board with three extra cells. à ÃÁÁ4.B.10.ÁÁMU TOREREÄ Ä Ðа¤˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐÁÁThis is a Maori game which can be found in several books on board games. I have included it because it has been completely analysed. There are eight (or 2n) points around a central area. Each player has four (or n) markers, originally placed on consecutive points. One can move from a point to an adjacent point or to the centre, or one can move from the centre to a point, provided the position moved to is empty. The first player who cannot move is the loser. To prevent the game becoming trivial, it is necessary to require that the first two (or one) moves of each player involve his end pieces, though other restrictions are sometimes given. ÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐ Marcia Ascher. Mu Torere: An analysis of a Maori game. MM 60 (1987) 90©100. Analyses the game with 2n points. For n = 1, there are 6 inequivalent positions (where equivalence is by rotation or reflection of the board) and play is trivially cyclic. For n = 2, there are 12 inequivalent positions, but there are no winning positions. For n = 3, there are 30 inequivalent positions, some of which are wins, but the game is a tie. Obtains the number of positions for general n. For the traditional version with n = 4, there are 92 inequivalent positions, some of which are wins, but the game is a tie, though this is not at all obvious to an inexperienced player. In 1856, it was reported that no foreigner could win against a Maori. For n = 5, there are 272 inequivalent positions, but the game is a easy win for the first player ©© the constraints on first moves need to be revised. Ascher gives references to the ethnographic literature for descriptions of the game. Marcia Ascher. Ethnomathematics. Brooks/Cole Publishing, Pacific Grove, California, 1991. Sections 4.4©4.7, pp. 95©109 & Notes 4©7, pp. 118©119. Amplified version of her MM article. à ÃÁÁ4.B.11.ÁÁMASTERMIND, ETC.Ä Ä Ðа¤˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐÁÁThere were a number of earlier guessing games of the Mastermind type before the popular version devised by Marco Meirovitz in 1973 ©© see: Reddi. One of these was the English Bulls and Cows, but I haven't seen anything written on this and it doesn't appear in Bell, Falkener or Gomme. Since 1975 there have been several books on the game and a number of papers on optimal strategies. I include a few of the latter. ÁÁNOTATION. If there are h holes and c choices at each hole, then I abbreviate this as cÃÃhÄÄ. ÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐA. K. Austin. How do You play 'Master Mind'. MTg 71 (Jun 1975) 46©47. How to state the rules correctly. S. S. Reddi. A game of permutations. JRM 8:1 (1975) 8©11. Mastermind type guessing of a permutation of 1,2,3,4 can win in 5 guesses. Donald E. Knuth. The computer as Master Mind. JRM 9:1 (1976©77) 1©6. 6ÃÃ4ÄÄ can be won in 5 guesses. Robert W. Irving. Towards an optimum Mastermind strategy, JRM 11:2 (1978©79) 81©87. Knuth's algorithm takes an average of 5804/1296 = 4.478 guesses. The author presents a better strategy that takes an average of 5662/1296 = 4.369 guesses, but requires six guesses in one case. A simple adaptation eliminates this, but increases the average number of guesses to 5664/1296 = 4.370. An intelligent setter will choose a pattern with a single repetition, for which the average number of guesses is 3151/720 = 4.376. A. K. Austin. Strategies for Mastermind. G&P 71 (Winter 1978) 14©16. Presents Knuth's results and some other work. Merrill M. Flood. Mastermind strategy. JRM 18:3 (1985©86) 194©202. Cites five earlier papers on strategy, including Knuth and Irving. He considers it as a two©person game and considers the setter's strategy. He has several further papers in JRM developing his ideas. Antonio M. Lopez, Jr. A PROLOG Mastermind program. JRM 23:2 (1991) 81©93. Cites Knuth, Irving, Flood and two other papers on strategy. Kenji Koyama and Tony W. Lai. An optimal Mastermind strategy. JRM 25:4 (1994) 251-256. Using exhaustive search, they find the strategy that minimizes the expected number of guesses, getting expected number 5625/1296 = 4.340. However, the worst case in this problem requires 6 guesses. By a slight adjustment, they find the optimal strategy with worst case requiring 5 guesses and its expected number of guesses is 5626/1296 = 4.341. 10 references to previous work, not including all of the above. JÀ?Àrg Bewersdorff. GlÀGÀck, Logik und Bluff Mathematik im Spiel ©© Methoden, Ergebnisse und Grenzen. Vieweg, 1998. Section 2.15 Mastermind: Auf Nummer sicher, pp. 227©234 & Section 3.13 Mastermind: Farbcodes und Minimax, pp. 316©319. Surveys the work on finding optimal strategies. Then studies Mastermind as a two©person game. Finds the minimax strategy for the 3ÃÃ2ÄÄ game and describes Flood's approach. ÙÙ ÁÁà Ã4.B.12.ÁÁRITHMOMACHIA = THE PHILOSOPHERS' GAMEÄ Ä Ðа¤˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐÁÁI have generally not tried to include board games in any comprehensive manner, but I have recently seen some general material on this which will be useful to anyone interested in the game. The game is one of the older and more mathematical of board games, dating from c1000, but generally abandoned about the end of the 16C along with the Neo©Pythagorean number theory of Boethius on which the game was based. ÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐ Arno Borst. Das mittelalterliche Zahlenkampfspiel. Sitzungsberichten der Heidelberger Akademie der Wissenschaften, Philosophisch©historische Klasse 5 (1986) Supplemente. Available separately: Carl Winter, Heidelberg, 1986. Edits the surviving manuscripts on the game. ??NYS ©© cited by Stigter & Folkerts. Detlef Illmer, Nora GÀÀdeke, Elisabeth Henge, Helen Pfeiffer & Monika Spicker©Beck. Rhythmomachia. Hugendubel, Munich, 1987. Jurgen Stigter. Emanuel Lasker: A Bibliography AND Rithmomachia, the Philosophers' Game: a reference list. Corrected, 1988 with annotations to 1989, 1 + 15 + 16pp preprint available from the author, Molslaan 168, NL-2611 CZ Delft, Netherlands. Bibliography of the game. Jurgen Stigter. The history and rules of Rithmomachia, the Philosophers' Game. 14pp preprint available from the author, as above. Menso Folkerts. 'Rithmimachia'. In: Die deutsche Litteratur des Mittelalters: Verfasserlexikon; 2nd ed., De Gruyter, Berlin, 1990; vol. 8, pp. 86©94. Sketches history and describes the 10 oldest texts. Menso Folkerts. Die ÃÃRithmachiaÄÄ des Werinher von Tegernsee. In: Vestigia Mathematica, ed. by M. Folkerts & J. P. Hogendijk, Rodopi, Amsterdam, 1993, pp. 107©142. Discusses Werinher's work (12C), preserved in one MS of c1200, and gives an edition of it. ÙÙ ÁÁà Ã4.B.13.ÁÁMANCALA GAMESÄ Ä Ðа¤˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐÁÁThis is a very broad field and I will only mention a few early items. Four row mancala games are played in south and east Africa. Three row games are played in Ethiopia and adjacent parts of Somaliland. Two row games are played everywhere else in Africa, the Middle East and south and south©east Asia. See the standard books by R. C. Bell and Falkener for many examples. Many general books mention the game, but I only know a few specific books on the game ©© these are listed first below. ÁÁOne article says that game boards have been found in the pyramids of Khamit (©1580) and there are numerous old boards carved in rocks in several parts of Africa. ÁÁAn anonymous article, by a member of the Oware Society in London, [Wanted: skill, speed, strategy; West Africa (16©22 Sep 1996) 1486©1487] lists the following names for variants of the game: Aditoe (Volta region of Ghana), Awaoley (CÀ=Àte d'Ivoire), Ayo (Nigeria), Chongkak (Johore), Choro (Sudan), Congclak (Indonesia), Dakon (Philippines), Guitihi (Kenya), Kiarabu (Zanzibar), Madji (Benin), Mancala (Egypt), Mankaleh (Syria), Mbau (Angola), Mongola (Congo), Naranji (Sri Lanka), Qai (Haiti), Ware (Burkina Faso), Wari (Timbuktu), Warri (Antigua), ÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐ Stewart Culin. Mancala, The National Game of Africa. IN: US National Museum Annual Report 1894, Washington, 1896, pp. 595©607. Chief A. O. Odeleye. Ayo A Popular Yoruba Game. University Press Ltd., Ibadan, Nigeria, 1979. No history. Laurence Russ. Mancala Games. Reference Publications, Algonac, Michigan, 1984. Photocopy from Russ, 1995. Kofi Tall. Oware The Abapa Version. Kofi Tall Enterprise, Kumasi, Ghana, 1991. Salimata Doumbia & J. C. Pil. Les Jeux de Cauris. Institut de Recherches MathÀ)Àmatiques, 08 BP 2030, Abidjan 08, CÀ=Àte d'Ivoire, 1992. Pascal Reysset & FranÀ'Àois Pingaud. L'AwÀ)ÀlÀ)À. Le jeu des semailles africaines. 2nd ed., Chiron, Paris, 1995 (bought in Dec 1994). Not much history. FranÀ'Àois Pingaud. L'awÀ)ÀlÀ)À jeu de strategie africain. Bornemann, 1996. Alexander J. de Voogt. Mancala Board Games. British Museum Press, 1997. ??NYR. Larry (= Laurence) Russ. The Complete Mancala Games Book How to Play the World's Oldest Board Games. Foreword by Alex de Voogt. Marlowe & Co., NY, 2000. His 1984 book is described as an earlier edition of this. William Flinders Petrie. Objects of Daily Use. (1929); Aris & Phillips, London??, 1974. P. 55 & plate XLVII. ??NYS ©© described with plate reproduced in Bell, below. Shows and describes a 3 x 14 board from Memphis, ancient Egypt, but with no date given, but Bell indicates that the context implies it is probably earlier than -1500. Petrie calls it 'The game of forty©two and pool' because of the 42 holes and a large hole on the side, apparently for storing pieces either during play or between games. R. C. Bell. Games to Play. Michael Joseph (Penguin), 1988. Chap. 4, pp. 54©61, Mancala games. On pp. 54©55, he shows the ancient Egyptian board from Petrie and his own photo of a 3 x 6 board cut into the roof of a temple at Deir©el©Medina, probably about -87. Thomas Hyde. Historia Nerdiludii, hoc est dicere, Trunculorum; .... (= Vol. 2 of De Ludis Orientalibus, see 7.B for vol. 1.) From the Sheldonian Theatre (i.e. OUP), Oxford, 1694. De Ludo Mancala, pp. 226©232. Have X of part of this. R. H. Macmillan. Wari. Eureka 13 (Oct 1950) 12. 2 x 6 board with each cup having four to start. Says it is played on the Gold Coast. Vernon A. Eagle. On some newly described mancala games from Yunnan province, China, and the definition of a genus in the family of mancala games. IN: Alexander J. de Voogt, ed.; New Approaches to Board Games Research: Asian Origins and Future Perspectives; International Institute for Asian Studies, Leiden, 1995; pp. 48©62. Discusses the game in general, with many references. Attempts a classification in general. Describes six forms found in Yunnan. Ulrich SchÀÀdler. Mancala in Roman Asia Minor? Board Games Studies International Journal for the Study of Board Games 1 (1998) 10©25. Notes that mancala could have been played on a flat board of two parallel rows of squares, i.e. something like a 2 x n chessboard, but that archaeologists have tended to view such patterns as boards for race games, etc. Describes 52 examples from Asia Minor. Some general discussion of Greek and Roman games. John Romein & Henri E. Bal (Vrije Universiteit, Amsterdam). New computer cluster solves 3500©year old game. Posted on www.alphagalileo.org on 29 Aug 2002. They show that Awari is a tie game. They determined all 889,063,398,406 possible positions and stored them in a 778 GByte database. They then used a 144 processor cluster to analyse the graph, which 'only' took 51 hours. ÁÁà Ã4.B.14.ÁÁDOMINOES, ETC.Ä Ä R. C. Bell. Games to Play. 1988. Op. cit. in 4.B.13. P. 136 gives some history. The AcadÀ)Àmie FranÀ'Àais adopted the word for both the pieces and the game in 1790 and it was generally thought that they were an 18C invention. However, a domino was found on the ÃÃMary RoseÄÄ, which sank in 1545, and a record of Henry VIII (reigned 1509©1547) losing À À450 at dominoes has been found. Bell, p. 131, describes the modern variant Tri©Ominos which are triangular pieces with values at the corners. They were marketed c1970 and marked ÀÀ Pressman Toy Corporation, NY. Hexadoms are hexagonal pieces with numbers on the edges ©© opposite edges have the same numbers. These were also marketed in the early 1970s ©© I have a set made by Louis Marx, Swansea, but there is no date on it. ÁÁà Ã4.B.15.ÁÁSVOYI KOSIRIÄ Ä Anonymous [R. S. & J. M. B[rew ?]]. Svoyi kosiri is an easy game. Eureka 16 (Oct 1953) 8-12. This is an intriguing game of pure strategy commonly played in Russia and introduced to Cambridge by Besicovitch. It translates roughly as 'One's own trumps'. There are two players and the hands are exposed, with one's spades and clubs being the same as the other's hearts and diamonds. At Cambridge, the cards below 6 are removed, leaving 36 cards in the deck. The article doesn't explain how trumps are chosen, but if one has spades as trumps, then the other has hearts as trumps! Players alternate playing to a central discard pile. A player can take the pile and start a new pile with any card, or he can 'cover' the top card and then play any card on that. 'Covering' is done by playing a higher card of the same suit or one of the player's own trumps ©© if this cannot be done, e.g. if the ace of the player's own trumps has been played, the player has to take the pile. The object is to get rid of all one's cards. à Ã5.ÁÁCOMBINATORIAL RECREATIONSÄ Ä Ðа¤˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐ ÁÁ7.AZ is actually combinatorial rather than arithmetical and I may shift it. ÁÁà Ã5.A.ÁÁTHE 15 PUZZLE, ETC.Ä Ä ÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐPictorial versions: The Premier (1880), Lemon (1890), Stein (1898), King (1927). Double©sided versions: The Premier (1880), Brown (1891). Relation to Magic Squares: Loyd (1896), Cremer (1880), Tissandier (1880 & 1880?), Cassell's (1881), Hutchison (1891). Making a magic square with the Fifteen Puzzle: Dudeney (1898), Anon & Dudeney (1899), Loyd (1914), Dudeney (1917), Gordon (1988). See also:  Ollerenshaw & Bondi in 7.N. ÁÁÁÁà ÃGENERALÄ Ä Peter Hajek. 1995 report of his 1992 visit to the Museum of Money, Montevideo, Uruguay, with later pictures by Jaime Poniachik. In this Museum is a metal chest made in England in 1870 for the National State Bank of Uruguay. The front has a 7 x 7 array of metal squares with bolt heads. These have to be slid in a 12 move sequence to reveal the three keyholes for opening the chest. This opens up a whole new possible background for the 15 Puzzle ©© can anyone provide details of other such sliding devices? S&B, pp. 126-129, shows several versions of the puzzle. L. Edward Hordern. Sliding Piece Puzzles. OUP, 1986. Chap. 2: History of the sliding block puzzle, pp. 18-30. This is the most extensive survey of the history. He concludes that Loyd did not invent the general puzzle where the 15 pieces are placed at random, which became popular in 1879(?). Loyd may have invented the 14-15 version or he may have offered the $1000 prize for it, but there is no evidence of when (1881??) or where. However, see the entries for Loyd's Tit-Bits article and Dudeney's 1904 article which seem to add weight to Loyd's claims. Most of the puzzles considered here are described by Hordern and have code numbers beginning with a letter, e.g. E23, which I will give. ÁÁÁÁI contributed a note about computer techniques of solving such puzzles and hoping that programmers would attack them as computer power increased. In 1993©1995, I produced four Sliding Block Puzzle Circulars, totalling 24 pages (since reformatted to 21), largely devoted to reporting on computer solutions of puzzles in Hordern. Since then, a large number of solution programs have appeared and many more puzzles have appeared. The best place to look is on Nick Baxter's Sliding Block Home Page: http://www.johnrausch.com/slidingblockpuzzles/index.html . ÁÁÁÁà ÃEARLY ALPHABETIC VERSIONSÄ Ä Embossing Co. Puzzle labelled "No. 2 Patent Embossed puzzle of Fifteen and Magic Sixteen. Manufactured by the Embossing Co. Patented Oct 24 1865". Illustrated in S&B, p. 127. Examples are in the collections of Slocum and Hordern. Hordern, p. 25, says that searching has not turned up such a patent. Edward F. [but drawing gives E.] Gilbert. US Patent 91,737 ©© Alphabetical Instruction Puzzle. Patented 22 Jun 1869. 1p + 1p diagrams. Described by Hordern, p. 26. This is not really a puzzle ©© it has the sliding block concept, but along several tracks and with many blank spaces. I recall a similar toy from c1950. Ernest U. Kinsey. US Patent 207,124 ©© Puzzle©Blocks. Applied: 22 Nov 1877; patented: 20 Aug 1878. 2pp + 1p diagrams. Described by Hordern, p. 27. 6 x 6 square sliding block puzzle with one vacant space and tongue & grooving to prevent falling out. Has letters to spell words. He suggests use of triangular and diamond-shaped pieces. This seems to be the most likely origin of the Fifteen Puzzle craze. Montgomery Ward & Co. Catalogue. 1889. Reproduced in: Joseph J. Schroeder, Jr.; The Wonderful World of Toys, Games & Dolls 1860ÀÀÀÀ1930; DBI Books, Northfield, Illinois, 1977?, p. 34. Spelling Boards. Like Gilbert's idea, but a more compact layout. ÙÙ ÁÁÁÁà ÃLOYDÄ Ä Loyd prize puzzle: One hundred pounds. Tit©Bits (14 Oct 1893) 25 & (18 Nov 1893) 111. Loyd is described as "author of "Fifteen Puzzle," ...." Loyd. Tit-Bits 31 (24 Oct 1896) 57. Loyd asserts he developed the 15 puzzle from a 4 x 4 magic square. "[The fifteen block puzzle] had such a phenomenal run some twenty years ago. ... There was one of the periodical revivals of the ancient Hindu "magic square" problem, and it occurred to me to utilize a set of movable blocks, numbered consecutively from 1 to 16, the conditions being to remove one of them and slide the others around until a magic square was formed. The "Fifteen Block Puzzle" was at once developed and became a craze. ÁÁÁÁI give it as originally promulgated in 1872 ..." and he shows it with the 15 and 14 interchanged. "The puzzle was never patented" so someone used round blocks instead of square ones. He says he would solve such puzzles by turning over the 6 and the 9. "Sphinx" [= Dudeney] says he well remembers the sensation and hopes "Mr. Loyd is duly penitent." Dudeney. Great puzzle crazes. Op. cit. in 2. 1904. "... the "Fifteen Puzzle" that in 1872 and 1873 was sold by millions, .... When this puzzle was brought out by its inventor, Mr. Sam Loyd, ... he thought so little of it that he did not even take any steps to protect his idea, and never derived a penny profit from it.... We have recently tried all over the metropolis to obtain a single example of the puzzle, without success." Dudeney says the puzzle came with 16 pieces and you removed the 16. He also says he recently could not find a single example in London. Loyd. The 14-15 puzzle in puzzleland. Cyclopedia, 1914, pp. 235 & 371 (= MPSL1, prob. 21, pp. 19-20 & 128). He says he introduced it 'in the early seventies'. One problem asks to move from the wrong position to a magic square with sum = 30 (i.e. the blank is counted as 0). This is c= SLAHP, pp. 17-18 & 89. G. G. Bain. Op. cit. in 1, 1907. Story of Loyd being unable to patent it. Anonymous & Sam Loyd. Loyd's puzzles, op. cit. in 1, 1896. Loyd "owns up to the great sin of having invented the "15 block puzzle"", but doesn't refer to the patent story or the date. W. P. Eaton. Loc. cit. in 1, 1911. Loyd refers to it as the 'Fifteen block' puzzle, but doesn't say he couldn't patent it. Loyd Jr. SLAHP. 1928. Pp. 1-3 & 87. "It was in the early 80's, ... that the world-disturbing "14-15 Puzzle" flashed across the horizon, and the Loyds were among its earliest victims." He gives many of the stories in the Cyclopedia and two of the same problems. He doesn't mention the patent story. ÁÁÁÁà ÃTHE 15 PUZZLEÄ Ä W. W. Johnson. Notes on the 15-Puzzle ©© I. Amer. J. Math. 2 (1879) 397-399. W. E. Story. Notes on the 15-Puzzle ©© II. Ibid., 399-404. J. J. Sylvester. Editorial comment. Ibid., 404. ÁÁ(This issue may have been delayed to early 1880?? Johnson & Story are not terribly readable, but Sylvester is interesting, asserting that this is the first time that the parity of a permutation has become a popular concept.) Anonymous. Untitled editorial. New York Times (23 Feb 1880) 4. "... just now the chief amusement of the New York mind, ... a mental epidemic .... In a month from now, the whole population of North America will be at it, and when the 15 puzzle crosses the seas, it is sure to become an English mania." Anonymous. EUREKA! The Popular but Perplexing Problem Solved at Last. "THIRTEEN ª© FOURTEEN ©© FIFTEEN" New York Herald (28 Feb 1880) 8. ""Fifteen" is a puzzle of seeming simplicity, but is constructed with diabolical cunning. At first sight the victim feels little or no interest; but if he stops for a single moment to try it, or to look at any one else who is trying it, the mania strikes him. ... As to the last two numbers, it depends entirely upon the way in which the blocks happen to fall in the first place .... Two or three enterprising gamblers took up the puzzle and for a time made an excellent living.... The subject was brought up in the Academy of Sciences by the veteran scientist Dr. P. H. Vander Weyde", who showed it could not be solved. The Herald reporter discovered that the problem is solvable if one turns the board 90ÃÃoÄÄ, i.e. runs the numbers down instead of across, and Vander Weyde was impressed. The article implies the puzzle had already been widely known for some time. Mary T. Foote. US Patent 227,159 ©© Game apparatus. Filed: 4 Mar 1880; patented: 4 May 1880. 1p + 1p diagrams. The patent is for a box with sliding numbered blocks for teaching the multiplication tables. Lines 57©63: "I am aware that it is not novel to produce a game apparatus in which blocks are to be mixed and then replaced by a series of moves; also, that it is not novel to number such blocks, as in the "game of 15," so called, where the fifteen numbers are first mixed and then moved into place." Persifor Frazer Jr. Three methods and forty-eight solutions of the Fifteen Problem. Proc. Amer. Philos. Soc. 18 (1878-1880) 505-510. Meeting of 5 Mar 1880. Rather cryptic presentation of some possible patterns. Asserts his 26 Feb article in the Bulletin (??NYS ©© ??where ©© Philadelphia??) was the first "solution for the 13, 15, 14 case". J. A. Wales. 15 © 14 © 13 ©© The Great Presidential Puzzle. Puck 7 (No. 158) (17 Mar 1880) back cover. Anonymous. Editorial: "Fifteen". New York Times (22 Mar 1880) 4. "No pestilence has ever visited this or any other country which has spread with the awful celerity of what is popularly called the "Fifteen Puzzle." It is only a few months ago that it made its appearance in Boston, and it has now spread over the entire country." Asserts that an unregenerate Southern sympathiser has introduced it into the White House and thereby disrupted a meeting of President Hayes' cabinet. Sch. [H. Schubert]. The Boss Puzzle. Hamburgischer Correspondent (= Staats© und Gelehrte Zeitung des Hamburgischen unpartheyeischen Correspondent) No. 82 (6 Apr 1880) 11, with response on 87 (11 Apr 1880) 12 (Sprechsaal). Gives a fairly careful description of odd and even permutations and shows the puzzle is solvable if and only if it is in an even permutation. The response is signed X and says that when the problem is insoluble, just turn the box by 90ÃÃoÄÄ to see another side of the problem! Gebr. Spiro, Hofliefer (Court supplier), Jungfernsteig 3(?©©hard to read), Hamburg. Hamburgischer Correspondent (= Staats© und Gelehrte Zeitung des Hamburgischen unpartheyeischen Correspondent) No. 88 (13 Apr 1880) 7. Advertises Boss Puzzles: "Kaiser©Spiel 50Pf. Bismarck©Spiel 50 Pf. Spiel der 15 u. 16, 50 Pf. Spiel der 16 separat, 15 Pf. System und LÀ?Àsung, 20 Pf." G. W. Warren. Letter: Clew to the Fifteen Puzzle. The Nation 30 (No. 774) (29 Apr 1880) 326. Anon. Shavings. The London Figaro (1 May 1880) 12. "The "15 Puzzle," which has for some months past been making a sensation in New York equal to that aroused by "H. M. S. Pinafore" last year, has at length reached this country, and bids fair to become the rage here also." (Complete item!) George Augustus Sala. Echoes of the Week. Illustrated London News 76 (No. 2138) (22 May 1880) 491. Mary T. Foote. US Patent 227,159 ©© Game Apparatus. Applied: 4 Mar 1880; patented: 4 May 1880. 1p + 1p diagrams. Described in Hordern, p. 27. 3 x 12 puzzles based on multiplication tables. Refers to the "game of 15" and Kinsey. Arthur Black. ?? Brighton Herald (22 May 1880). ??NYS ©© mentioned by Black in a letter to Knowledge 1 (2 Dec 1881) 100. Anonymous. Our latest gift to England. From the London Figaro. New York Times (11 Jun 1880) 2(?). ??page The Premier. First (?) double-sided version, with pictures of Gladstone and Beaconsfield, apparently produced for the 1880 UK election. Described in Hordern, pp. 32-33 & plate I. Ahrens. MUS II 227. 1918. Story of Reichstag being distracted in 1880. P. G. Tait. Note on the Theory of the "15 Puzzle". Proc. Roy. Soc. Edin. 10 (1880) 664-665. Brief but valid analysis. Mentions Johnson & Story. First mention of the possibility of a 3D version. T. P. Kirkman. Question 6489 and Note on the solution of the 15-puzzle in question 6489. Mathematical Questions with their Solutions from the Educational Times 34 (1880) 113-114 & 35 (1881) 29-30. The question considers the n x n problem. The note is rather cryptic. (No use??) Messrs. Cremer (210 Regent St. and 27 New Bond St., London). Brilliant Melancholia. Albrecht Durer's Game of the Thirty Four and "Boss" Game of the Fifteen. 1880. Small booklet, 16pp + covers, apparently instructions to fit in a box with pieces numbered 1 to 16 to be used for making magic squares as well as for the 15 puzzle. Explains that only half the positions of the 15 puzzle are obtainable and describes them by examples. (Photo in The Hordern Collection of Hoffmann Puzzles, p. 74, and in Hordern, op. cit. above, plate IV.) Possibly written by "Cavendish" (Henry Jones). H. Schubert. Theoretische Entscheidung ÀGÀber das Boss-Puzzle Spiel. 2nd ed., Hamburg, 1880. ??NYS (MUS, II, p. 227) Gaston Tissandier. Les carrÀ)Às magiques ©© À!À propos du "Taquin," jeu mathÀ)Àmatique. La Nature 8 (No. 371) (10 Jul 1880) 81-82. Simple description of the puzzle called 'Taquin' which came from America and has had a very great success for several weeks. Says it had 16 squares and was usable as a sliding piece puzzle or a magic square puzzle. Cites FrÀ)Ànicle's 880 magic squares of order 4. Anon. & C. Henry. Gaz. Anecdotique LittÀ)Àraire, Artistique et Bibliographique. (Pub. by G. d'Heylli, Paris) Year 5, t. II, 1880, pp. 58-59 & 87-92. ??NYS Piarron de MondÀ)Àsir. Le dernier mot du taquin. La Nature 8 (No. 382) (25 Sep 1880) 284-285. Simple description of parity decision for the 15 puzzle. Says 'la Presse illustrÀ)Àe' offered 500 francs for achieving the standard pattern from a random pattern, but it was impossible, or rather it was possible in only half the cases. Jasper W. Snowdon. The "Fifteen" Puzzle. Leisure Hour 29 (1880) 493-495. Gwen White. Antique Toys. Batsford, London, 1971; reprinted by Chancellor Press, London, nd [1982?]. On p. 118, she says: "The French game of Taquin was played in 1880, in which 15 pieces had to be moved into 16 compartments in as few moves as possible; the word 'taquin' means 'a teaser'." She gives no references. Tissandier. RÀ)ÀcrÀ)Àations Scientifiques. 1880? ÁÁÁÁ2nd ed., 1881 ©© unlabelled section, pp. 143©153. As: Le taquin et les carrÀ)Às magiques; seen in 1883 ed., ??NX; 1888: pp. 208©215. Adapted from the 1880 La Nature articles of Tissandier and de MondÀ)Àsir. 1881 says it came from America ©© 'rÀ)Àcemment une nouvelle apparition', but this is dropped in 1888 ©© otherwise the two versions are the same. ÁÁÁÁTranslated in Popular Scientific Recreations, nd [c1890], pp. 731-735. Text says "Mathematical games, ..., have recently obtained a new addition .... ... from America, ...." The references to contemporary reactions are deleted and the translation is confused. E.g. the newspaper is now just "a French paper" and the English says the problem is impossible in nine cases out of ten! Lucas. RÀ)ÀcrÀ)Àations scientifiques sur l'arithmÀ)Àtique et sur la gÀ)ÀomÀ)Àtrie de situation. SixiÀ/Àme rÀ)ÀcrÀ)Àation: Sur le jeu du taquin ou du casse-tÀ+Àte amÀ)Àricain. Revue scientifique de France et de l'À)Àtranger (3) 27 (1881) 783-788. c= Le jeu du taquin, RM1, 1882, pp. 189-211. Revue says that Sylvester told him that it was invented 18 months ago by an American deaf-mute. RM1 says "vers la fin de 1878". Cf Schubert, 1895. Cassell's. 1881. Pp. 96-97: American puzzles "15" and "34". = Manson, pp. 246©248. Says "articles ... have appeared in many periodicals, but no one has ... publish[ed] a solution." Then sketches the parity concept and its application. Richard A. Proctor. The fifteen puzzle. Gentlemen's Magazine 250 (No. 1801) (1881) 30-45. "Boss". Letter: The fifteen puzzle. Knowledge 1 (11 Nov 1881) 37©38, item 13. This magazine was edited by Proctor. The letter starts: "I am told that in a magazine article which appeared some time ago, you have attempted to show that there are positions in the Fifteen Puzzle from which the won position can never be obtained." I suspect the letter was produced by Proctor. The response is signed Ed. and begins: "I thought the Fifteen Puzzle was dead, and hoped I had had some share in killing the time©absorbing monster." Notes that many people get to the position starting blank, 1, 2, 3 and view this as a win. Sketches parity argument and suggests "Boss" work on the 3 x 3 or 3 x 2 or even the 2 x 2 version. Editorial comment. The fifteen puzzle. Knowledge 1 (25 Nov 1881) 79. "I supposed every one knew the Fifteen Puzzle." Proceeds to explain, obviously in response to readers who didn't know it. Arthur Black. Letter: The fifteen puzzle. Knowledge 1 (2 Dec 1881) 100, item 80. Sketches a proof which he says he published in the Brighton Herald of 22 May 1880. "Yawnups". Letter: The fifteen puzzle. Knowledge 1 (30 Dec 1881) 185. Solution from the 15©14 position obtained by turning the box. Editorial comment says the solution uses 102 moves and the editor gets an easy solution in 57 moves. Adds that a 60 move solution has been received. Arthur Black. Letter: The fifteen puzzle. Knowledge 1 (13 Jan 1882) 230. Finds a solution from the 15©14 position in 39 moves by turning the box and asserts no shorter solution is possible. Says he also gave this in the Brighton Herald in May 1880. An addition says J. Watson has provided a similar solution, which takes 38 moves?? A. B. Letter: The fifteen puzzle. Knowledge 2 (20 Oct 1882) 345, item 598. Finds a boxªturning solution in 39 moves. C. J. Malmsten. GÀ?Àteborg Handl 1882, p. 75. ??NYS ©© cited by Ahrens in his Encyklopadie article, op. cit. in 3.B, 1904. Anonymous. Enquire Within upon Everything. Houlston and Sons, London. This was a popular book with editions almost every year ©© I don't know when the following material was added. Section 2591: Boss; or the Fifteen Puzzle, p. 363. Place the pieces 'indifferently' in the box. Half the positions are unsolvable. Cites Cavendish for the solution by turning the box 90ÃÃoÄÄ but notes this only works with round pieces. Goes on to The thirty©four puzzle, citing DÀGÀrer. I found this material in the 66th ed., 862nd thousand, of 1883, but I didn't find the material in the 86th ed of 1892. Letters received and short answers. Knowledge 4 (16 Nov 1883) 310. 'Impossible'. P. G. Tait. Listing's ÃÃTopologieÄÄ. Philosophical Mag. (5) 17 (No. 103) (Jan 1884) 30-46 & plate opp. p. 80. Section 11, p. 39. Simple but cryptic solution. Letters received and short answers. Letter from W. S. B. asks how to solve the problem when the last row has 13, 14, 15 [sic!]; Answer by Ed. points out the misprint and says the easiest solution is to remove the 15 and put it after the 14, or to invert the 6 and 9. Knowledge 6 (No. 159) (14 Nov 1884) 412 & 6 (No. 160) (21 Nov 1884) 429. Don Lemon. Everybody's Pocket Cyclopedia .... Saxon & Co., London, (1888), revised 8th ed., 1890. P. 137: The fifteen puzzle. Brief description, with pieces placed randomly in the box ©© "to get the last three into order is often a puzzle indeed". John D. Champlin & Arthur E. Bostwick. The Young Folk's Cyclopedia of Games and Sports. 1890. ??NYS Cited in Rohrbough; Brain Resters and Testers; c1935; Fifteen Puzzle, p. 20. Describes idea of parity of number of exchanges. [Another reference provided more details of Champlin & Bostwick.] Lemon. 1890. A trick puzzle, no. 202, pp. 31 & 105 (= Sphinx, no. 422, pp. 60 & 112). 15 puzzle with lines on the pieces to arrange as "a representation of a president with only one eye". The solution is a spelling of the word 'president'. Attributed to Golden Days ª© ??. After The Premier puzzle of c1880, this is the second suggestion of using a picture and the first publication of the idea that I have seen. G. A. Hutchison, ed. Indoor Games and Recreations. The Boy's Own Bookshelf. (1888); New ed., Religious Tract Society, London, 1891. (See M. Adams; Indoor Games for a much revised version, but which doesn't contain this material.) Chap. 19: The American Puzzles., pp. 240-241. "These puzzles, known as the 'Thirty-four Game' and the 'Fifteen Game,' on their introduction amongst us some years ago ...." "The '15' puzzle would appear to have been, on its coming to England a few years ago, strictly a new introduction ...." He sketches the parity concept. [NOTE. I have seen a reference to the editor as Hutchinson, but the book definitely omits the first n.] Daniel V. Brown. US Patent 471,941 ©© Puzzle. Applied: 23 Apr 1891; patented: 29 Mar 1892, 2pp + 1p diagrams. Double©sided 16 block puzzle to spell George Washington on one side and Benjamin Harrison on the other. No sliding involved. Berkeley & Rowland. Card Tricks and Puzzles. 1892. American fifteen puzzle, pp. 105©107. "The Fifteen Puzzle was introduced by a shrewd American some ten years ago, ...." Refers to Tait's 1880 paper. Says half the positions are impossible, but solves them by turning the box 90ÃÃoÄÄ or by inverting the 6 and the 9. Hoffmann. 1893. Chap IV, no. 69: The "Fifteen" or "Boss" puzzles, pp. 161-162 & 217-218 = Hoffmann©Hordern, pp. 142©144, with photo of five early examples, two or three of which also are thirty©four puzzles. (Hordern Collection, p. 74, has a photo of a version by Cremer, cf above.) "This, like a good many of the best puzzles, hails from America, where, some years ago, it had an extraordinary vogue, which a little later spread to this country, the British public growing nearly as excited over the mystic "Fifteen" as they did at a later date over the less innocent "Missing Word" competitions." He distinguishes between the ordinary Fifteen where one puts the pieces in at random, and the Boss or Master puzzle which has the 14 and 15 reversed. "Notwithstanding the enormous amount of energy that has been expended over the "Fifteen" Puzzle, no absolute rule for its solution has yet been discovered and it appears to be now generally agreed by mathematicians that out of the vast number of haphazard positions ... about half admit [of solution]. To test whether ... the following rule has been suggested." He then says to count the parity of the number of transpositions. Hoffmann. 1893. Chap. IV, no. 70: The peg-away puzzle, pp. 163 & 218 = Hoffmann-Hordern, p. 145. This is a 3 x 3 version of the Fifteen puzzle, made by Perry & Co. Start with a random pattern and get to standard form. "The possibility of success in solving this puzzle appears to be governed by precisely the same rule as the "Fifteen" Puzzle." Hoffmann©Hordern has no photo of this ©© do any examples exist?? H. Schubert. ZwÀ?Àlf Geduldspiele. DÀGÀmmler, Berlin, 1895. [Taken from his columns in Naturwissenschaftlichen Wochenschrift, 1891©1894.] Chap. VII: Boss©Puzzle oder FÀGÀnfzehner©Spiel, pp. 75©94?? Pp. 75©77 sketches the history, saying it was called "Jeu du Taquin" (Neck©Spiel) in France and was popular in 1879©1880 in Germany. Cites Johnson & Story and his own 1880 booklet. Gives the story of a deaf and dumb American inventing it in Dec 1878, saying "Sylvester communicated this at the annual meeting of the Association FranÀ'Àaise pour l'Avancement des Sciences at Reims". Cf Lucas, 1881. [There is a second edition, Teubner??, Leipzig, 1899, ??NYS. However this material is almost identical to the beginning of Chap. 15 in Schubert's Mathematische Mussestunden, 3rd ed., GÀ?Àschen, Leipzig, 1909, vol. 2. The later version omits only some of the Hamburg details of 1879©1880. Hence the 2nd ed. of ZwÀ?Àlf Geduldspiele is probably very close to these versions.] Dudeney. Problem 49: The Victoria Cross puzzle. Tit-Bits 32 (4 & 25 Sep 1897) 421 & 475. = AM, 1917, prob. 218, pp. 60 & 194. B7. 3 x 3 board with letters Victoria going clockwise around the edges, leaving the middle empty, and starting with V in a corner. Slide to get Victoria starting at an edge cell, in the fewest moves. Does it in 18 moves, by interchanging the i's and says there are 6 such solutions. Dudeney. Problem 65: The Spanish dungeon. Tit-Bits 33 (1 Jan & 5 Feb 1898) 257 & 355. = AM, 1917, prob. 403, pp. 122©123 & 244. B14. Convert 15 Puzzle, with pieces in correct order, into a magic square. Does it in 37 moves. Conrad F. Stein. US Design 29,649 ©© Design for a Game©Board. Applied: 29 Sep 1898; patented: 8 Nov 1898 as No. 692,242. 1p + 1p diagrams. This appears to be a 3 x 4 puzzle with a picture of a city with a Spanish flag on a tower. Apparently the object is to move an American flag to the tower. Anon. & Dudeney. A chat with the Puzzle King. The Captain 2 (Dec? 1899) 314©320; 2:6 (Mar 1900) 598©599 & 3:1 (Apr 1900) 89. The eight fat boys. 3 x 3 square with pieces: 1 2 3; 4 X 5; 6 7 8 to be shifted into a magic square. Two solutions in 19 moves. Cf Dudeney, 1917. Addison Coe. US Patent 785,665 ©© Puzzle or Game Apparatus. Applied: 17 Nov 1904; patented: 1 Mar 1905. 4pp + 3pp diagrams. Gives a 3 x 5 flat version and a 3-dimensional version ©© cf 5.A.2. Dudeney. AM. 1917. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐProb. 401: Eight jolly gaol birds, pp. 122 & 243. E23. Same as 'The eight fat boys' (see Anon. & Dudeney, 1899) with the additional condition that one person refuses to move, which occurs in one of the two previous solutions. Prob. 403: The Spanish dungeon, pp. 122©123 & 244. = Tit©Bits prob. 65 (1898). B14. Prob. 404: The Siberian dungeons, pp. 123 & 244. B16. 2 x 8 array with prisoners 1, 2, ..., 8 in top row and 9, 10, ..., 16 in bottom row. Two extra rows of 4 above the right hand end (i.e. above 5, 6, 7, 8) are empty. Slide the prisoners into a magic square. Gives a solution in 14 moves, due to G. Wotherspoon, which they feel is minimal. This allows long moves ©© e.g. the first move moves 8 up two and left 3. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐ"H. E. Licks" [pseud. of Mansfield Merriman]. Recreations in Mathematics. Van Nostrand, NY, 1917. Art. 28, pp. 20-21. 'About the year 1880 ... invented in 1878 by a deaf and dumb man....' ÁÁÁÁ[From sometime in the 1980s, I suspected the author's name was a pseudonym. On pp. 132©138, he discusses the Diaphote Hoax, from a Pennsylvania daily newspaper of 10 Feb 1880, which features the following people: H. E. Licks, M. E. Kannick, A. D. A. Biatic, L. M. Niscate. The diaphote was essentially a television. He says this report was picked up by the ÃÃNew York TimesÄÄ and the ÃÃNew York WorldÄÄ. An email from Col. George L. Sicherman on 5 Jun 2000 agrees that the name is false and suggested that the author was "the eminent statistician Mansfield Merriman" who wrote the article on The Cattle Problem of Archimedes in ÃÃPopular Science MonthlyÄÄ (Nov 1905), which is abridged on pp. 33©39 of the book, but omitting the author's name. Sichermann added that Merriman was one of the authors of Pillsbury's List. William Hartston says this was an extraordinary list of some 30 words which Pillsbury, who did memory feats, was able to commit to memory quite rapidly. Sicherman continued to investigate Merriman and got Prof. Andri Lange interested and Lange corresponded with a James A. McLennan, author of a history of the physics department at Lehigh University where Merriman had been. McLennan found Merriman's obituary from the American Society of Civil Engineers which states that Merriman used H. E. Licks as a pseudonym. [Email from Sicherman on 25 Feb 2002.]] Stephen Leacock. Model Memoirs and Other Sketches from Simple to Serious. John Lane, The Bodley Head, 1939, p. 300. "But this puzzle stuff, as I say, is as old as human thought. As soon as mankind began to have brains they must have loved to exercise them for exercise' sake. The 'jig©saw' puzzles come from China where they had them four thousand years ago. So did the famous 'sixteen puzzle' (fifteen movable squares and one empty space) over which we racked our brains in the middle eighties." G. Kowalewski. Boss-Puzzle und verwandte Spiele. K. F. Kohler Verlag, Leipzig, 1921 (reprinted 1939). Gives solution of general polygonal versions, i.e. on a graph with a Hamilton circuit and one or more diagonals. ÐФ˜Œ € th\ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐHummerston. Fun, Mirth & Mystery. 1924. 1 2ÁÁÁÁÁÁÁÁ 9ÁÁÁÁÁÁÁÁPush, pp. 22 & 25. This is played on the board 3 10 11 4ÁÁÁÁshown at the left with its orthogonal lines, like 12 13ÁÁÁÁÁÁ3, 10, 11, 4, and its diagonal lines, like 5 14 15 6ÁÁÁÁ1, 9, 11, 13, 6. 10, 15 and 11, 14 are ÃÃnotÄÄ 16ÁÁÁÁÁÁÁÁconnected, so this is an octagram. Take 16 7 8ÁÁÁÁÁÁÁÁnumbered counters and place them at random on ÁÁÁÁÁÁÁÁÁÁÁÁÁÁthe board and remove counter 16. Move the pieces ÁÁÁÁto their correct locations. He asserts that 'unlike the original ["Sixteen" Puzzle], ÁÁÁÁno position can be set up in "Push" that cannot be solved'. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐThe six bulls puzzle, Puzzle no. 34, pp. 90 & 177. This uses the 2 x 3 + 1 Á`Á0 ÁÁboard shown at the right, where the 0 is the blank space. ExchangeÁ`Á1 2 3 ÁÁ3 and 6 and 4 and 5. He does it in 20 moves. [This is Hordern'sÁ`Á4 5 6 ÁÁB3, first known from 1977 under the name Bull Pen, but is a variant of ÁÁHordern's B2, first known from 1973.] Q. E. D. ©© The sergeant's problem, Puzzle no. 40, pp. 106 & 178. Take a 2 x 3 board, with the centre of one long side blank. Interchange the men along one short side. He does this in 17 moves, but the blank is not in its initial position nor are the other men. [This is Hordern's B1, first known from Loyd's Cyclopedia, 1914.] Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐKing. Best 100. 1927. No. 26, p. 15. = Foulsham's, no. 9, pp. 7 & 10. "An entertaining variation ... is to draw, and colour, if you like, a small picture; then cut it into sixteen squares and discard the lower right hand square." G. Kowalewski. Alte und neue mathematische Spiele. Teubner, Leipzig, 1930, pp. 61-81. Gives solution of general polygonal versions. Dudeney. PCP. 1932. The Angelica puzzle, prob. 253, pp. 76 & 167. = 435, prob. 378, pp. 136 & 340. B8. 3 x 3 problem ©© convert: A C I L E G N A X to A N G E L I C A X. Requires interchanging the As. Solution in 36 moves. In the answer in 435, Gardner notes that it can be done in 30 moves. H. V. Mallison. Note 1454: An array of squares. MG 24 (No. 259) (May 1940) 119-121. Discusses 15 Puzzle and says any legal position can be achieved in at most about 150 moves. But if one fixes cells 6, 7, 11, then a simple problem requires about 900 moves. McKay. At Home Tonight. 1940. Prob. 44: Changing the square, pp. 73 & 88. In the usual formation, colour the pieces alternately blue and red, as on a chessboard, with the blank at the lower right position 16 being a missing red, so there are 7 reds. Move so the colours are still alternating but the blank is at the lower left, i.e. position 13. Takes 15 moves. Sherley Ellis Stotts. US Patent 3,208,753 ©© Shiftable Block Puzzle Game. Filed: 7 Oct 1963; patented: 28 Sep 1965. 4pp + 2pp diagrams. Described in Hordern, pp. 152©153, F10-12. Rectangular pieces of different sizes. One can also turn a piece. Gardner. SA (Feb 1964) = 6th Book, chap. 7. Surveys sliding©block puzzles with non©square pieces and notes there is no theory for them. Describes a number of early versions and the minimum number of moves for solution, generally done by hand and then confirmed by computer. Pennant Puzzle, C19; L'ÀÀne Rouge, C27d; Line Up the Quinties, C4; Ma's Puzzle, D1; a form of Stotts' Baby Tiger Puzzle, F10. Gardner. SA (Mar & Jun 1965) c= 6th Book, chap. 20. Prob. 9: The eight©block puzzle. B5. 3 x 3 problem ©© convert: 8 7 6 5 4 3 2 1 X to 1 2 3 4 5 6 7 8 X. Compares it with Dudeney's Angelica puzzle (1932, B8) but says it can be done if fewer than 36 moves. Many readers found solutions in 30 moves; two even found all 10 minimal solutions by hand! Says Schofield (see next entry) has been working on this and gives the results below, but this did not quite resolve Gardner's problem. William F. Dempster, at Lawrence Radiation Laboratory, programmed a IBM 7094 to find all solutions, getting 10 solutions in 30 moves; 112 in 32 moves and 512 in 34 moves. Notes it is unknown if any problem with the blank in a side or corner requires more than 30 moves. (The description of Schofield's work seems a bit incorrect in the SA solution, and is changed in the book.) Peter D. A. Schofield. Complete solution of the 'Eight-Puzzle'. Machine Intelligence 1 (1967) 125-133. This is the 3 x 3 version of the 15 Puzzle, with the blank space in the centre. Works with the corner twists which take the blank around a 2 x 2 corner in four moves. Shows that the 5©puzzle, which is the 3 x 2 version, has every position reachable in at most 20 moves, from which he shows that an upper bound for the 8©puzzle is 48 moves. Since the blank is in the middle, the 8!/2 = 20160 possible positions fall into 2572 equivalence classes. He also considers having inverse permutations being equivalent, which reduces to 1439 classes, but this was too awkward to implement. An ATLAS program found that the maximum number of moves required was 30 and 60 positions of 12 classes required this maximum number, but no example is given ©© but see previous entry. A. L. Davies. Rotating the fifteen puzzle. MG 54 (No. 389) (Oct 1970) 237-240. Studies versions where the numbers are printed diagonally so one can make a 90ÃÃoÄÄ turn of the puzzle. Then any pattern can be brought to one of two 'natural' patterns. He then asks when this is true for an m x n board and obtains a complicated solution. For an n x n board, n must be divisible by 4. R. M. Wilson. Graph puzzles, homotopy and the alternating group. J. Combinatorial Thy., Ser. B, 16 (1974) 86-96. Shows that a sliding block puzzle, on any graph of n + 1 points which is non-separable and not a cycle, has at least AÃÃnÄÄ as its group ©© except for one case on 7 points. Alan G. & Dagmar R. Henney. Systematic solutions of the famous 15-14 puzzles. Pi Mu Epsilon J. 6 (1976) 197-201. They develop a test-value which significantly prunes the search tree. Kraitchik gave a problem which took him 114 moves ©© the authors show the best solution has 58 moves! David Levy. Computer Gamesmanship. Century Publishing, London, 1983. [Most of the material appeared in Personal Computer World, 1980-1981.] Pp. 16-29 discusses 8-puzzle and uses the Henney's test-value as an evaluation function. Cites Schofield. Nigel Landon & Charles Snape. A Way with Maths. CUP, 1984. Cube moving, pp. 23 & 46. Consider a 9©puzzle in the usual arrangement: 1 2 3, 4 5 6, 7 8 x. Move the 1 to the blank position in the minimal number of moves, ignoring what happens to the other pieces. Generalise. Their answer only says 13 is minimal for the 3 x 3 board. ÁÁÁÁMy student Tom Henley asked me the m x n problem in 1993 and gave a conjectural minimum, which I have corrected to: if m = n, then it can be done in 8m - 11 moves; but if n < m, then it can be done in 6m + 2n © 13 moves, using a straightforward method. However, I don't see how to show this is minimal, though it seems pretty clear that it must be. I call this a one©piece problem. See also Ransom, 1993. Len Gordon. Sliding the 15-1 [sic, but 15-14 must have been meant] puzzle to magic squares. G&PJ 4 (Mar 1988) 56. Reports on computer search to find minimal moves from either ordinary or 15-14 forms to a magic square. However, he starts with the blank before the 1, i.e. as a 0 rather than a 16. Leonard J. Gordon. The 16-15 puzzle or trapezeloyd. G&PJ 10 (1989) 164. Introduces his puzzle which has a trapezoidal shape with a triangular wedge in the 2nd and 3rd row so the last row can hold 5 pieces, while the other rows hold four pieces. Reversing the last two pieces can be done in 85 moves, but this may not be minimal. George T. Gilbert & Loren C. Larson. A sliding block problem. CMJ 23:4 (Sep 1992) 315-319. Essentially the same results as obtained by R. Wilson (1974). Guy points this out in 24:4 (Sep 1993) 355©356. P. H. R. [Peter H. Ransom]. Adam's move. Mathematical Pie 128 (Spring 1993) 1017 & Notes, p. 3. Considers the one piece problem of Langdon & Snape, 1984. Solution says the minimal solution on a n x n board is 8n © 11, but doesn't give the answer for the m x n board. Bernhard Wiezorke. Sliding caution. CFF 32 (Aug 1993) 24©25 & 33 (Feb 1994) 32. In 1986, the German games company ASS (Altenburg Stralsunder Spielkarten AG) produced a game called Vorsicht (= Caution). Basically this is a 3 x 3 board considered as a doubly crossed square. It has pieces marked with + or x. The + pieces can only move orthogonally; the x pieces can only move diagonally. The pieces are coloured and eight are placed on the board to be played as a sliding piece puzzle from given starts to given ends. The diagonal moves are awkward to make and Wiezorke suggests the board be spread out enough for diagonal moves to be made. A note at the end says he has received two similar games made by Y. A. D. Games in Israel. Bala Ravikumar. The Missing Link and the Top©Spin. Report TR94©228, Department of Computer Science and Statistics, University of Rhode Island, Jan 1994. The Missing Link is a cylindrical form of the Fifteen Puzzle, with four layers and four pieces in each layer. The middle two layers are rigidly joined, but that makes little difference in solving the puzzle. After outlining the relevant group theory and solving the Fifteen Puzzle, he shows the state space of the Missing Link is SÃÃ15ÄÄ. Richard E. Korf & Ariel Felner. Disjoint pattern database heuristics. Artificial Intelligence 134 (2002) 9©22. Discusses heuristic methods of solving the Fifteen Puzzle, Rubik's Cube, etc. The authors applied their method to 1000 random positions of the Fifteen Puzzle. The optimal solution length averaged 52.522 and the average time required was 27 msec. They also did 50 random positions of the Twenty©Four Puzzle and found an average optimal solution length of 100.78, with average time being two days on a 440MHz machine. à ÃÁÁ5.A.1.ÁÁNON-SQUARE PIECESÄ Ä S&B, pp. 130-133, show many versions. See Kinsey, 1878, above, for mention of triangular and diamond-shaped pieces. Henry Walton. US Patent 516,035 ©© Puzzle. Applied: 14 Mar 1893; patented: 6 Mar 1894. 1p + 1p diagrams. Described in Hordern, pp. 27 & 68-69, C1. 4 x 4 area with five 1 x 2 & two 2 x 1 pieces. Lorman P. Shriver. US Patent 526,544 ©© Puzzle. Applied: 28 Jun 1894; patented: 25 Sep 1894, 2pp + 1p diagrams. Described in Hordern, p. 27. 4 x 5 area with two 2 x 1 & 15  1 x 1 pieces. Because there is only one vacant space, the rectangles can only move lengthwise and so this is a dull puzzle. Frank E. Moss. US Patent 668,386 ©© Puzzle. Applied: 8 Jun 1900; patented: 19 Feb 1901. 2pp + 1p diagrams. Described in Hordern, pp. 27-28 & 75, C14. 4 x 4 area with six 1 x 1, two 1 x 2 & two  2 x 1 pieces, allowing sideways movement of the rectangles. William H. E. Wehner. US Patent 771,514 ©© Game Apparatus. Applied: 15 Feb 1904; patented: 4 Oct 1904. 2pp + 1p diagrams. First to use L©shaped pieces. Described in Hordern, pp. 28 & 107, D5. Lewis W. Hardy. US Patent 1,017,752 ©© Puzzle. Applied: 14 Dec 1907; patented: 20 Feb 1912. 3pp + 1p diagrams. Described in Hordern, pp. 29 & 89-90, C43©45. 4 x 5 area with one 2 x 2, two 1 x 2, three 2 x 1 & four 1 x 1 pieces. L. W. Hardy. Pennant Puzzle. Copyright 1909. Made by OK Novelty Co., Chicago. No known patent. Described in Gardner, SA (Feb 1964) = 6th Book, chap. 7 and in Hordern, pp. 28-29 & 78©79, C19. 4 x 5 area with one  2 x 2, two 1 x 2, four 2 x 1, two  1 x 1 pieces. Nob Yoshigahara designed Rush Hour in the late 1970s and it was produced in Japan as Tokyo Parking Lot. Binary Arts introduced it to the US in 1996 and it became very popular. Winning Ways. 1982. Pp. 769©777: A trio of sliding block puzzles. This covers Dad's Puzzler (c19, with piece 8 moved two places to the right), The Donkey (C27d, with all the central pieces moved down one position) and The Century (C42), showing how one can examine partial problems which allow one to consider many positions the same and much reduce the number of positions to be studied. This allows the graph to be written on a large sheet and solutions to be readily found. Andrew N. Walker. Checkmate and other sliding©block puzzles. Mathematics Preprint Series, University of Nottingham, no. 95©32, 1995, 8pp + covers. Describes a version by W. G. H. [Wil] Strijbos made by Pussycat. 4 x 4 with an extra position below the left column. Pieces are alternately black and white and have a black king, a white king and a white rook on them and the object is to produce checkmate, but all positions must be legal in chess, except that the black and white markings do not have to be correct in the intermediate positions. However, one soon finds that one tile is fixed in place and two other tiles are joined together. He discusses general computer solving techniques and finds there are five optimal solutions in 68 moves. He then discusses other problems, citing Winning Ways, Hordern and my Sliding Block Puzzle Circulars. He gives the UNIX shell scripts that he used. Ivars Peterson. Simple puzzles can give computers an unexpectedly strenuous workout. Science News 162:7 (17 Aug 2002) 6pp PO from their website, http:''sciencenews.org . Reports on recent work by Gary W. Flake & Eric B. Baum that Nob Yoshigahara's Rush Hour puzzle is PSPACE complete, but is not polynomial time. Robert A. Hearn and Erik D. Demaine have verified and extended this, showing other sliding block puzzles are PSPACE complete, including the case where all pieces are dominoes and can slide sideways as well as front and back. à ÃÁÁ5.A.2.ÁÁTHREE DIMENSIONAL VERSIONSÄ Ä See Hordern, pp. 27, 156-160 & plates IX & X. P. G. Tait. Note on the Theory of the "15 Puzzle". Proc. Roy. Soc. Edin. 10 (1880) 664-665. "... conceivable, but scarcely realisable ..." Charles I. Rice. US Patent 416,344 ©© Puzzle. Applied: 9 Sep 1889; patented: 3 Dec 1889. 2pp + 1p diagrams. Described in Hordern, pp. 27 & 157-158, G2. 2 x 2 x 2 version with peepholes in the faces. Ball. MRE, 1st ed., 1892, p. 78. Mentions possibility. Hoffmann. 1893. Chap. X, No. 1: The John Bull political puzzle, pp. 331 & 357©358 = Hoffmann©Hordern, pp. 215©216. A 3 x 3 board in the form of a cylinder, with an extra cell attached to one bottom cell. Pieces can move back and forth around each level, but the connections from one level to the next are all parallel to one of the diagonals ©© though this isn't really a complication compared to having vertical connections. The pieces have two markings: three colours and three letters. When they are randomly placed on the board, you have to move them so they form a pair of orthogonal 3 x 3 Latin squares. Fortunately there are such arrangements which differ by an odd permutation, so the puzzle can be solved from any random starting point. Two examples done. Says the game is produced by Jaques & Son. Addison Coe. US Patent 785,665 ©© Puzzle or Game Apparatus. Applied: 17 Nov 1904; patented: 1 Mar 1905. 4pp + 3pp diagrams. Mentioned in Hordern, pp. 158-159, G3. Gives a 3 x 5 flat version and a 3 x 3 x 3 cubical version with 3 x 3 arrays of holes in the six faces (in order to push the pieces) and a 3 x 5 cylindrical version. Burren Loughlin & L. L. Flood. Bright©Wits Prince of Mogador. H. M. Caldwell Co., NY, 1909. The nine disks, pp. 29©34 & 60. Same as Hoffmann except pieces have colour and shape. Guy thinks Hein patented Bloxbox, but I have not found any US patent of it ©© ??CHECK. Gardner. SA (Feb 1973). First mention of Hein's Bloxbox. Daniel Kosarek. US Patent 3,845,959 ©© Three©Dimensional Block Puzzle. Filed: 14 Nov 1973; patented: 5 Nov 1974. 3pp + 1p diagrams (+ 1p abstract). Mentioned in Hordern, pp. 158-159, G3. 3 x 3 x 3 box with 3 x 3 array of portholes on each face. Mentions 4 x 4 x 4 and larger versions. Gabriel Nagorny. US Patent 4,428,581 ©© Tri©dimensional Puzzle. Filed: 16 Jun 1981; patented: 31 Jan 1984. Cover page + 3pp + 3pp diagrams. Three dimensional sliding cube puzzles with central pieces joined together. A 3 x 3 x 3 version was made in Hungary and marketed as a Varikon Box. Inventor's address is in France and he cites earlier French applications of 19 Jun 1980 and 19 Nov 1980. He also describes a 3 x 4 x 4 version with the central areas of each face joined to a 1 x 2 x 2 block in the middle. à ÃÁÁ5.A.3.ÁÁROLLING PIECE PUZZLESÄ Ä Ðа¤˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐÁÁHere one has a set of solid pieces in a tray and one tilts or rolls a piece into the blank space. ÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐ Thomas Henry Ward. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐUK Patent 2,870 ©© Apparatus for Playing Puzzle or Educational Games. Provisional: 8 Jun 1883; Complete as: An Improved Apparatus to be Employed in Playing Puzzle or Educational Games, 6 Dec 1883. 3pp + 1p diagrams. US Patent 287,352 ©© Game Apparatus. Applied: 13 Sep 1883; patented: 23 Oct 1883. 1p + 1p diagrams. Hexagonal board of 19 triangles with 18 tetrahedra to tilt. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐGeorge Mitchell & George Springfield. UK Patent 6867 ©© A novel puzzle, and improvements in the construction of apparatus therefor. Applied: 16 Mar 1897; accepted: 5 Jun 1897. 2pp + 1p diagrams. Rolling cubes puzzle, where the cube faces are hollowed and fit onto domes in the tray. Basic form has four cubes in a row with two extra spaces above the middle cubes, but other forms are shown. Sven Bergling invented the rolling ball labyrinth puzzle/game and they began to be produced in 1946. [Kenneth Wells; Wooden Puzzles and Games; David & Charles, Newton Abbot, 1983, p. 114.] Ronald Sprague. Unterhaltsame Mathematik. Vieweg, Braunschweig, 1961. Translated by T. H. O'Beirne as: Recreations in Mathematics, Blackie, London, 1963. Problem 3: Schwere Kiste, pp. 3©4 & 22©23 (= Heavy boxes, pp. 4©5 & 25©26). Three problems with 5 boxes some of which are so heavy that one has to tilt or roll them. Gardner. SA (Dec 1963). = Sixth Book, chap. 8. Gives Sprague's first problem. Gardner. SA (Nov 1965). c= Carnival, chap. 9. Prob. 1: The red©faced cube. Two problems of John Harris involving one cube with one red face rolling on a chessboard. Gardner says that the field is new and that only Harris has made any investigations of the problem. The book chapter cites Harris's 1974 article, below, and a 1971 board game called Relate with each player having four coloured cubes on a 4 x 4 board. Charles W. Trigg. Tetrahedron rolled onto a plane. JRM 3:2 (Apr 1970) 82©87. A tetrahedron rolled on the plane forms the triangular lattice with each cell corresponding to a face of the tetrahedron. He also considers rolling on a mirror image tetrahedron and rolling octahedra. John Harris. Single vacancy rolling cube problems. JRM 7:3 (1974) 220©224. This seems to be the first appearance of the problem with one vacant space. He considers cubes rolling on a chessboard. Any even permutation of the pieces with the blank left in place is easily obtained. From the simple observation that each roll is an odd permutation of the pieces and an odd rotation of the faces of a cube, he shows that the parity of the rotation of a cube is the same as the parity of the number of spaces it has moved. He shows that any such rotation can be achieved on a 2 x 3 board. Rotating one cube 120ÃÃoÄÄ about a diagonal takes 32 moves. If the blank is allowed to move, the the parity of the permutation of the pieces is the parity of the number of spaces the blank moves, but each cube still has to have the parity of its rotation the same as the parity of the number of spaces it has moved. If the identical pieces are treated as indistinguishable, the parity of the permutation is only shown by the location of the blank space. He suggests the use of ridges on the board so that the cube will roll automatically ©© this was later used in commercial versions. He gives a number of problems with different colourings of the cubes. Gardner. SA (Mar 1975). = Time Travel, chap. 9. Prob. 8: Rolling cubes. This is the first of Harris's problems. Computer analysis has found that it can be done in fewer moves than Harris had. Gardner also reports on the last of Harris's problems, which has also been resolved by computer. A 3 x 3 array with 8 coloured cubes was available from Taiwan in the early 1980s. It was called Color Cube Mental Game ©© I called it 'Rolling Cubes'. The cubes had thick faces, producing grooved edges which fit into ridges in the bottom of the plastic frame, causing automatic rolling quite nicely. I wonder if this was inspired by Harris's article. John Ewing & Czes KoÀ¯Àniowski. Puzzle it Out ©© Cubes, Groups and Puzzles. CUP, 1982. The 8 Cubes Puzzle, pp. 58©59. Analysis of the Rolling Cubes puzzle. The authors show how to rotate a single cube about a diagonal in 36 moves. ÁÁà Ã5.A.4.ÁÁPANEX PUZZLEÄ Ä Ðа¤˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐÁÁInvented by Toshio Akanuma (??SP). Manufactured by Tricks Co., Japan, in 1983. Described in Hordern, pp. 144©145 & 220, E35, and in S&B, p. 135. This looks like a Tower of Hanoi (cf 7.M.2) with two differently coloured piles of 10 pieces on the outside two tracks of three tracks of height 12 joined like a letter E. This is made as a sliding block puzzle, but with blockages ©© a piece cannot slide down a track further than its original position. ÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐ Mark Manasse, Danny Sleator & Victor K. Wei. Some Results on the Panex Puzzle. Preprint sent by Jerry Slocum, 23pp, nd [1983, but S&B gives 1985]. For piles of size n, the minimum number of moves, T(n), to move one pile to the centre track is determined by means of a 2nd order, non©homogeneous recurrence which has different forms for odd and even n. Compensating for this leads to a 2nd order non-homogeneous recurrence, giving T(10) = 4875 and T(n) À À  C(1 + ÀÀ2)ÃÃnÄÄ. This solution doesn't ever move the other pile. The minimum number of moves, X(n), to exchange the piles is bounded above and below and determined exactly for n ÀÀ 6 by computer search. X(5) = 343, compared to bounds of 320 and 343. X(6) = 881, compared to the bounds of 796 and 881. For n = 7, the bounds are 1944 and 2189, For n = 10, the bounds are 27,564 and 31,537. The larger bounds are considered as probably correct. Christoph Hausammann. US Patent 5,261,668 ©© Logic Game. Filed: 6 Aug 1992; patented: 16 Nov 1993. 1p abstract + 2pp text + 3pp diagrams. Essentially identical to Panex. Vladimir Dubrovsky. Nesting Puzzles ©© Part I: Moving oriental towers. Quantum 6:3 (Jan/Feb 1996) 53©59 & 49©51. Says Panex was produced by the Japanese Magic Company in the early 1980s. Discusses it and cites S&B for the bounds given above. Sketches a number of standard configurations and problems, leading to "Problem 9. Write out a complete solution to the Panex puzzle." He says his method is about 1700 moves longer than the upper bound given above. Nick Baxter. Recent results for the Panex Puzzle. 4pp handout at G4G5, 2002. Describes the puzzle and its history. David Bagley wrote a program to implement the Manasse, Sleator & Wei methods. On 7 Feb 2002, this confirmed the conjecture that X(7) = 2189. On 26 Mar 2002, it obtained X(8) = 5359, compared to bounds of 4716 and 5359. It is estimated that the cases n = 9 and 10 will take 10 and 1200 years! If Moore's Law on the increase of computing power continues for another 20 years, the latter answer may be available by then. He gives a simplified version of the algorithm for the upper bound, which gets 31,544 for n = 10. He has a Panex page: www.baxterweb.com/puzzles/panex/ and will be publishing an edited and annotated version of the Manasse, Sleator & Wei paper on it. ÁÁà Ã5.B.ÁÁCROSSING PROBLEMSÄ Ä See MUS I 1©13, Tropfke 658 and also 5.N. Wolf, goat and cabbages: Alcuin, Abbot Albert, Columbia Algorism, Munich 14684, Folkerts, Chuquet, Pacioli, Tartaglia, van Etten, Merry Riddles, Ozanam, Dilworth, Wingate/Dodson, Jackson, Endless Amusement II, Boy's Own Book, Nuts to Crack, Taylor; The Riddler, Child, Fireside Amusements, Magician's Own Book, Book of 500 Puzzles, Boy's Own Conjuring Book, Secret Out (UK), Mittenzwey, Carroll 1873, Kamp, Carroll 1878, Berg, Lemon, Hoffmann, Brandreth Puzzle Book, Carroll 1899, King, Voggenreiter, Stein, Stong, Zaslavsky, Ascher, Weismantel (a film), Verse version: Taylor, Version with only one pair of incompatibles: Voggenreiter Extension to four items: Gori, Phillips, M. Adams, Gibbs, Ascher Adults and children: Alcuin, Kamp, Hoffmann, Parker?, Voggenreiter, Gibbs Three jealous husbands: Alcuin, Abbot Albert, Columbia Algorism, Munich 14684, Folkerts, Rara, Chuquet, Pacioli, Cardan, Tartaglia, H&S - Trenchant, Gori, Bachet, van Etten, Wingate/Kersey, Ozanam, MinguÀ)Àt, Dilworth, Les Amusemens, Wingate/Dodson, Jackson, Endless Amusement II, Nuts to Crack, Young Man's Book, Family Friend, Magician's Own Book, The Sociable, Book of 500 Puzzles, Boy's Own Conjuring Book, Vinot, Secret Out (UK), Lemon, Hoffmann, Fourrey, H. D. Northrop, Mr. X, Loyd, Williams, Clark, Goodstein, O'Beirne, Doubleday, Allen, Verse mnemonic: Abbot Albert, Munich 14684, Verse solution: Ozanam, Vinot, Four or more jealous husbands: Pacioli, Filicaia, Tartaglia, Bachet, Delannoy, Ball, Carroll©Collingwood, Dudeney, O'Beirne Jealous husbands, with island in river: De Fontenay, Dudeney, Ball, Loyd, Dudeney, Pressman & Singmaster Missionaries and cannibals: Jackson, Mittenzwey, Cassell's, Lemon, Pocock, Hoffmann, Brandreth Puzzle Book, H. D. Northrop, Schubert, Arbiter, H&S, Abraham, Bile Beans, Goodstein, Beyer, O'Beirne, Pressman & Singmaster. With only one cannibal who can row: Brandreth Puzzle Book, Abraham, Beyer. Bigger boats: Pacioli, Filicaia?, Bachet(©Labosne), Delannoy, Ball, Dudeney, Abraham?, Goodstein, Kaplan, O'Beirne, Alcuin. 9C. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐProb. 17: Propositio de tribus fratribus singulas habentibus sorores. 3 couples, rather earthily expressed. Prob. 18: Propositio de lupo et capra et fasciculo cauli. Wolf, goat, cabbages. Prob. 19: Propositio de viro et muliere ponderantibus plaustrum. Man, wife and two small children. Prob. 20: Propositio de ericiis. Rewording of Prob. 19. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐAhrens. MUS II 315-318, cites many sources, mostly from folklore and riddle collections, with one from the 12C and several from the 14C. ??NYS. Abbot Albert. c1240. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐProb. 5, p. 333. Wolf, goat & cabbages. Prob. 6, p. 334. 3 couples, with verse mnemonic. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐColumbia Algorism. c1350. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐNo. 122, pp. 130-131 & 191: wolf, goat, bundle of greens. See also Cowley 402 & plate opposite. P. 191 and the Cowley plate are reproductions of the text with a crude but delightful illustration. P. 130 gives a small sketch of the illustration. I have a colour slide from the MS. No. 124, p. 132: 3 couples. See also Cowley 403 & plate opposite. The plate shows another crude but delightful illustration. I have a colour slide from the MS. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐMunich 14684. 14C. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐProb. XXVI, pp. 82-83: 3 couples, with verse mnemonic. Prob. XXVII, p. 83: wolf, goat, cabbage. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐFolkerts. Aufgabensammlungen. 13©15C. 11 sources with wolf, goat, cabbage. 12 sources with three jealous couples. Rara, 459-465, cites two Florentine MSS of c1460 which include 'the jealous husbands'. ??NYS. Chuquet. 1484. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐProb. 163: wolf, goat & cabbages. FHM 233 says that a 12C MS claims that every boy of five knows this problem. Prob. 164: 3 couples. FHM 233. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐPacioli. De Viribus. c1500. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐFf. 103v © 105v. LXI. C(apitolo). de .3. mariti et .3. mogli gelosi (About 3 husbands and 3 wives). = Peirani 146©148. 3 couples. Says that 4 or 5 couples requires a 3 person boat. F. IIIv. = Peirani 6. The Index lists the above as Problem 66 and lists a Problem 65: Del modo a salvare la capra el capriolo dal lupo al passar de un fiume ch' non siano devorati (How to save the goat and the kid from the wolf in crossing a river so they are not eaten). Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐPiero di Nicolao d'Antonio da Filicaia. Libro dicto giuochi mathematici. Early 16C ©© ??NYS, mentioned in Franci, op. cit. in 3.A. Franci, p. 23, says Pacioli and Filicaia deal with the case of four or five couples and that Pacioli considers bigger boats, but I'm not clear if Filicaia also does so. Cardan. Practica Arithmetice. 1539. Chap. 66, section 73, f. FF.v.v (p. 157). (The 73 is not printed in the Opera Omnia). Three jealous husbands. Tartaglia. General Trattato, 1556, art. 141-143, p. 257r- 257v. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐArt. 141: wolf, goat and cabbages. Art. 142: three couples. Art. 143: four couples ©© erroneously ©© see Bachet. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐH&S 51 says 3 couples occurs in Trenchant (1566), ??NYS. Gori. Libro di arimetricha. 1571. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐFf. 71r-71v (p. 77). 3 couples. F. 80v (p. 77). Dog, wolf, sheep, horse to cross river in boat which holds 2, but each cannot abide his neighbours in the given list, so each cannot be alone with such a neighbour. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐBachet. Problemes. 1612. Addl. prob. IV: Trois maris jaloux ..., 1612: 140©142; 1624: 212-215; 1884: 148-153. Three couples; four couples ©© notes that Tartaglia is wrong by showing that one can never get five persons on the far side. Labosne gives a solution with a 3 person boat and does n couples with an n-1 person boat. van Etten. 1624. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐProb. 14: Des trois maistres & trois valets, p. 14. 3 men and 3 valets. (The men hate the other valets and will beat them if given a chance.) (Not in English editions.) Prob. 15: Du loup, de la chevre & du chou, pp. 14-15. Wolf, goat & cabbages. (Not in English editions.) Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐBook of Merry Riddles. 1629 72 Riddle, pp. 43©44. "Over a water I must passe, and I must carry a lamb, a woolfe, and a bottle of hay if I carry any more than one at once my boat will sink." Tony Augarde; The Oxford Guide to Word Games; OUP, 1984; p. 6 says wolf, goat, cabbage appears in the 1629 ed. Wingate/Kersey. 1678?. Prob. 6., p. 543. Three jealous couples. Cf 1760 ed. Ozanam. 1725. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐProb. 2, 1725: 3-4. Prob. 18, 1778: 171; 1803: 171; 1814: 150. Prob. 17, 1840: 77. Wolf, goat and cabbage. Prob. 3, 1725: 4-5. Prob. 19, 1778: 171©172; 1803: 171©172; 1814: 150©151. Prob. 18, 1840: 77. Jealous husbands. Latin verse solution. He also discusses three masters and valets: "none of the the masters can endure the valets of the other two; so that if any one of them were left with any of the other two valets, in the absence of his master, he would infallibly cane him." Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐMinguet. 1733. Pp. 158©159 (1755: 114©115; 1822: 175©176; 1864: 151). Three jealous couples. Dilworth. Schoolmaster's Assistant. 1743. Part IV: Questions: A short Collection of pleasant and diverting Questions, p. 168. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐProblem 6: Fox, goose and peck of corn. = D. Adams; Scholar's Arithmetic; 1801, p. 200, no. 8. Problem 7: Three jealous husbands. (Dilworth cites Wingate for this ©© but this is in Kersey's additions ©© cf Wingate/Kersey, 1678? above.) = D. Adams; Scholar's Arithmetic; 1801, p. 200, no. 9. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐLes Amusemens. 1749. Prob. 14, p. 136: Les Maris jaloux. Solution is incorrect and has been corrected by hand in my copy. Edmund Wingate (1596©1656). A Plain and Familiar Method for Attaining the Knowledge and Practice of Common Arithmetic. .... 19th ed., previous ed. by John Kersey (1616ª1677) and George Shell(e)y, now by James Dodson. C. Hitch and L. Hawes, et al., 1760. ÁÁÁÁArt. 749. Prob. VI. P. 379. Three jealous husbands. As in 1678? ed. ÁÁÁÁArt. 750. Prob. VII. P. 379. Fox, goose and corn. Jackson. Rational Amusement. 1821. Arithmetical Puzzles. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐNo. 7, pp. 2 & 52. Fox, goose and corn. One solution. No. 13, pp. 4 & 54. Three jealous husbands. No. 21, pp. 5 & 56. Three masters and servants, where the servants will murder the masters if they outnumber them ©© i.e. missionaries and cannibals. First appearance of this type. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐEndless Amusement II. 1826? Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐProb. 17, pp. 198©199. Wolf, goat and cabbage. Prob. 25, pp. 201©202. Three jealous husbands. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐBoy's Own Book. The wolf, the goat and the cabbages. 1828: 418-419; 1828©2: 423; 1829 (US): 214; 1855: 570; 1868: 670. Nuts to Crack III (1834). Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐNo. 209. Fox, goose and peck of corn. No. 214. Three jealous husbands. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐThe Riddler. 1835. The wolf, the goat and the cabbages, pp. 5©6. Identical to Boy's Own Book. Young Man's Book. 1839. Pp. 39©40. Three jealous Husbands ..., identical to Wingate/Kersey. Child. Girl's Own Book. 1842: Enigma 49, pp. 237©238; 1876: Enigma 40, p. 200. Fox, goose and corn. Says it takes four trips instead of three ©© but the solution has 7 crossings. Walter Taylor. The Indian Juvenile Arithmetic, or Mental Calculator; to which is added an appendix, containing arithmetical recreations and amusements for leisure hours .... For the author at the American Press, Bombay, 1849. [Quaritch catalogue 1224, Jun 1996, says their copy has a note in French that Ramanujan learned arithmetic from this and that it is not in BMC nor NUC. Graves 14.c.35.] P. 211, No. 8. Wolf, goat and cabbage in verse! No solution. ÁÁÁÁUpon a river's brink I stand, it is both deep and wide; ÁÁÁÁWith a wolf, a goat, and cabbage, to take to the other side. ÁÁÁÁTho' only one each time can find, room in my little boat; ÁÁÁÁI must not leave the goat and wolf, not the cabbage and the goat. ÁÁÁÁLest one should eat the other up, ©© now how can it be done ©© ÁÁÁÁHow can I take them safe across without the loss of one? Fireside Amusements. 1850: No. 24, pp. 111 & 181; 1890: No. 24, p. 100. Fox, goose and basket of corn. Family Friend 3 (1850) 344 & 351. Enigmas, charades, etc. ©© No. 17: The three jealous husbands. Magician's Own Book. 1857. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐThe three jealous husbands, p. 251. The fox, goose, and corn, p. 253. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐThe Sociable. 1858. Prob. 33: The three gentlemen and their servants, pp. 296 & 314©315. "None of the gentlemen shall be left in company with any of the servants, except when his own servant is present" ©© so this is like the Jealous Husbands. = Book of 500 Puzzles, 1859, prob. 33, pp. 14 & 32©33. = Illustrated Boy's Own Treasury, 1860, prob. 11, pp. 427©428 & 431. Book of 500 Puzzles. 1859. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐProb. 33: The three gentlemen and their servants, pp. 14 & 32©33. As in The Sociable. The three jealous husbands, p. 65. The fox, goose and corn, p. 67. ÁÁBoth identical to Magician's Own Book. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐBoy's Own Conjuring Book. 1860. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐThe three jealous husbands, pp. 222-223. The fox, goose, and corn, pp. 225. ÁÁBoth identical to Magician's Own Book. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐVinot. 1860. Art. XXXVII: Les trois maris jaloux, pp. 56©57. Three jealous husbands, with verse solution taken from Ozanam. The Secret Out (UK). c1860. A comical dilemma, p. 27. Wolf, goat and cabbage. Varies it as fox, goose and corn and then as gentlemen and servants, which is jealous husbands, rather than the same problem. Lewis Carroll. Letter of 15 Mar 1873 to Helen Feilden. Pp. 212©215 (Collins: 154©155). Fox, goose and bag of corn. "I rashly proposed to her to try the puzzle (I daresay you know it) of "the fox, and goose, and bag of corn."" Cf Carroll©Collingwood, pp. 212©215 (Collins: 154©155); Carroll©Wakeling, prob. 28, pp. 36©37 and Carroll©Gardner, p. 51. Cf Carroll, 1878. Wakeling writes that this does not appear elsewhere in Carroll. Bachet©Labosne. 1874. For details, see Bachet, 1612. Jens Kamp. Danske Folkeminder, Aeventyr, Folksagen, Gaader, Rim og Folketro, Samlede fra Folkemende. R. Neilsen, Odense, 1877. Marcia Ascher has kindly sent me a photocopy of the relevant material with a translation by Viggo Andressen. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐNo. 18, pp. 326-327: Fox, lamb and cabbage. No. 19, p. 327: Husband, wife and two half-size sons. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐLewis Carroll. Letter of 22 Jan 1878 to Jessie Sinclair. Fox, goose and bag of corn. Cf Carroll©Collingwood, pp. 205©207 (Collins: 150); Carroll©Wakeling, prob. 26: The fox, the goose and the bag of corn, pp. 34 & 72. Cf Carroll. 1872. Mittenzwey. 1880. Prob. 227©228, pp. 42 & 92; 1895?: 254©255, pp. 46 & 94; 1917: 254-255, pp. 42 & 90. Bear, goat and cabbage, mentioning second solution; three kings and three servants, where the servants will rob the kings if they outnumber them, i.e. like missionaries and cannibals. Cassell's. 1881. P. 105: The dishonest servants. The servants are rogues who will murder masters if they outnumber them, so this is equivalent to the missionaries and cannibals version. Lucas. RM1. 1882. Pp. 1©18 is a general discussion of the problem. De Fontenay. Unknown source and date ©© 1882?? Described in RM1, 1882, pp. 15-18 (check 1st ed.??). n > 3 couples, 2 person boat, island in river, can be done in 8n - 8 passages. Lucas says this was suggested at the CongrÀ/Às de l'Association franÀ'Àaise pour l'avancement des sciences at Montpellier in 1879, ??NYS. (De Fontenay is unclear ©© sometimes he permits bank to bank crossings, other times he only permits bank to island crossings. His argument really gives 8n © 6 if bank to bank crossings are prohibited. See Pressman & Singmaster, below, for clarification.) Albert Ellery Berg, ed. Op. cit. in 4.B.1. 1883. P. 377: Fox, goose & peck of corn. Lemon. 1890. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐGentlemen and their servants, no. 101, pp. 17-18 & 101. This is the same as missionaries and cannibals. The three jealous husbands, no. 151, pp. 24 & 103 (= Sphinx, no. 478, pp. 66 & 114.) The solution mentions Alcuin. Crossing the river, no. 450, pp. 59 & 114. English travellers and native servants = missionaries and cannibals. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐDon Lemon. Everybody's Pocket Cyclopedia. Revised 8th ed., 1890. Op. cit. in 5.A. P. 136, no. 14. Fox, goose and corn. No solution. Herbert Llewelyn Pocock. UK Patent 15,358 ©© Improvements in Toy Puzzles. Applied: 29 Sep 1890; complete specification: 29 Jun 1891; accepted: 22 Aug 1891. 2pp + 1p diagrams. Three whites and three blacks and the blacks must never outnumber the whites, i.e. same as missionaries and cannibals. He describes the puzzle as "well known". Delannoy. Described in RM1, 1891, Note 1: Sur le jeu des traversÀ)Àes, pp. 221-222. ??check 1882 ed. Shows n couples can cross in an x person boat in N trips, for n, x, N  =  2, 2, 5; 3, 2, 11; 4, 3, 9; 5, 3, 11; n > 5, 4, 2n - 3. (He has 2n - 1 by mistake. Simple modification shows we also have 5, 4, 7; 6, 5, 9; 7, 6, 5; 8, 7, 7; n > 8, n - 1, 5.) Ball. MRE, 1st ed., 1892, pp. 45-47, says Lucas posed the problem of minimizing x for a given n and quotes the Delannoy solution (with erroneous 2n - 1) and also gives De Fontenay's version and solution. (He spells it De Fonteney as does his French translator, though Ahrens gives De Fontenay and the famous abbey in Burgundy is Fontenay ©© ??) The Ballybunnion and Listowel Railway in County Kerry, Ireland, was a late 19C railway using the Lartigue monorail system. This had a single rail, about three feet off the ground, with a carriage hanging over both sides of the rail. The principle job of the conductor/guard to make sure the passengers and goods were equally distributed on both sides. Kerry legend asserts that a piano had to be sent on this railway and there were not enough passengers or goods to balance it. So a cow was sent on the other side. At the far end, the piano was unloaded and replaced with two large calves and the carriage sent back. The cow was then unloaded and one calf moved to the other side, so the carriage could be sent back to the far end and everyone was happy. Hoffmann. 1893. Chap. IV, pp. 157-158 & 211-213 = Hoffmann©Hordern, pp. 136©138, with photos. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐNo. 56: The three travellers. Masters and servants, equivalent to missionaries and cannibals. Solution says Jaques & Son make a puzzle version with six figures, three white and three black. Photos in Hoffmann©Hordern, pp. 136 & 137 ©© the latter shows Caught in the Rain, 1880©1905, where Preacher, Deacon, Janitor and their wives have to get somewhere using one umbrella. No. 57: The wolf, the goat, and the cabbages. Photo on p. 136 of La Chevre et le Chou. with box, by Watilliaux, 1874©1895. Hordern Collection, p. 72, and S&B, p. 134, show the same puzzle.) No. 58: The three jealous husbands. No. 59: The captain and his company. This is Alcuin's prop. 19 with many adults. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐBrandreth Puzzle Book. Brandreth's Pills (The Porous Plaster Co., NY), nd [1895]. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐP. 7: The wolf, the goat and the cabbages. Identical to Hoffmann No. 57, with nice colour picture. No solution. P. 9: The missionaries' and cannibals' puzzle. Usual form, with nice colour picture, but only one cannibal can row. No solution. This seems to be the first to use the context of missionaries and cannibals and the first to restrict the number of rowers. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐLucas. L'ArithmÀ)Àtique Amusante. 1895. Les vilains maris jaloux, pp. 125©144 & Note II, pp. 198©202. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐProb. XXXVI: La traversÀ)Àe des trois mÀ)Ànages, pp. 125©130. 3 couples. Gives Bachet's 1624 reasoning for the essentially unique solution ©© but attributes it to 1613. Prob. XXXVII: La traversÀ)Àe des quatre mÀ)Ànages, pp. 130©132. 4 couples in a 3 person boat done in 9 crossings. L'erreur de Tartaglia, pp. 133©134. Discusses Tartaglia's error and Bachet's notice of it and gives an easy proof that 4 couples cannot be done with a 2 person boat. Prob. XXXVIII: La station dans une À3Àle, pp. 135©140. 4 couples, 2 person boat, with an island. Gives De Fontenay's solution in 24 crossings. Prob. XXXIX: La traversÀ)Àe des cinq mÀ)Ànages, pp. 141©143. 5 couples, 3 person boat in 11 crossings. À(ÀnoncÀ)À gÀ)ÀnÀ)Àral du problÀ/Àme des traversÀ)Àes, pp. 143©144. n couples, x person boat, can be done in N crossings as given by Delannoy above. He corrects 2n © 1 to 2n © 3 here. Note II: Sur les traversÀ)Àes, pp. 198©202. Gives Tarry's version with an island and with n men having harems of size m, where the women are obviously unable to row. He gives solutions in various cases. For the ordinary case, i.e. m = 1, he finds a solution for 4 couples in 21 moves, using the basic ferrying technique that Pressman and Singmaster found to be optimal, but the beginning and end take longer because the women cannot row. He says this gives a solution for n couples in 4n + 5 crossings. He then considers the case of n © 1 couples and a mÀ)Ànage with m wives and finds a solution in 8n + 2m + 7 crossings. I now see that this solution has the same defects as those in Pressman & Singmaster, qv. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐBall. MRE, 3rd ed., 1896, pp. 61-64, repeats 1st ed., but adds that Tarry has suggested the problem for harems ©© see above. Dudeney. Problem 68: Two rural puzzles. Tit-Bits 33 (5 Feb & 5 Mar 1898) 355 & 432. Three men with sacks of treasure and a boat that will hold just two men or a man and a sack, with additional restrictions on who can be trusted with how much. Solution in 13 crossings. Carroll©Collingwood. 1899. P. 317 (Collins: 231 or 232 (missing in my copy)) Cf CarrollªWakeling II, prob. 10: Crossing the river, pp. 17 & 66. Four couples ©© only posed, no solution. Wakeling gives a solution, but this is incorrect. After one wife is taken across, he has another couple coming across and from Bachet onward, this is considered improper as the man could get out of the boat and attack the first, undefended, wife. E. Fourrey. Op. cit. in 4.A.1, 1899. Section 211: Les trois maÀ3Àtres et les trois valets. Says a master cannot leave his valet with the other masters for fear that they will intimidate him into revealing the master's secrets. Hence this is the same as the jealous couples. H. D. Northrop. Popular Pastimes. 1901. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐNo. 5: The three gentlemen and their servants, pp. 67 & 72. = The Sociable. No. 12: The dishonest servants, pp. 68 & 73. "... the servants on either side of the river should not outnumber the masters", so this is the same as missionaries and cannibals. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐMr. X [cf 4.A.1]. His Pages. The Royal Magazine 10:2 (Jun 1903) 140©141. A matrimonial difficulty. Three couples. No answer given. Dudeney. Problem 523. Weekly Dispatch (15 & 29 Nov 1903), both p. 10, (= AM, prob. 375, pp. 113 & 236-237). 5 couples in a 3 person boat. Johannes Bolte. Der Mann mit der Ziege, dem Wolf und dem Kohle. Zeitschrift des Vereins fÀGÀr Volkskunde 13 (1903) 95©96 & 311. The first part is unaware of Alcuin and Albert. He gives a 12C Latin solution: It capra, fertur olus, redit hec, lupus it, capra transit [from Wattenbach; Neuen Archiv fÀGÀr ÀÀltere deutsche Geschichtskunde 2 (1877) 402, from Vorauer MS 111, ??NYS] and a 14C solution: O natat, L sequitur, redit O, C navigat ultra, / Nauta recurrit ad O, bisque natavit ovis (= ovis, lupus, ovis, caulis, ovis) [from Mone; Anzeiger fÀGÀr Kunde der deutschen Vorzeit 45 (No. 105) (1838), from Reims MS 743, ??NYS]. Cites Kamp and several other versions, some using a fox, a sheep, or a lamb. The addendum cites and quotes Alcuin and Albert as well as relatively recent French and Italian versions. H. Parker. Ancient Ceylon. Op. cit. in 4.B.1. 1909. Crossing the river, p. 623. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐA King, a Queen, a washerman and a washerwoman have to cross a river in a boat that holds two. However the King and Queen cannot be left on a bank with the low caste persons, though they can be rowed by the washerperson of the same sex. Solution in 7 crossings. Ferry©man must transport three leopards and three goats in a boat which holds himself and two others. If leopards ever outnumber goats, then the goats get eaten. So this is like missionaries and cannibals, but with a ferry©man. Solution in 9 crossings. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐH. Schubert. Mathematische Mussestunde. Vol. 2, 3rd ed., GÀ?Àschen, Leipzig, 1909. Pp. 160-162: Der drei Herren und der drei Sklaven. (Same as missionaries and cannibals.) Arbiter Co. (Philadelphia). 1910. Capital and Labor Puzzle. Shown in S&B, p. 134. Equivalent to missionaries and cannibals. Ball. MRE, 5th ed., 1911, pp. 71©73, repeats 3rd ed., but omits the details of De Fonteney's solution in 8(n©1) crossings. Loyd. Cyclopedia, 1914. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐSummer tourists, pp. 207 & 366. 3 couples, 2 person boat, with additional complications ©© the women cannot row and there have been some arguments. Solution in 17 crossings. The four elopements, pp. 266 & 375. 4 couples, 2 person boat, with an island and the stronger constraint that no man is to get into the boat alone if there is a girl alone on either the island or the other shore. "The [problem] presents so many complications that the best or shortest answer seems to have been overlooked by mathematicians and writers on the subject." "Contrary to published answers, ... the feat can be performed in 17 trips, instead of 24." Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐBall. MRE, 6th ed., 1914, pp. 71©73, repeats 5th ed., but adds that 6n - 7 trips suffices for n couples with an island, though he gives no reference. Williams. Home Entertainments. 1914. Alcuin's riddle, pp. 125©126. "This will be recognized as perhaps the most ancient British riddle in existence, though there are several others conceived on the same lines." Three jealous couples. Clark. Mental Nuts. 1916, no. 67. The men and their wives. "... no man shall be left alone with another's wife." Dudeney. AM. 1917. Prob. 376: The four elopements, pp. 113 & 237. 4 couples, 2 person boat, with island, can be done in 17 trips and that this cannot be improved. This is the same solution as given by Loyd. (See Pressman and Singmaster, below.) Ball. MRE, 8th ed., 1919, pp. 71©73 repeats 6th ed. and adds a citation to Dudeney's AM prob. 376 for the solution in 6n - 7 trips for n couples. Hummerston. Fun, Mirth & Mystery. 1924. Crossing the river puzzles, Puzzle no, 52, pp. 128 & 180. 'Puzzles of this type ... interested people who lived more than a thousand years ago'. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐNo. 1: The eight travellers. Six men and two boys who weigh half as much. No. 2: White and black. = Missionaries and cannibals. No. 3: The fox, the goose, and the corn. No. 4: the jealous husbands. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐH&S, 1927, p. 51 says missionaries and cannibals is 'a modern variant'. King. Best 100. 1927. No. 10, pp. 10 & 40. Dog, goose and corn. Heinrich Voggenreiter. Deutsches Spielbuch Sechster Teil: Heimspiele. Ludwig Voggenreiter, Potsdam, 1930. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐP. 106: Der Wolf, die Ziege und der Kohlkopf. Usual wolf, goat, cabbage. Pp. 106©107: Die 100 Pfund©Familie. Parents weigh 100 pounds; the two children weigh 100 pounds together. P. 107: Der LandjÀÀger and die Strolche [The policeman and the vagabonds]. Two of the vagabonds hate each other so much that they cannot be left together. As far as I recall, this formulation is novel and I was surprised to realise that it is essentially equivalent to the wolf, goat and cabbage version. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐPhillips. Week-End. 1932. Time tests of intelligence, no. 41, pp. 22 & 194. Rowing explorer with 4 natives: A, B, C, D, who cannot abide their neighbours in this list. A can row. They get across in seven trips. Abraham. 1933. Prob. 54 ©© The missionaries at the ferry, pp. 18 & 54 (14 & 115). 3 missionaries and 3 cannibals. Doesn't specify boat size, but says 'only one cannibal can row'. 1933 solution says 'eight double journeys', 1964 says 'seven crossings'. This seems to assume the boat holds 3. (For a 2 man boat, it takes 11 crossings with one missionary and two cannibals who can row or 13 crossings with one missionary and one cannibal who can row.) The Bile Beans Puzzle Book. 1933. No. 34: Missionaries & cannibals. Three of each but only one of each can row. Done in 13 crossings. Phillips. Brush. 1936. Prob. L.2: Crossing the Limpopo, pp. 39-40 & 98. Same as in Week-End, 1932. M. Adams. Puzzle Book. 1939. Prob. C.63: Going to the dance, pp. 139 & 178. Same as Week-End, 1932, phrased as travelling to a dance on a motorcycle which carries one passenger. R. L. Goodstein. Note 1778: Ferry puzzle. MG 28 (No. 282) (1944) 202-204. Gives a graphical way of representing such problems and considers m soldiers and m cannibals with an n person boat, 3 jealous husbands and how many rowers are required. David Stein. Party and Indoor Games. P. M. Productions, London, nd [c1950?]. P. 98, prob. 5: Man with cat, parrot and bag of seeds. C. L. Stong. The Amateur Scientist. Ill. by Roger Hayward. S&S, 1960. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐA puzzle©solving machine, pp. 377©384. Describes how Paul Bezold made a logic machine from relays to solve the fox, goose, corn problem. How to design a "Pircuit" or Puzzle circuit, pp. 388©394. On pp. 391©394, Harry Rudloe describes relay circuits for solving the three jealous couples problem, which he attributes to Tartaglia, and the missionaries and cannibals problem. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐNathan Altshiller Court. Mathematics in Fun and in Earnest. Mentor (New American Library), NY, 1961. [John Fauvel sent some pages from a different printing which has much different page numbers than my copy.] "River crossing" problems, pp. 168-171. Discusses various forms of the problem and adds a problem with two parents weighing 160, two children weighing 80 and a dog weighing 12, with a boat holding 160. E. A. Beyer, proposer; editorial solution. River-crossing dilemma. RMM 4 (Aug 1961) 46 & 5 (Oct 1961) 59. Explorers and natives (= missionaries and cannibals), with all the explorers and one native who can row. Solves in 13 crossings, but doesn't note that only one rowing explorer is needed. (See note at Abraham, 1933, above.) Philip Kaplan. Posers. (Harper & Row, 1963); Macfadden Books, 1964. Prob. 36, pp. 41 & 91. 5 men and a 3 person boat on one side, 5 women on the other side. One man and one woman can row. Men are not allowed to outnumber women on either side nor in the boat. Exchange the men and the women in 7 crossings. T. H. O'Beirne. Puzzles and Paradoxes, 1965, op. cit. in 4.A.4, chap. 1, One more river to cross, pp. 1-19. Shows 2n - 1 couples (or 2n - 1 each of missionaries and cannibals ?) can cross in a n person boat in 11 trips. 2n - 2 can cross in 9 trips. He also considers variants on Gori's second version. Doubleday © 2. 1971. Family outing, pp. 49©50. Three couples, but one man has quarrelled with the other men and his wife has quarrelled with the other women, so this man and wife cannot go in the boat nor be left on a bank with others of their sex. Further men cannot be outnumbered by women on either bank. Gives a solution in 9 crossings, but I find the conditions unworkable ©© e.g. the initial position is prohibited! Claudia Zaslavsky. Africa Counts. Prindle, Weber & Schmidt, Boston, 1973. Pp. 109-110 says that leopard, goat and pile of cassava leaves is popular with the Kpelle children of Liberia. However, Ascher's Ethnomathematics (see below), p. 120, notes that this is based on an ambiguous description and that an earlier report of a Kpelle version has the form described below. Ball. MRE, 12th ed., 1974, p. 119, corrects Delannoy's 2n - 1 to 2n - 3 and corrects De Fontenay's 8n - 8 to 8n - 6, but still gives the solution for n = 4 with 24 crossings. W. Gibbs. Pebble Puzzles ©© A Source Book of Simple Puzzles and Problems. Curriculum Development Unit, Solomon Islands, 1982. ??NYS, o/o??. Excerpted in: Norman K. Lowe, ed.; Games and Toys in the Teaching of Science and Technology; Science and Technology Education, Document Series No. 29, UNESCO, Paris, 1988, pp. 54-57. On pp. 56-57 is a series of river crossing problems. E.g. get people of weights 1, 2, 3 across with a boat that holds a weight of at most 3. Also people numbered 1, 2, 3, 4, 5 such that no two consecutive people can be in the boat or left together. In about 1986, James Dalgety designed interactive puzzles for Techniquest in Cardiff. Their version has a Welshman with a dragon, a sheep and a leek! Ian Pressman & David Singmaster. Solutions of two river crossing problems: The jealous husbands and the missionaries and the cannibals. Extended Preprint, April 1988, 14pp. MG 73 (No. 464) (Jun 1989) 73-81. (The preprint contains historical and other detail omitted from the article as well as some further information.) Observes that De Fontenay seems to be excluding bank to bank crossings and that Lucas' presentation is cryptic. Shows that De Fontenay's method should be 8n - 6 crossings for n > 3 and that this is minimal. If bank to bank crossings are permitted, as by Loyd and Dudeney, a computer search revealed a solution with 16 crossings for n = 4, using an ingenious move that Dudeney could well have ignored. For n > 4, there is a simple solution in 4n + 1 crossings, and these numbers are minimal. [When this was written, I had forgotten that Loyd had done the problem for 4 couples in 17 moves, which changes the history somewhat. However, I now see that Loyd was copying from Dudeney's Weekly Dispatch problem 270 of 23 Apr 1899 & 11 Jun 1899. Loyd states what appears to be a stronger constraint but all the methods in our article do obey the stronger constraint. However, one could make the constraints stronger ©© e.g. our solutions have a husband taking the boat from bank to bank while his wife and another wife are on the island ©© the solution of Loyd & Dudeney avoids this and may be minimal in this case ©ª??.] ÁÁÁÁFor the missionaries and cannibals problem, the 16 crossing solution reduces to 15 and gives a general solution in 4n - 1 crossings, which is shown to be minimal. If bank to bank crossings are not permitted, then De Fontenay's amended 8n - 6 solution is still optimal. Marcia Ascher. A river-crossing problem in cultural perspective. MM 63 (1990) 26-28. Describes many appearances in folklore of many cultures. Discusses African variants of the wolf, goat and cabbage problem in which the man can take two of the items in the boat. This is much easier, requiring only three crossings, but some versions say that the man cannot control the items in the boat, so he cannot have the wolf and goat or the goat and cabbage in the boat with him. This still only takes three crossings. Various forms of these problems are mentioned: fox, fowl and corn; tiger, sheep and reeds; jackal, goat and hay; caged cheetah, fowl and rice; leopard, goat and leaves ©© see below for more details. ÁÁÁÁShe also discusses an Ila (Zambia) version with leopard, goat, rat and corn which is unsolvable! Marcia Ascher. Ethnomathematics. Op. cit. in 4.B.10. 1991. Section 4.8, pp. 109©116 & Note 8, pp. 119©121. Good survey of the problem and numerous references to the folklore and ethnographic literature. Amplifies the above article. A version like the Wolf, goat and cabbage is found in the Cape Verde Islands, in Cameroon and in Ethiopia. The African version is found as far apart as Algeria and Zanzibar, but with some variations. An Algerian version with jackal, goat and hay allows one to carry any two in the boat, but an inefficient solution is presented first. A Kpelle (Liberia) version with cheetah, fowl and rice adds that the man cannot keep control while rowing so he cannot take the fowl with either the cheetah or the rice in the boat. A Zanzibar version with leopard, goat and leaves adds instead that no two items can be left on either bank together. (A similar version occurs among African©Americans on the Sea Islands of South Carolina.) Ascher notes that Zaslavsky's description is based on an ambiguous report of the Kpelle version and probably should be like the Algerian or Kpelle version just described. Liz Allen. Brain Sharpeners. New English Library (Hodder & Stoughton), London, 1991. Crossing the river, pp. 62 & 125. Three mothers and three sons. The sons are unwilling to be left with strange mothers, so this is a rephrasing of the jealous husbands. Yuri B. Chernyak & Robert S. Rose. The Chicken from Minsk. BasicBooks, NY, 1995. Chap. 1, probs. 4©6: The knights and the pages; More knights and pages; Yet more knights and pages: no man is an island, pp. 4©5 & 100©102. Equivalent to the jealous couples. Prob. 4 is three couples, solved in 11 crossings. Prob. 5 is four couples ©© "There is no solution unless one of the four pages is sacrificed. (In medieval times, this was not a problem.)" Prob. 6 is four couples with an island in the river, solved in general by moving all pages to the island, then having the pages go back and accompany his knight to other side, then return to the island. After the last knight is moved, the pages then move from the island to the other side. This takes 7n © 6 steps in general. It satisfies the jealousy conditions used by Pressman & Singmaster, but not those of Loyd & Dudeney. John P. Ashley. Arithmetickle. Arithmetic Curiosities, Challenges, Games and Groaners for all Ages. Keystone Agencies, Radnor, Ohio, 1997. P. 16: The missionaries and the pirates. Politically correct rephrasing of the missionaries and the cannibals version. All the missionaries, but only one pirate, can row. Solves in 13 crossings. Prof. Dr. Robert Weismantel, Otto©von©Guericke©UniversitÀÀt Magdeburg, FakultÀÀt fÀGÀr Mathematik, PSF 3120, D©39016 Magdeburg, Germany; tel: 0391/67©18745; email: weismantel@imo.math.uni©magdeburg.de; has produced a 45 min. film: "Der Wolf, die Ziege und KohlkÀ?Àpfe Transportprobleme von Karl dem Grossen bis heute", suitable for the final years of school. ÁÁà Ã5.B.1.ÁÁLOWERING FROM TOWER PROBLEMÄ Ä Ðа¤˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐ ÁÁThe problem is for a collection of people (and objects or animals) to lower themselves from a window using a rope over a pulley, with baskets at each end. The complication is that the baskets cannot contain very different weights, i.e. there is a maximum difference in the weights, otherwise they go too fast. This is often attributed to Carroll. ÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐ Carroll©Collingwood. 1899. P. 318 (Collins: 232©233 (232 is lacking in my copy)). = Carroll©Wakeling II, prob. 4: The captive queen, pp. 8 & 65©66. 3 people of weights 195, 165, 90 and a weight of 75, with difference at most 15. He also gives a more complex form. No solutions. Although the text clearly says 165, the prevalence of the exact same problem with 165 replaced by 105 makes me wonder if this was a misprint?? Wakeling says there is no explicit evidence that Carroll invented this, and neither book assigns a date, but Carroll seems a more original source than the following and he was more active before 1890 than after. ÁÁÁÁAn addition is given in both books: add three animals, weighing 60, 45, 30. Lemon. 1890. The prisoners in the tower, no. 497, pp. 65 & 116. c= Sphinx, The escape, no. 113, pp. 19 & 100-101. Three people of weights 195, 105, 90 with a weight of 75. The difference in weights cannot be more than 15. Hoffmann. 1893. Chap. IV, no. 28: The captives in the tower, pp. 150 & 196 = Hoffmann-Hordern, p. 123. Same as Lemon. Brandreth Puzzle Book. Brandreth's Pills (The Porous Plaster Co., NY), nd [1895]. P. 3: The captives in the tower. Same as Lemon. Identical to Hoffmann. With colour picture. No solution. Loyd. The fire escape puzzle. Cyclopedia, 1914, pp. 71 & 348. c= MPSL2, prob. 140, pp. 98-99 & 165. = SLAHP: Saving the family, pp. 59 & 108. Simplified form of Carroll's problem. Man, wife, baby & dog, weighing a total of 390. Williams. Home Entertainments. 1914. The escaping prisoners, pp. 126©127. Same as Lemon. Rudin. 1936. No. 92, pp. 31©32 & 94. Same as Lemon. Haldeman©Julius. 1937. No. 150: Fairy tale, pp. 17 & 28. Same as Lemon, except the largest weight is printed as 196, possibly an error. Kinnaird. Op. cit. in 1 ©© Loyd. 1946. Pp. 388-389 & 394. Same as Lemon. Simon Dresner. Science World Book of Brain Teasers. Scholastic Book Services, NY, 1962. Prob. 61: Escape from the tower, pp. 29 & 99-100. Same as Lemon. Robert Harbin [pseud. of Ned Williams]. Party Lines. Oldbourne, London, 1963. Escape, p. 29. As in Lemon. Howard P. Dinesman. Superior Mathematical Puzzles. Allen & Unwin, London, 1968. No. 60: The tower escape, pp. 78 & 118. Same as Carroll. Answer in 15 stages. He cites Carroll, noting that Carroll did not give a solution and he asks if a shorter solution can be found. F. Geoffrey Hartswick. In: H. O. Ripley & F. G. Hartswick; Detectograms and Other Puzzles; Scholastic Book Services, NY, 1969. No. 15: Stolen treasure puzzle, pp. 54-55 & 87. Same as Lemon. ÁÁà Ã5.B.2.ÁÁCROSSING A BRIDGE WITH A TORCHÄ Ä Ðа¤˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐÁÁNew section. ÁÁFour people have to get across a bridge which is dark and needs to be lit with the torch. The torch can serve for at most two people and the gap is too wide to throw the torch across, so the torch has to be carried back and forth. The various people are of different ages and require 5, 10, 20, 25 minutes to cross and when two cross, they have to go at the speed of the slower. But the torch (= flashlight) battery will only last an hour. Can it be done? I heard this about 1997, when it was claimed to be used by Microsoft in interviewing candidates. I never found any history of it, until I recently found a discussion on Torsten Sillke's site: Crossing the bridge in an hour (www.mathematik.uni-bielefeld.de/~sillke/PUZZLES/crossing©bridge), starting in Jun 1997 and last updated in Sep 2001. This cites the 1981 source and the other references below. Denote the problem with speeds a, b, c, d and total time t by (a, b, c, d; t), etc. t is sometimes given, sometimes not. ÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐSaul X. Levmore & Elizabeth Early Cook. Super Strategies for Puzzles and Games. Doubleday, 1981, p. 3 ©© ??NYS. (5, 10, 20, 25; 60), as in the introduction to this section.. Heinrich Hemme. Das Problem des ZwÀ?Àlf©Elfs. Vandenhoeck & Ruprecht, 1998. Prob. 81: Die Flucht, pp. 40 & 105©106, citing a web posting by Gunther Lientschnig on 4 Dec 1996. (2, 4, 8, 10; t). Dick Hess. Puzzles from Around the World. Apr 1997. Prob. 107: The Bridge. ÁÁ(1, 2, 5, 10; 17). Poses versions with more people: (1, 3, 4, 6, 8, 9; 31) and, with a three©person bridge, (1, 2, 6, 7, 8, 19, 10; 25). Quantum (May/Jun 1997) 13. Brainteaser B 205: Family planning. Problem (1, 3, 8, 10; 20). Karen Lingel. Email of 17 Sep 1997 to rec.puzzles. Careful analysis, showing that the 'trick' solution is better than the 'direct' solution if and only if a + c > 2b. [Indeed, a + c © 2b is the time saved by the 'trick' solution.] She cites (2, 3, 5, 8; 19) and (2, 2, 3, 3; 11) to Sillke and (1, 3, 6, 8, 12; 30), from an undated website. Expressing the solution for more people seems to remain an open question. ÁÁà Ã5.C.ÁÁFALSE COINS WITH A BALANCEÄ Ä Ðа¤˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐÁÁSee 5.D.3 for use of a weighing scale. ÁÁThere are several related forms of this problem. Almost all of the items below deal with 12 coins with one false, either heavy or light, and its generalizations, but some other forms occur, including the following. ÁÁ 8 coins, ÀÀ1 light: Schell, Dresner ÁÁ26 coins, ÀÀ1 light: Schell ÁÁ 8 coins, 1 light: Bath (1959) ÁÁ 9 coins, 1 light: Karapetoff, Meyer (1946), Meyer (1948), M. Adams, Rice ÁÁI have been sent an article by Jack Sieburg; Problem Solving by Computer Logic; Data Processing Magazine, but the date is cut off ©© ??ÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐ E. D. Schell, proposer; M. Dernham, solver. Problem E651 ©© Weighed and found wanting. AMM 52:1 (Jan 1945) 42 & 7 (Aug/Sep 1945) 397. 8 coins, at most one light ©© determine the light one in two weighings. Benjamin L. Schwartz. Letter: Truth about false coins. MM 51 (1978) 254. States that Schell told Michael Goldberg in 1945 that he had originated the problem. Emil D. Schell. Letter of 17 Jul 1978 to Paul J. Campbell. Says he did NOT originate the problem, nor did he submit the version published. He first heard of it from Walter W. Jacobs about Thanksgiving 1944 in the form of finding at most one light coin among 26 good coins in three weighings. He submitted this to the AMM, with a note disclaiming originality. The AMM problem editor published the simpler version described above, under Schell's name. Schell says he has heard Eilenberg describe the puzzle as being earlier than Sep 1939. Campbell wrote Eilenberg, but had no response. ÁÁÁÁSchell's letter is making it appear that the problem derives from the use of 1, 3, 9, ... as weights. This usage leads one to discover that a light coin can be found in 3ÃÃnÄÄ coins using n weighings. This is the problem mentioned by Karapetoff. If there is at most one light coin, then n weighings will determine it among 3ÃÃnÄÄ - 1 coins, which is the form described by Schell. The problem seems to have been almost immediately converted into the case with one false coin, either heavy or light. Walter W. Jacobs. Letter of 15 Aug 1978 to Paul J. Campbell. Says he heard of the problem in 1943 (not 1944) and will try to contact the two people who might have told it to him. However, Campbell has had no further word. V. Karapetoff. The nine coin problem and the mathematics of sorting. SM 11 (1945) 186-187. Discusses 9 coins, one light, and asks for a mathematical approach to the general problem. (?? ©© Cites AMM 52, p. 314, but I cannot find anything relevant in the whole volume, except the Schell problem. Try again??) Dwight A. Stewart, proposer; D. B. Parkinson & Lester H. Green, solvers. The counterfeit coin. In: L. A. Graham, ed.; Ingenious Mathematical Problems and Methods; Dover, 1959; pp. 37-38 & 196-198. 12 coins. First appeared in Oct 1945. Original only asks for the counterfeit, but second solver shows how to tell if it is heavy or light. R. L. Goodstein. Note 1845: Find the penny. MG 29 (No. 287) (Dec 1945) 227-229. Non-optimal solution of general problem. Editorial Note. Note 1930: Addenda to Note 1845. Ibid. 30 (No. 291) (Oct 1946) 231. Comments on how to extend to optimal solution. Howard D. Grossman. The twelve-coin problem. SM 11:3/4 (Sep/Dec 1945) 360-361. Finds counterfeit and extends to 36 coins. Lothrop Withington, Jr. Another solution of the 12-coin problem. Ibid., 361-362. Finds also whether heavy or light. Donald Eves, proposer; E. D. Schell & Joseph Rosenbaum, solvers. Problem E712 ©© The extended coin problem. AMM 53:3 (Mar 1946) 156 & 54:1 (Jan 1947) 46-48. 12 coins. Jerome S. Meyer. Puzzle Paradise. Crown, NY, 1946. Prob. 132: The nine pearls, pp. 94 & 132. Nine pearls, one light, in two weighings. N. J. Fine, proposer & solver. Problem 4203 ©© The generalized coin problem. AMM 53:5 (May 1946) 278 & 54:8 (Oct 1947) 489-491. General problem. H. D. Grossman. Generalization of the twelve-coin problem. SM 12 (1946) 291-292. Discusses Goodstein's results. F. J. Dyson. Note 1931: The Problem of the Pennies. MG 30 (No. 291) (Oct 1946) 231-234. General solution. C. A. B. Smith. The Counterfeit Coin Problem. MG 31 (No. 293) (Feb 1947) 31-39. C. W. Raine. Another approach to the twelve-coin problem. SM 14 (1948) 66-67. 12 coins only. K. Itkin. A generalization of the twelve-coin problem. SM 14 (1948) 67-68. General solution. Howard D. Grossman. Ternary epitaph on coin problems. SM 14 (1948) 69-71. Ternary solution of Dyson & Smith. Jerome S. Meyer. Fun©to©do. A Book of Home Entertainment. Dutton, NY, 1948. Prob. 40: Nine pearls, pp. 41 & 188. Nine pearls, one light, in two weighings. Blanche Descartes [pseud. of Cedric A. B. Smith]. The twelve coin problem. Eureka 13 (Oct 1950) 7 & 20. Proposal and solution in verse. J. S. Robertson. Those twelve coins again. SM 16 (1950) 111-115. Article indicates there will be a continuation, but Schaaf I 32 doesn't cite it and I haven't found it yet. E. V. Newberry. Note 2342: The penny problem. MG 37 (No. 320) (May 1953) 130. Says he has made a rug showing the 120 coins problems and makes comments similar to Littlewood's, below. J. E. Littlewood. A Mathematician's Miscellany. Methuen, London, 1953; reprinted with minor corrections, 1957 (& 1960). [All the material cited is also in the later version: Littlewood's Miscellany, ed. by B. BollobÀÀs, CUP, 1986, but on different pages. Since the 1953 ed. is scarce, I will also cite the 1986 pages in (  ).] Pp. 9 & 135 (31 & 114). "It was said that the 'weighing-pennies' problem wasted 10,000 scientist-hours of war-work, and that there was a proposal to drop it over Germany." John Paul Adams. We Dare You to Solve This! Berkley Publishing, NY, nd [1957?]. [This is apparently a collection of problems used in newspapers. The copyright is given as 1955, 1956, 1957.] Prob. 18: Weighty problem, pp. 13 & 46. 9 equal diamonds but one is light, to be found in 2 weighings. Hubert Phillips. Something to Think About. Revised ed., Max Parrish, London, 1958. Foreword, p. 6 & prob. 115: Twelve coins, pp. 81 & 127-128. Foreword says prob. 115 has been added to this edition and "was in oral circulation during the war. So far as I know, it has only appeared in print in the Law Journal, where I published both the problem and its solution." This may be an early appearance, so I should try and track this down. ??NYS Dan Pedoe. The Gentle Art of Mathematics. (English Universities Press, 1958); Pelican (Penguin), 1963. P. 30: "We now come to a problem which is said to have been planted over here during the war by enemy agents, since Operational Research spent so many man-hours on its solution." Philip E. Bath. Fun with Figures. The Epworth Press, London, 1959. No. 7: No weights ©© no guessing, pp. 8 & 40. 8 balls, including one light, to be determined in two weighings. Method actually works for ÀÀ 1 light. M. R. Boothroyd & J. H. Conway. Problems drive, 1959. Eureka 22 (Oct 1959) 15©17 & 22©23. No. 9. Five boxes of sugar, but some has been taken from one box and put in another. Determine which in least number of weighings. Does by weighing each division of A, B, C, D into two pairs. Nathan Altshiller Court. Mathematics in Fun and in Earnest. Op. cit. in 5.B. 1961. The "False Coin" problem, pp. 178©182. Sketches history and solution. Simon Dresner. Science World Book of Brain Teasers. 1962. Op. cit. in 5.B.1. Prob. 46: Dud reckoning, pp. 21 & 94. Find one light among eight in two weighings. Philip Kaplan. More Posers. (Harper & Row, 1964); Macfadden©Bartell Books, 1965. Prob. 55, pp. 57 & 98. Six identical appearing coins, three of which are identically heavy. In two weighings, identify two of the heavy coins. Charlie Rice. Challenge! Hallmark Editions, Kansas City, Missouri, 1968. Prob. 7, pp. 22 & 54©55. 9 pearls, one light. Jonathan Always. Puzzling You Again. Tandem, London, 1969. Prob. 86: Light-weight contest, pp. 51-52 & 106-107. 27 weights of sizes 1, 2, ..., 27, except one is light. Find it in 3 weighings. He divides into 9 sets of three having equal weights. Using two weighings, one locates the light weight in a set of three and then weighing two of these with good weights reveals the light one. [3 weights 1, 2, 3 cannot be done in one weighing, but 9 weights 1, 2, ..., 9 can be done in two weighings.] Robert H. Thouless. The 12-balls problem as an illustration of the application of information theory. MG 54 (No. 389) (Oct 1970) 246-249. Uses information theory to show that the solution process is essentially determined. Ron Denyer. Letter. G&P, No. 37 (Jun 1975) 23. Asks for a mnemonic for the 12 coins puzzles. He notes that one can use three predetermined weighings and find the coin from the three answers. Basil Mager & E. Asher. Letters: Coining a mnemonic. G&P, No. 40 (Sep 1975) 26. One mnemonic for a variable method, another for a predetermined method. N. J. Maclean. Letter: The twelve coins. G&P, No. 45 (Feb 1976) 28©29. Exposits a ternary method for predetermined weighings for (3ÃÃnÄÄ©3)/2 in n weighings. Each weighing determines one ternary digit and the resulting ternary number gives both the coin and whether it is heavy or light. Tim Sole. The Ticket to Heaven and Other Superior Puzzles. Penguin, 1988. Weighty problems ©© (iii), pp. 124 & 147. Nine equal pies, except someone has removed some filling from one and inserted it in a pie, possibly the same one. Determine which, if any, are the heavy and light ones in 4 balancings. Calvin T. Long. Magic in base 3. MG 76 (No. 477) (Nov 1992) 371©376. Good exposition of the base 3 method for 12 coins. Ed Barbeau. After Math. Wall & Emerson, Toronto, 1995. Problems for an equal©arm balance, pp. 137©141. ÁÁÁÁ1. Six balls, two of each of three colours. One of each colour is lighter than normal and all light weights are equal. Determine the light balls in three weighings. ÁÁÁÁ2. Five balls, three normal, one heavy, one light, with the differences being equal, i.e. the heavy and the light weigh as much as two normals. Determine the heavy and light in three weighings. ÁÁÁÁ3. Same problem with nine balls and seven normals, done in four weighings. ÙÙ ÁÁà Ã5.C.1ÁÁRANKING COINS WITH A BALANCEÄ Ä Ðа¤˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐ ÁÁIf one weighs only one coin against another, this is the problem of sorting except that we don't actually put the objects in order. If one weighs pairs, etc., this is a more complex problem. ÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐ J. Schreier. Mathesis Polska 7 (1932) 154-160. ??NYS ©© cited by Steinhaus. Hugo Steinhaus. Mathematical Snapshots. Not in Stechert, NY, 1938, ed. OUP, NY: 1950: pp. 36-40 & 258; 1960: pp. 51-55 & 322; 1969 (1983): pp. 53-56 & 300. Shows n objects can be ranked in M(n) = 1 + kn - 2ÃÃkÄÄ steps where k = 1 + [logÃÃ2ÄÄ n]. Gets M(5) = 8. Lester R. Ford Jr. & Selmer M. Johnson. A tournament problem. AMM 66:5 (May 1959) 387-389. Note that ÀIÀlogÃÃ2ÄÄ n!ÀJÀ = L(n) is a lower bound from information theory. Obtain a better upper bound than Steinhaus, denoted U(n), which is too complex to state here. For convenience, I give the table of these values here. ÁÁÁÁ nÁÁÁÁ1 2 3 4 5 6 7 8 9 10 11 12 13 ÁÁÁÁM(n)ÁÁ0 1 3 5 8 11 14 17 21 25 29 33 37 ÁÁÁÁU(n)ÁÁÁÁ0 1 3 5 7 10 13 16 19 22 26 30 34 ÁÁÁÁL(n)ÁÁÁÁ0 1 3 5 7 10 13 16 19 22 26 29 33 ÁÁU(n) = L(n) also holds at n = 20 and 21. Roland Sprague. Unterhaltsame Mathematik. Op. cit. in 4.A.1. 1961. Prob. 22: Ein noch ungelÀ?Àstes Problem, pp. 16 & 42-43. (= A still unsolved problem, pp. 17 & 48-49.) Sketches Steinhaus's method, then does 5 objects in 7 steps. Gives the lower bound L(n) and says the case n = 12 is still unsolved. Kobon Fujimura, proposer; editorial comment. Another balance scale problem. RMM 10 (Aug 1962) 34 & 11 (Oct 1962) 42. Eight coins of different weights and a balance. How many weighings are needed to rank the coins? In No. 11, it says the solution will appear in No. 13, but it doesn't appear there or in the last issue, No. 14. It also doesn't appear in the proposer's Tokyo Puzzles. Howard P. Dinesman. Superior Mathematical Puzzles. Op. cit. in 5.B.1. 1968. No. 6: In the balance, pp. 18 & 85©86. Rank five balls in order in seven weighings. John Cameron. Establishing a pecking order. MG 55 (No. 394) (Dec 1971) 391-395. Reduces Steinhaus's M(n) by 1 for n ÀÀ 5, but this is not as good as Ford & Johnson. W. Antony Broomhead. Letter: Progress in congress? MG 56 (No. 398) (Dec 1972) 331. Comments on Cameron's article and says Cameron can be improved. States the values U(9) and U(10), but says he doesn't know how to do 9 in 19 steps. Cites Sprague for numerical values, but these don't appear in Sprague ©© so Broomhead presumably computed L(9) and L(10). He gets 10 in 23 steps, which is better than Cameron. Stanley Collings. Letter: More progress in congress. MG 57 (No. 401) (Oct 1973) 212-213. Notes the ambiguity in Broomhead's reference to Sprague. Improves Cameron by 1 (or more??) for n ÀÀ 10, but still not as good as Ford & Johnson. L. J. Upton, proposer; Leroy J. Myers, solver. Problem 1138. CM 12 (1986) 79 & 13 (1987) 230-231. Rank coins weighing 1, 2, 3, 4 with a balance in four weighings. ÁÁà Ã5.D.ÁÁMEASURING PROBLEMSÄ Ä ÁÁà Ã5.D.1.ÁÁJUGS & BOTTLESÄ Ä Ðа¤˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐ ÁÁSee MUS I 105©124, Tropfke 659. ÁÁNOTATION: I©(a, b, c) means we have three jugs of sizes a, b, c with a full and we want to divide a in half using b and c. We normally assume a ÀÀ b ÀÀ c and GCD(a, b, c) = 1. Halving a is clearly impossible if GCD(b, c) does not divide a/2 or if b+c < a/2, unless one has a further jug or one can drink some. If a ÀÀ b+c ÀÀ a/2 and GCD(b, c) divides a/2, then the problem is solvable. ÁÁMore generally, the question is to determine what amounts can be produced, i.e. given a, b, c as above, can one measure out an amount d? We denote this by II©(a, b, c; d). Since this also produces a©d, we can assume that d ÀÀ a/2. Then we must have d ÀÀ b+c for a solution. When a ÀÀ b+c ÀÀ d, the condition GCD(b, c) À À d guarantees that d can be produced. This also holds for a = b+c-1 and a = b+c-2. The simplest impossible cases are I-(4, 4, 3) = II©(4, 4, 3; 2) and II-(5, 5, 3; 1). Case I-(a, b, c) is the same as II©(a, b, c; a/2). ÁÁIf a is a large source, e.g. a stream or a big barrel, we have the problem of measuring d using b and c without any constraint on a and we denote this II©(ÀÀ, b, c; d). However, the solution may not use the infiniteness of the source and such a problem may be the same as II-(b+c, b, c; d). ÁÁThe general situation when a < b+c is more complex and really requires us to consider the most general three jug problem: III-(A; a, b, c; d) means we have three jugs of sizes a, b, c, containing a total amount of liquid A (in some initial configuration) and we wish to measure out d. In our previous problems, we had A = a. Clearly we must have a+b+c ÀÀ A. Again, producing d also produces A©d, so we can assume d ÀÀ A/2. By considering the amounts of empty space in the containers, the problem III©(A; a, b, c; d) is isomorphic to III-(a+b+c-A; a, b, c; d') for several possible d'. ÁÁ ÁÁNOTES. I have been re©examining this problem and I am not sure if I have reached a final interpretation and formulation. Also, I have recently changed to the above notation and I may have made some errors in so doing. I have long had the problem in my list of projects for students, but no one looked at it until 1995©1996 when Nahid Erfani chose it. She has examined many cases and we have have discovered a number of properties which I do not recall seeing. E.g. in case I©(a,b,c) with a ÀÀ b ÀÀ c and GCD(b,c) = 1, there are two ways to obtain a/2. If we start by pouring into b, it takes b + c © 1 pourings; if we start by pouring into c, it takes b + c pourings; so it is always best to start pouring into the larger jug. A number of situations II©(a,b,c;d) are solvable for all values of d, except a/2. E.g. II-(a,b,c;a/2) with b+c > a and c > a/2 is unsolvable. ÁÁFrom about the mid 19C, I have not recorded simple problems. ÐФH tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐ I©( 8, 5, 3):ÁÁalmost all the entries below I©(10, 6, 4):ÁÁPacioli, Court I©(10, 7, 3):ÁÁYoshida I©(12, 7, 5):ÁÁPacioli, van Etten/Henrion, Ozanam, Bestelmeier, Jackson, Manuel des Sorciers, Boy's Own Conjuring Book I©(12, 8, 4):ÁÁPacioli I©(12, 8, 5):ÁÁBachet, Arago I©(16, 9, 7):ÁÁBachet©Labosne I©(16,11, 6):ÁÁBachet©Labosne I©(16,12, 7):ÁÁBachet©Labosne I©(20,13, 9):ÁÁBachet©Labosne I©(42,27,12):ÁÁBachet©Labosne II©(10,3,2;6)ÁÁLeacock = II(10,3,2;4) II©(11,4,3;9):ÁÁMcKay = II(11,4,3;2) II©( ÀÀ,5,3;1):ÁÁWood, Serebriakoff, Diagram Group II©( ÀÀ,5,3;4):ÁÁChuquet, Wood, Fireside Amusements, II©( ÀÀ,7,4;5):ÁÁMeyer, Stein, Brandes II©( ÀÀ,8,5;11):ÁÁYoung World, III©(20;19,13,7;10):ÁÁDevi ÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐ General problem, usually form I, sometimes form II: Bachet-Labosne, Schubert, Ahrens, Cowley, Tweedie, Grossman, Buker, Goodstein, Browne, Scott, Currie, Sawyer, Court, O'Beirne, Lawrence, McDiarmid & Alfonsin. Versions with 4 or more jugs: Tartaglia, Anon: Problems drive (1958), Anon (1961), O'Beirne. Impossible versions: Pacioli, Bachet, Anon: Problems drive (1958). Abbot Albert. c1240. Prob. 4, p. 333. I©(8,5,3) ©© one solution. Columbia Algorism. c1350. Chap. 123: I©(8,5,3). Cowley 402-403 & plate opposite 403. The plate shows the text and three jars. I have a colour slide of the three jars from the MS. Munich 14684. 14C. Prob. XVIII & XXIX, pp. 80 & 83. I©(8,5,3). Folkerts. Aufgabensammlungen. 13©15C. 16 sources with I©(8,5,3). Pseudo©dell'Abbaco. c1440. Prob. 66, p.62. I©(8,5,3) ©© one solution. "This problem is of little utility ...." I have a colour slide of this. Chuquet. 1484. Prob. 165. Measure 4 from a cask using 5 and 3. You can pour back into the cask, i.e. this is II©(ÀÀ,5,3;4). FHM 233 calls this the tavern©keeper's problem. HB.XI.22. 1488. P. 55 (= Rath 248). Same as Abbot Albert. Pacioli. De Viribus. c1500. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐFf. 97r © 97v. LIII. C(apitolo). apartire una botte de vino fra doi (To divide a bottle of wine between two). = Peirani 137©138. I©(8,5,3). One solution. Ff. 97v © 98v. LIIII. C(apitolo). a partire unaltra botte fra doi (to divide another bottle between two). = Peirani 138©139. I©(12,7,5). Dario Uri points out that the solution is confused and he repeats himself so it takes him 18 pourings instead of the usual 11. He then says one can divide 18 among three brothers who have containers of sizes 5, 6, 7, which he does by filling the 6 and then the problem is reduced to the previous problem. [He could do it rather more easily by pouring the 6 into the 7 and then refilling the 6!] Ff. 98v © 99r. LV. (Capitolo) de doi altri sotili divisioni. de botti co'me se dira (Of two other subtle divisions of bottles as described). = Peirani 139©140. I-(10,6,4) and I©(12,8,4). Pacioli suggests giving these to idiots. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐGhaligai. Practica D'Arithmetica. 1521. Prob. 20, ff. 64v©65r. I-(8,5,3). One solution. Cardan. Practica Arithmetice. 1539. Chap. 66, section 33, f. DD.iiii.v (p. 145). I©(8,5,3). Gives one solution and says one can go the other way. H&S 51 says I©(8,5,3) case is also in Trenchant (1566). ??NYS Tartaglia. General Trattato, 1556, art. 132 & 133, p. 255v-256r. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐArt. 132: I©(8,5,3). Art. 133: divide 24 in thirds, using 5, 11, 13. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐButeo. Logistica. 1559. Prob. 73, pp. 282©283. I©(8,5,3). Gori. Libro di arimetricha. 1571. Ff. 71r-71v (p. 76). I©(8,5,3). Bachet. Problemes. 1612. Addl. prob. III: Deux bons compagnons ont 8 pintes de vin À!À partager entre eux À)Àgalement, ..., 1612: 134©139; 1624: 206©211; 1884: 138-147. I-(8,5,3)  ©© both solutions; I©(12,8,5) (omitted by Labosne). Labosne adds I-(16,9,7); I-(16,11,6); I-(42,27,12); I©(20,13,9); I©(16,12,7) (an impossible case!) and discusses general case. (This seems to be the first discussion of the general case.) van Etten. 1624. Prob. 9 (9), pp. 11 & fig. opp. p. 1 (pp. 22-23). I-(8,5,3) ©© one solution. Henrion's Nottes, 1630, pp. 11-13, gives the second solution and poses and solves I-(12,7,5). Hunt. 1631 (1651). P. 270 (262). I©(8,5,3). One solution. Yoshida (Shichibei) KÀ¥ÀyÀÁÀ (= Mitsuyoshi Yoshida) (1598©1672). JinkÀ¥À-ki. 2nd ed., 1634 or 1641??. ??NYS The recreational problems are discussed in Kazuo Shimodaira; Recreative Problems on "JingÀ¥Àki", a 15 pp booklet sent by Shigeo Takagi. [This has no details, but Takagi says it is a paper that Shimodaira read at the 15th International Conference for the History of Science, Edinburgh, Aug 1977 and that it appeared in Japanese Studies in the History of Science 16 (1977) 95©103. I suspect this is a copy of a preprint.] This gives both JingÀ¥Àki and JinkÀ¥Àki as English versions of the title and says the recreational problems did not appear in the first edition, 4 vols., 1627, but did appear in the second edition of 5 vols. (which may be the first use of coloured wood cuts in Japan), with the recreational problems occurring in vol. 5. He doesn't give a date, but Mikami, p. 179, indicates that it is 1634, with further editions in 1641, 1675, though an earlier work by Mikami (1910) says 2nd ed. is 1641. Yoshida (or Suminokura) is the family name. Shimodaira refers to the current year as the 350th anniversary of the edition and says copies of it were published then. I have a recent transcription of some of Yoshida into modern Japanese and a more recent translation into English, ??NYR, but I don't know if it is the work mentioned by Shimodaira. ÁÁÁÁShimodaira discusses a jug problem on p. 14: I©(10,7,3) ©© solution in 10 moves. Shimodaira thinks Yoshida heard about such puzzles from European contacts, but without numerical values, then made up the numbers. I certainly can see no other example of these numbers. The recent transcription includes this material as prob. 7 on pp. 69©70. Wingate/Kersey. 1678?. Prob. 7, pp. 543©544. I©(8,5,3). Says there is a second way to do it. Witgeest. Het Natuurlyk Tover©Boek. 1686. Prob. 38, p. 308. I©(8,5,3). Ozanam. 1694. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐProb. 36, 1696: 91©92; 1708: 82-83. Prob. 42, 1725: 238-240. Prob. 21, 1778: 175-177; 1803: 174©176; 1814: 153©154. Prob. 20, 1840: 79. I©(8,5,3) ©© both solutions. Prob. 43, 1725: 240-241. Prob. 22, 1778: 177©178; 1803: 176©177; 1814: 154©155. Prob. 21, 1840: 79-80. I©(12,7,5) ©© one solution. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐDilworth. Schoolmaster's Assistant. 1743. Part IV: Questions: A short Collection of pleasant and diverting Questions, p. 168. Problem 8. I©(8,5,3). (Dilworth cites Wingate for this ©© cf in 5.B.) = D. Adams; Scholar's Arithmetic; 1801, p. 200, no. 10. Les Amusemens. 1749. Prob. 17, p. 139: Partages À)Àgaux avec des Vases inÀ)Àgaux. I©(8,5,3) ª© both solutions. Bestelmeier. 1801. Item 416: Die 3 Maas-GefÀÀss. I©(12,7,5). Badcock. Philosophical Recreations, or, Winter Amusements. [1820]. Pp. 48©49, no. 75: How to part an eight gallon bottle of wine, equally between two persons, using only two other bottles, one of five gallons, and the other of three. Gives both solutions. Jackson. Rational Amusement. 1821. Arithmetical Puzzles. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐNo. 14, pp. 4 & 54. I©( 8,5,3). One solution. No. 52, pp. 12 & 67. I©(12,7,5). One solution. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐRational Recreations. 1824. Exer. 10, p. 55. I©(8,5,3) one way. Manuel des Sorciers. 1825. ??NX Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐPp. 55©56, art. 27©28. I©(8,5,3) two ways. P. 56, art. 29. I©(12,7,5). Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐEndless Amusement II. 1826? Prob. 7, pp. 193©194. I©(8,5,3). One solution. = New Sphinx, c1840, p. 133. Nuts to Crack III (1834), no. 212. I©(8,5,3). 8 gallons of spirits. Young Man's Book. 1839. Pp. 43©44. I©(8,5,3). Identical to Wingate/Kersey. The New Sphinx. c1840. P. 133. I©(8,5,3). One solution. Boy's Own Book. 1843 (Paris): 436 & 441, no. 7. The can of ale: 1855: 395; 1868: 432. I-(8,5,3). One solution. The 1843 (Paris) reads as though the owners of the 3 and 5 kegs both want to get 4, which would be a problem for the owner of the 3. = Boy's Treasury, 1844, pp. 425 & 429. Fireside Amusements. 1850. Prob. 9, pp. 132 & 184. II©(ÀÀ,5,3;4). One solution. Arago. [Biographie de] Poisson (16 Dec 1850). Oeuvres, Gide & Baudry, Paris, vol. 2, 1854, pp. 593-??? P. 596 gives the story of Poisson's being fascinated by the problem I-(12,8,5). "Poisson rÀ)Àsolut À!À l'instant cette question et d'autres dont on lui donna l'À)ÀnoncÀ)À. Il venait de trouver sa vÀ)Àritable vocation." No solution given by Arago. Parlour Pastime, 1857. = Indoor & Outdoor, c1859, Part 1. = Parlour Pastimes, 1868. Arithmetical puzzles, no. 8, pp. 174©175 (1868: 185©186). I©(8,5,3). Milkmaid with eight quarts of milk. Magician's Own Book. 1857. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐP. 223©224: Dividing the beer: I©(8,5,3). P. 224: The difficult case of wine: I©(12,7,5). Pp. 235©236: The two travellers: I©(8,5,3) posed in verse. ÁÁEach problem gives just one solution. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐBoy's Own Conjuring Book. 1860. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐP. 193: Dividing the beer: I©(8,5,3). P. 194: The difficult case of wine: I©(12,7,5). Pp. 202-203: The two travellers: I©(8,5,3) posed in verse. ÁÁEach problem gives just one solution. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐIllustrated Boy's Own Treasury. 1860. Prob. 21, pp. 428©429 & 433. I-(8,5,3). "A man coming from the Lochrin distillery with an 8©pint jar full of spirits, ...." Vinot. 1860. Art. XXXVIII: Les cadeaux difficiles, pp. 57©58. I©(8,5,3). Two solutions. The Secret Out (UK). c1860. To divide equally eight pints of wine ..., pp. 12©13. Bachet©Labosne. 1874. For details, see Bachet, 1612. Labosne adds a consideration of the general case which seems to be the first such. Kamp. Op. cit. in 5.B. 1877. No. 17, p. 326: I©(8,5,3). Mittenzwey. 1880. Prob. 106, pp. 22 & 73©74; 1895?: 123, pp. 26 & 75©76; 1917: 123, pp. 24 & 73©74. I©(8,5,3). One solution. Don Lemon. Everybody's Pocket Cyclopedia. Revised 8th ed., 1890. Op. cit. in 5.A. P. 135, no. 1. I©(8,5,3). No solution. Loyd. Problem 11: "Two thieves of Damascus". Tit-Bits 31 (19 Dec 1896 & 16 Jan 1897) 211 & 287. Thieves found with 2 & 2 quarts in pails of size 3 & 5. They claim the merchant measured the amounts out from a fresh hogshead. Solution is that this could only be done if the merchant drained the hogshead, which is unreasonable! Loyd. Problem 13: The Oriental problem. Tit-Bits 31 (19 Jan, 30 Jan & 6 Feb 1897) 269, 325 & 343. = Cyclopedia, 1914, pp. 188 & 364: The merchant of Bagdad. Complex problem with hogshead of water, barrel of honey, three 10 gallon jugs to be filled with 3 gallons of water, of honey and of half and half honey & water. There are a 2 and a 4 gallon measure and also 13 camels to receive 3 gallons of water each. Solution takes 521 steps. 6 Feb reports solutions in 516 and 513 steps. Cyclopedia gives solution in 506 steps. Dudeney. The host's puzzle. London Magazine 8 (No. 46) (May 1902) 370 & 8 (No. 47) (Jun 1902) 481-482 (= CP, prob. 6, pp. 28-29 & 166-167). Use 5 and 3 to obtain 1 and 1 from a cask. One must drink some! H. Schubert. Mathematische Mussestunden, 3rd ed., GÀ?Àschen, Leipzig, 1907. Vol. 1, chap. 6, UmfÀGÀllungs-Aufgaben, pp. 48-56. Studies general case and obtains some results. (The material appeared earlier in ZwÀ?Àlf Geduldspiele, 1895, op. cit. in 5.A, Chap. IX, pp. 110©119. The 13th ed. (De Gruyter, Berlin, 1967), Chap. 9, pp. 62-70, seems to be a bit more general (??re©read).) Ahrens. MUS I, 1910, chap. 4, UmfÀGÀllungsaufgaben, pp. 105-124. Pp. 106-107 is Arago's story of Poisson and this problem. He also extends and corrects Schubert's work. Dudeney. Perplexities: No. 141: New measuring puzzle. Strand Magazine 45 (Jun 1913) 710 & 46 (Jul 1913) 110. (= AM, prob. 365, pp. 110 & 235.) Two 10 quart vessels of wine with 5 and 4 quart measures. He wants 3 quarts in each measure. (Dudeney gives numerous other versions in AM.) Loyd. Cyclopedia. 1914. Milkman's puzzle, pp. 52 & 345. (= MPSL2, prob. 23, pp. 17 & 127-128 = SLAHP: Honest John, the milkman, pp. 21 & 90.) Milkman has two full 40 quart containers and two customers with 5 and 4 quart pails, but both want 2 quarts. (Loyd Jr. says "I first published [this] in 1900...") Williams. Home Entertainments. 1914. The measures puzzle, p. 125. I©(8,5,3). Hummerston. Fun, Mirth & Mystery. 1924. A shortage of milk, Puzzle no. 75, pp. 164 & 183. I©(8,5,3), one solution. Elizabeth B. Cowley. Note on a linear diophantine equation. AMM 33 (1926) 379-381. Presents a technique for resolving I©(a,b,c), which gives the result when a = b+c. If a < b+c, she only seems to determine whether the method gets to a point with A empty and neither B nor C full and it is not clear to me that this implies impossibility. She mentions a graphical method of Laisant (Assoc. FranÀ'À. Avance. Sci, 1887, pp. 218©235) ??NYS. Wood. Oddities. 1927. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐProb. 15: A problem in pints, pp. 16©17. Small cask and measures of size 5 and 3, measure out 1 in each measure. Starts by filling the 5 and the 3 and then emptying the cask, so this becomes a variant of II©(ÀÀ,5,3;1). Prob. 26: The water©boy's problem, pp. 28©29. II©(ÀÀ;,5,3;4). Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐErnest K. Chapin. Scientific Problems and Puzzles. In: S. Loyd Jr.; Tricks and Puzzles, Vol. 1 (only volume to appear); Experimenter Publishing Co., NY, nd [1927] and Answers to Sam Loyd's Tricks and Puzzles, nd [1927]. [This book is a selection of pages from the Cyclopedia, supplemented with about 20 pages by Chapin and some other material.] P. 89 & Answers p. 8. You have a tablet that has to be dissolved in 7ÀÀ quarts of water, though you only need 5 quarts of the resulting mixture. You have 3 and 5 quart measures and a tap. Stephen Leacock. Model Memoirs and Other Sketches from Simple to Serious. John Lane, The Bodley Head, 1939, p. 298. "He's trying to think how a farmer with a ten©gallon can and a three©gallon can and a two©gallon can, manages to measure out six gallons of milk." II©(10,3,2;6) = II©(10,3,2;4). M. C. K. Tweedie. A graphical method of solving Tartaglian measuring puzzles. MG 23 (1939) 278-282. The elegant solution method using triangular coordinates. H. D. Grossman. A generalization of the water-fetching puzzle. AMM 47 (1940) 374-375. Shows II©(ÀÀ,b,c;d) with GCD(b,c) = 1 is solvable. McKay. Party Night. 1940. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐNo. 18, p. 179. II©(11,4,3;9). No. 19, pp. 179©180. I©(8,5,3). Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐMeyer. Big Fun Book. 1940. No. 10, pp. 165 & 753. II©(ÀÀ,7,4,5). W. E. Buker, proposer. Problem E451. AMM 48 (1941) 65. ??NX. General problem of what amounts are obtainable using three jugs, one full to start with, i.e. I©(a,b,c). See Browne, Scott, Currie below. Eric Goodstein. Note 153: The measuring problem. MG 25 (No. 263) (Feb 1941) 49-51. Shows II©(ÀÀ,b,c;d) with GCD(b,c) = 1 is solvable. D. H. Browne & Editors. Partial solution of Problem E451. AMM 49 (1942) 125-127. W. Scott. Partial solution of E451 ©© The generalized water-fetching puzzle. AMM 51 (1944) 592. Counterexample to conjecture in previous entry. J. C. Currie. Partial solution of Problem E451. AMM 53 (1946) 36-40. Technical and not complete. W. W. Sawyer. On a well known puzzle. SM 16 (1950) 107-110. Shows that I©(b+c,b,c) is solvable if b & c are relatively prime. David Stein. Party and Indoor Games. Op. cit. in 5.B. c1950. Prob. 13, pp. 79-80. Obtain 5 from a spring using measures 7 and 4, i.e. II©(ÀÀ,7,4,5). Anonymous. Problems drive, 1958. Eureka 21 (Oct 1958) 14©16 & 30. No. 8. Given an infinite source, use: 6, 10, 15 to obtain 1, 6, 7 simultaneously; 4, 6, 9, 12 to obtain 1, 2, 3, 4 simultaneously; 6, 9, 12, 15, 21 to obtain 1, 3, 6, 8, 9 simultaneously. Answer simply says the first two are possible (the second being easy) and the third is impossible. Young World. c1960. P. 58: The 11 pint problem. II©(ÀÀ,8,5;11). This is the same as II-(13,8,5;11) or II©(13,8,5,2). Anonymous. Moonshine sharing. RMM 2 (Apr 1961) 31 & 3 (Jun 1961) 46. Divide 24 in thirds using cylindrical containers holding 10, 11, 13. Solution in No. 3 uses the cylindricity of a container to get it half full. Nathan Altshiller Court. Mathematics in Fun and in Earnest. Op. cit. in 5.B. 1961. "Pouring" problems ©© The "robot" method. General description of the problem. Attributes Tweedie's triangular 'bouncing ball' method to Perelman, with no reference. Does I-(8,5,3) two ways, also I©(12,7,5) and I©(16,9,7), then considers type II questions. Considers the problem with II©(10,6,4;d) and extends to II©(a,6,4;d) for a > 10, leaving it to the reader to "try to formulate some rule about the results." He then considers II-(7,6,4;d), noting that the parallelogram has a corner trimmed off. Then considers II©(12,9,7;d) and II©(9,6,3;d). Lloyd Jim Steiger. Letter. RMM 4 (Aug 1961) 62. Solves the RMM 2 problem by putting the 10 inside the 13 to measure 3. Irving & Peggy Adler. The Adler Book of Puzzles and Riddles. Or Sam Loyd Up©To©Date. John Day, NY, 1962. Pp. 32 & 46. Farmer has two full 10©gallon cans. Girls come with 5©quart and 4©quart cans and each wants 2 quarts. Philip Kaplan. More Posers. (Harper & Row, 1964); Macfadden©Bartell Books, 1965. Prob. 80, pp. 81 & 109. Tavern has a barrel with 15 pints of beer. Two customers, with 3 pint and 5 pint jugs appear and ask for 1 pint in each jug. Bartender finds it necessary to drink the other 13 pints! T. H. O'Beirne. Puzzles and Paradoxes. OUP, 1965. Chap. 4: Jug and bottle department, pp. 49-75. This gives an extensive discussion of Tweedie's method and various extensions to four containers, a barrel of unknown size, etc. P. M. Lawrence. An algebraic approach to some pouring problems. MG 56 (No. 395) (Feb 1972) 13-14. Shows II©(ÀÀ,b,c,d) with d ÀÀ b+c and GCD(b,c) = 1 is possible and extends to more jugs. Louis Grant Brandes. The Math. Wizard. revised ed., J. Weston Walch, Portland, Maine, 1975. Prob. 5: Getting five gallons of water: II-(ÀÀ,7,4,5). Shakuntala Devi. Puzzles to Puzzle You. Orient Paperbacks (Vision Press), Delhi, 1976. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐProb. 53: The three containers, pp. 57 & 110. III©(20;19,13,7;10). Solution in 15 steps. Looking at the triangular coordinates diagram of this, one sees that it is actually isomorphic to II©(19,13,7;10) and this can be seen by considering the amounts of empty space in the containers. Prob. 132: Mr. Portchester's problem, pp. 82 & 132. Same as Dudeney (1913). Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐVictor Serebriakoff. A Mensa Puzzle Book. Muller, London, 1982. (Later combined with A Second Mensa Puzzle Book, 1985, Muller, London, as: The Mensa Puzzle Book, Treasure Press, London, 1991.) Problem T.16: Pouring puttonos, part b, pp. 19©20 (1991: 37©38) & Answer 19, pp. 102©103 (1991: 118©119). II©( ÀÀ,5,3;1). The Diagram Group. The Family Book of Puzzles. The Leisure Circle Ltd., Wembley, Middlesex, 1984. Problem 161, with Solution at the back of the book. II©(ÀÀ,5,3;1), which can be done as II©(8,5,3;1). D. St. P. Barnard. 50 Daily Telegraph Brain Twisters. 1985. Op. cit. in 4.A.4. Prob. 4: Measure for measure, pp. 15, 79-80, 103. Given 10 pints of milk, an 8 pint bowl, a jug and a flask. He describes how he divides the milk in halves and you must deduce the size of the jug and the flask. Colin J. H. McDiarmid & Jorge Ramirez Alfonsin. Sharing jugs of wine. Discrete Mathematics 125 (1994) 279©287. Solves I©(b+c,b,c) and discusses the problem of getting from one state of the problem to another in a given number of steps, showing that GCD(b,c) = 1 guarantees the graph is connected. indeed essentially cyclic. Considers GCD(b,c) ÀcÀ 1. Notes that the work done easily extends to a > b + c. Says the second author's PhD at Oxford, 1993, deals with more cases. John P. Ashley. Arithmetickle. Arithmetic Curiosities, Challenges, Games and Groaners for all Ages. Keystone Agencies, Radnor, Ohio, 1997. P. 11: The spoon and the bottle. Given a 160 ml bottle and a 30 ml spoon, measure 230 ml into a bucket. ÁÁà Ã5.D.2.ÁÁRULER WITH MINIMAL NUMBER OF MARKSÄ Ä Dudeney. Problem 518: The damaged measure. Strand Mag. (Sep 1920) ??NX. Wants a minimal ruler for 33 inches total length. (=? MP 180) Dudeney. Problem 530: The six cottagers. Strand Mag. (Jan 1921) ??NX. Wants 6 points on a circle to give all arc distances 1, 2, ..., 20. (=? MP 181) Percy Alexander MacMahon. The prime numbers of measurement on a scale. Proc. Camb. Philos. Soc. 21 (1922-23) 651-654. He considers the infinite case, i.e. a(0) = 0, a(i+1) = a(i) + least integer which is not yet measurable. This gives: 0, 1, 3, 7, 12, 20, 30, 44, .... Dudeney. MP. 1926. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐProb. 180: The damaged measure, pp. 77 & 167. (= 536, prob. 453, pp. 173, 383-384.) Mark a ruler of length 33 with 8 (internal) marks. Gives 16 solutions. Prob. 181: The six cottagers, pp. 77-78 & 167. = 536, prob. 454, pp. 174 & 384. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐA. Brauer. A problem of additive number theory and its application in electrical engineering. J. Elisha Mitchell Sci. Soc. 61 (1945) 55-56. Problem arises in designing a resistance box. À À. À" ÀÀ ÀÀ ÀÀ ÀÀ À & À À. À" ÀÀ ÀÀ ÀÀ; ÀÀ À [L. Redei & A. Ren'i (RÀ)Ànyi)]. À À À! ÀÀ# ÀÀ ÀÀ ÀÀ% ÀÀ' ÀÀ ÀÀ ÀÀ ÀÀ ÀÀ ÀÀ ÀÀ À À1 ÀÀ ÀÀ% ÀÀ ÀÀ À 1, 2, ..., N À ÀÀ ÀÀ% ÀÀ# ÀÀ ÀÀ ÀÀ% ÀÀ' ÀÀ ÀÀ ÀÀ À À# ÀÀ ÀÀ ÀÀ ÀÀ ÀÀ% ÀÀ' ÀÀ ÀÀ À [O predstavlenin chisel 1, 2, ... , N losredstvom raznostei (On the representation of 1, 2, ..., N by differences)]. Mat. À$ ÀÀ ÀÀ ÀÀ# ÀÀ ÀÀ ÀÀ À [Mat. Sbornik] 66 (NS 24) (1949) 385-389. Anonymous. An unsolved problem. Eureka 11 (Jan 1949) 11 & 30. Place as few marks as possible to permit measuring integers up to n. For n = 13, an example is: 0, 1, 2, 6, 10, 13. Mentions some general results for a circle. John Leech. On the representation of 1, 2, ..., n by differences. J. London Math. Soc. 31 (1956) 160-169. Improves Redei & RÀ)Ànyi's results. Gives best examples for small n. Anon. Puzzle column: What's your potential? MTg 19 (1962) 35 & 20 (1962) 43. Problem posed in terms of transformer outputs ©© can we arrange 6 outputs to give every integral voltage up through 15? Problem also asks for the general case. Solution asserts, without real proof, that the optimum occurs with 0, 1, 4, 7, 10, ..., n-11, n-8, n-5, n-2 or its complement. T. H. O'Beirne. Puzzles and Paradoxes. OUP, 1965. Chap. 6 discusses several versions of the problem. Gardner. SA (Jan 1965) c= Magic Numbers, chap. 6. Describes 1, 2, 6, 10 on a ruler 13 long. Says 3 marks are sufficient on 9 and 4 marks on 12 and asks for proof of the latter and for the maximum number of distances that 3 marks on 12 can produce. How can you mark a ruler 36 long? Says Dudeney, MP prob. 180, believed that 9 marks were needed for a ruler longer than 33, but Leech managed to show 8 was sufficient up to 36. C. J. Cooke. Differences. MTg 47 (1969) 16. Says the problem in MTg 19 (1962) appears in H. L. Dorwart's The Geometry of Incidence (1966) related to perfect difference sets but with an erroneous definition which is corrected by references to H. J. Ryser's Combinatorial Mathematics. However, this doesn't prove the assertions made in MTg 20. Jonathan Always. Puzzles for Puzzlers. Tandem, 1971. Prob. 22: Starting and stopping, pp. 18 & 66. Circular track, 1900 yards around. How can one place marker posts so every multiple of 100 yards up to 1900 can be run. Answer: at 0, 1, 3, 9, 15. Gardner. SA (Mar 1972) = Wheels, Chap. 15. ÁÁà Ã5.D.3ÁÁFALSE COINS WITH A WEIGHING SCALEÄ Ä H. S. Shapiro, proposer; N. J. Fine, solver. Problem E1399 ©© Counterfeit coins. AMM 67 (1960) 82 & 697-698. Genuines weigh 10, counterfeits weigh 9. Given 5 coins and a scale, how many weighings are needed to find the counterfeits? Answer is 4. Fine conjectures that the ratio of weighings to coins decreases to 0. Kobon Fujimura & J. A. H. Hunter, proposers; editorial solution. There's always a way. RMM 6 (Dec 1961) 47 & 7 (Feb 1962) 53. (c= Fujimura's The Tokyo Puzzles (Muller, London, 1979), prob. 29: Pachinko balls, pp. 35 & 131.) Six coins, one false. Determine which is false and whether it is heavy or light in three weighings on a scale. In fact one also finds the actual weights. K. Fujimura, proposer; editorial solution. The 15-coin puzzle. RMM 9 (Jun 1962) & 10 (Aug 1962) 40-41. Same problem with fifteen coins and four weighings. ÁÁà Ã5.D.4.ÁÁTIMING WITH HOURGLASSESÄ Ä Ðа¤˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐÁÁI have just started these and they are undoubtedly older than the examples here. I don't recall ever seeing a general approach to these problems. ÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐ Simon Dresner. Science World Book of Brain Teasers. 1962. Op. cit. in 5.B.1. Prob. 17: Two-minute eggs, pp. 9 & 87. Time 2 minutes with 3 & 5 minute timers. Howard P. Dinesman. Superior Mathematical Puzzles. Op. cit. in 5.B.1. 1968. No. 21: The sands of time, pp. 35 & 93. Time 9 minutes with 4 & 7 minute timers. David B. Lewis. Eureka! Perigee (Putnam), NY, 1983. Pp. 73-74. Time 9 minutes with 4 & 7 minute timers. Yuri B. Chernyak & Robert S. Rose. The Chicken from Minsk. BasicBooks, NY, 1995. Chap. 1, prob. 8: Grandfather's breakfast, pp. 6 & 102. Time 15 minutes with 7 & 11 minute timers. ÁÁà Ã5.D.5.ÁÁMEASURE HALF A BARRELÄ Ä ÁÁI have just started this and there must be much older examples. Benson. 1904. The water-glass puzzle, p. 254. Dudeney. AM. 1917. Prob. 364: The barrel puzzle, pp. 109©110 & 235. King. Best 100. 1927. No. 1, pp. 7 & 38. Collins. Fun with Figures. 1928. The dairymaid's problem, pp. 29©30. William A. Bagley. Puzzle Pie. Vawser & Wiles, London, nd [BMC gives 1944]. [There is a revised edition, but it only affects material on angle trisection.] No. 14: 'Arf an' 'arf, p. 15. Anon. The Little Puzzle Book. Peter Pauper Press, Mount Vernon, NY, 1955. P. 52: The cider barrel. Jonathan Always. Puzzles for Puzzlers. Tandem, London, 1971. Prob. 87: But me no butts, pp. 42 & 88. Richard I. Hess. Email Christmas message to NOBNET, 24 Nov 2000. Solution sent by Nick Baxter on the same day. You have aquaria (assumed cuboidal) which hold 7 and 12 gallons and a water supply. The 12 gallon aquarium has dots accurately placed in the centre of each side face. How many steps are required to get 8 gallons into the 12 gallon aquarium? Fill the 12 gallon aquarium and tilt it on one corner so the water level passes through the centres of the two opposite faces. This leaves 8 gallons! Nick says this is two steps. ÁÁà Ã5.E.ÁÁEULER CIRCUITS AND MAZESÄ Ä Ðа¤˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐÁÁEuler circuits have been used in primitive art, often as symbols of the passage of the soul to the land of the dead. [MTg 110 (Mar 1985) 55] shows examples from Angola and New Hebrides. See Ascher (1988 & 1991) for many other examples from other cultures. ÙÙ ÁÁÁÁÁÁÁÁÀ ÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀ À ÁÁÁÁÁÁÁÁÀ À À À À À À À ÁÁÁÁÁÁÁÁÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀ ÁÁÁÁÁÁÁÁÀ À À À À À ÁÁÁÁÁÁÁÁÀ ÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀ À ÁÁAbove is the 'five-brick pattern'. See: Clausen, Listing, Kamp, White, Dudeney, Loyd Jr, Ripley, Meyer, Leeming, Adams, Anon., Ascher. Prior to Loyd Jr, the problem asked for the edges to be drawn in three paths, but about 1920 the problem changed to drawing a path across every wall. ÁÁTrick solutions: Tom Tit, Dudeney (1913), Houdini, Loyd Jr, Ripley, Meyer, Leeming, Adams, Gibson, Anon. (1986). ÁÁNon©crossing Euler circuits: Endless Amusement II, Bellew, Carroll 1869, Mittenzwey, Bile Beans, Meyer, Gardner (1964), Willson, Scott, Singmaster. ÁÁKÃÃnÄÄ denotes the complete graph on n vertices. ÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐMatthÀÀus Merian the Elder. Engraved map of KÀ?Ànigsberg. Bernhard Wiezorke has sent me a coloured reproduction of this, dated as 1641. He used an B&W version in his article: Puzzles und Brainteasers; OR News, Ausgabe 13 (Nov 2001) 52©54. BLW use a B&W version on their dust jacket and on p. 2 which they attribute to M. Zeiller; Topographia Prussiae et Pomerelliae; Frankfurt, c1650. I have seen this in a facsimile of the Cosmographica due to Merian in the volume on Brandenburg and Pomerania, but it was not coloured. There seem to be at least two versions of this picture ©©??CHECK. L. Euler. Solutio problematis ad geometriam situs pertinentis. (Comm. Acad. Sci. Petropol. 8 (1736(1741)) 128-140.) = Opera Omnia (1) 7 (1923) 1-10. English version: Seven Bridges of KÀ?Ànigsberg is in: BLW, 3-8; SA 189 (Jul 1953) 66-70; World of Mathematics, vol. 1, 573-580; Struik, Source Book, 183-187. My late colleague Jeremy Wyndham became interested in the seven bridges problem and made inquiries which turned up several maps of KÀ?Ànigsberg and a list of all the bridges and their dates of construction (though there is some ambiguity about one bridge). The first bridge was built in 1286 and until the seventh bridge of 1542, an Euler path was always possible. No further bridge was built until a railway bridge in 1865 which led to SaalschÀGÀtz's 1876 paper ©© see below. In 1905 and later, several more bridges were added, reaching a maximum of ten bridges in 1926 (with 4512 paths from the island), then one was removed in 1933. Then a road bridge was added, but it is so far out that it does not show on any map I have seen. Bombing and fighting in 1944©1945 apparently destroyed all the bridges and the Russians have rebuilt six or seven of them. I have computed the number of paths in each case ©© from 1865 until 1935 or 1944, there were always Euler paths. L. Poinsot. Sur les polygones et les polyÀ/Àdres. J. À(Àcole Polytech. 4 (Cah. 10) (1810) 16-48. Pp. 28-33 give Euler paths on KÃÃ2n+1ÄÄ and Euler's criterion. Discusses square with diagonals. Endless Amusement II. 1826? Prob. 34, p. 211. Pattern of two overlapping squares has a non©crossing Euler circuit. Th. Clausen. De linearum tertii ordinis propietatibus. Astronomische Nachrichten 21 (No. 494) (1844), col. 209-216. At the very end, he gives the five-brick pattern and says that its edges cannot be drawn in three paths. J. B. Listing. Vorstudien zur Topologie. GÀ?Àttinger Studien 1 (1847) 811-875. ??NYR. Gives five brick pattern as in Clausen. ?? Nouv. Ann. Math. 8 (1849?) 74. ??NYS. Lucas says this poses the problem of finding the number of linear arrangements of a set of dominoes. [For a double N set, N = 2n, this is (2n+1)(n+1) times the number of circular arrangements, which is nÃÃ2n+1ÄÄ times the number of Euler circuits on KÃÃ2n+1ÄÄ.] À(À. Coupy. Solution d'un problÀ/Àme appartenant a la gÀ)ÀomÀ)Àtrie de situation, par Euler. Nouv. Ann. Math. 10 (1851) 106-119. Translation of Euler. Translator's note on p. 119 applies it to the bridges of Paris. The Sociable. 1858. Prob. 7: Puzzle pleasure garden, pp. 288 & 303. Large maze©like garden and one is to pass over every path just once ©© phrased in verse. = Book of 500 Puzzles, 1859, prob. 7, pp. 6 & 21. = Illustrated Boy's Own Treasury, 1860, prob. 49, pp. 405 & 443. In fact, if one goes straight across every intersection, one finds the path, so this is really almost a unicursal problem. Leske. Illustriertes Spielbuch fÀGÀr MÀÀdchen. 1864? Prob. 587, pp. 297 & 410: AriadnerÀÀtsel. Three diagrams to trace with single lines. No attempt to avoid crossings. Frank Bellew. The Art of Amusing. Carleton, NY (& Sampson Low & Co., London), 1866 [C&B list a 1871]; John Camden Hotten, London, nd [BMC & NUC say 1870] and John Grant, Edinburgh, nd [c1870 or 1866?], with slightly different pagination. 1866: pp. 269©270; 1870: p. 266. Two overlapping squares have a non©crossing Euler circuit. Lewis Carroll. Letter of 22 Aug 1869 to Isabel Standen. Taken from: Stuart Dodgson Collingwood; The Life and Letters of Lewis Carroll; T. Fisher Unwin, London, (Dec 1898), 2nd ed., Jan 1899, p. 370: "Have you succeeded in drawing the three squares?" On pp. 369©370, the recipient is identified as Isabel Standen and she is writing Collingwood, apparently sending him the letter. Collingwood interpolates: "This puzzle was, by the way, a great favourite of his; the problem is to draw three interlaced squares without going over the same lines twice, or taking the pen off the paper". But no diagram is given. ÁÁÁÁDudeney; Some much-discussed puzzles; op. cit. in 2; 1908, quotes Collingwood, gives the diagram and continues: "This is sometimes ascribed to him [i.e. Carroll] as its originator, but I have found it in a little book published in 1835." This was probably a printing of Endless Amusement II, qv above and in Common References, though this has two interlaced squares. John Fisher; The Magic of Lewis Carroll; op. cit. in 1; pp. 58-59, says Carroll would ask for a non-crossing Euler circuit, but this is not clearly stated in Collingwood. Cf Carroll©Wakeling, prob. 29: The three squares, pp. 38 & 72, which clearly states that a non©crossing circuit is wanted and notes that there is more than one solution. Cf Gardner (1964). Carroll©Gardner, pp. 52©53. Mittenzwey. 1880. Prob. 269©279, pp. 47©48 & 98©100; 1895?: 298©308, pp. 51©52 & 100-102; 1917: 298©308, pp. 46©48 & 95©97. Straightforward unicursal patterns. The first is KÃÃ5ÄÄ, but one of the diagonals was missing in my copy of the 1st ed. ©© the path is not to use two consecutive outer edges. The third is the 'envelope' pattern. The fourth is three overlapping squares, where the two outer squares just touch in the middle. The last is a simple maze with no dead ends and the path is not to cross itself. See also the entry for Mittenzwey in 5.E.1, below. M. Reiss. À(Àvaluation du nombre de combinaisons desquelles les 28 dÀ)Às d'un jeu de dominos sont susceptibles d'aprÀ/Às la rÀ/Àgle de ce jeu. Annali di Matematica Pura ed Applicata (2) 5 (1871) 63-120. Determines the number of linear arrangements of a double-6 set of dominoes, which gives the number of Euler circuits on KÃÃ7ÄÄ. L. SaalschÀGÀtz. [Report of a lecture.] Schriften der Physikalisch-À>Àkonomischen Gesellschaft zu KÀ?Ànigsberg 16 (1876) 23-24. Sketches Euler's work, listing the seven bridges. Says that a recent railway bridge, of 1865, connecting regions B and C on Euler's diagram, can be considered within the walkable region. He shows there are 48 x 2 x 4 = 384 possible paths ©© the 48 are the lists of regions visited starting with A; the 2 corresponds to reversing these lists; the 4 (= 2 x 2) corresponds to taking each of the two pairs of bridges connecting the same regions in either order, He lists the 48 sequences of regions which start at A. I wrote a program to compute Euler paths and I tested it on this situation. I find that SaalschÀGÀtz has omitted two cases, leading to four sequences or 16 paths starting at A or 32 paths considering both directions. That is, his 48 should be 52 and his 384 should be 416. Kamp. Op. cit. in 5.B. 1877. Pp. 322-327 show several unicursal problems. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐNo. 8 is the five-brick pattern as in Clausen. No. 10 is two overlapping squares. No. 11 is a diagram from which one must remove some lines to leave an Eulerian figure. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐC. Hierholzer. Ueber die MÀ?Àglichkeit, einen Linienzug ohne Wiederholung und ohne Unterbrechung zu umfahren. Math. Annalen 6 (1873) 30-32. (English is in BLW, 11-12.) G. Tarry. GÀ)ÀomÀ)Àtrie de situation: Nombre de maniÀ/Àres distinctes de parcourir en une seule course toutes les allÀ)Àes d'un labyrinthe rentrant, en ne passant qu'une seule fois par chacune des allÀ)Àes. Comptes Rendus Assoc. FranÀ'À. Avance. Sci. 15, part 2 (1886) 49-53 & Plates I & II. General technique for the number of Euler circuits. Lucas.ÁÁRM2. 1883. Le jeu de dominos ©© Dispositions rectilignes, pp. 63-77 & Note 1: Sur le jeu de dominos, p. 229. ÁÁÁÁRM4. 1894. La gÀ)ÀomÀ)Àtrie des rÀ)Àseaux et le problÀ/Àme des dominos, pp. 123-151. ÁÁÁÁCites Reiss's work and says (in RM4) that it has been confirmed by Jolivald. The note in RM2 is expanded in RM4 to explain the connection between dominoes and KÃÃ2n+1ÄÄ. There are obviously 2 Euler circuits on KÃÃ3ÄÄ. He sketches Tarry's method and uses it to compute that KÃÃ5ÄÄ has 88 Euler circuits and KÃÃ7ÄÄ has 1299 76320. [This gives 28 42582 11840 domino rings for the double©6 set.] He says Tarry has found that KÃÃ9ÄÄ has 911 52005 70212 35200. Tom Tit, vol. 3. 1893. Le rectangle et ses diagonales, pp. 155©156. = K, no. 16: The rectangle and its diagonals, pp. 46-48. = R&A, The secret of the rectangle, p. 100. Trick solutions by folding the paper and making an arc on the back. Hoffmann. 1893. Chap. X, no. 9: Single-stroke figures, pp. 338 & 375 = Hoffmann©Hordern, pp. 230©231. Three figures, including the double crescent 'Seal of Mahomet'. Answer states Euler's condition. Dudeney. The shipman's puzzle. London Mag. 9 (No. 49) (Aug 1902) 88-89 & 9 (No. 50) (Sep 1902) 219 (= CP, prob. 18, pp. 40-41 & 173). Number of Euler circuits on KÃÃ5ÄÄ. Benson. 1904. A geometrical problem, p. 255. Seal of Mahomet. William F. White. A Scrap-Book of Elementary Mathematics. Open Court, 1908. [The 4th ed., 1942, is identical in content and pagination, omitting only the Frontispiece and the publisher's catalogue.] Bridges and isles, figure tracing, unicursal signatures, labyrinths, pp. 170-179. On p. 174, he gives the five-brick puzzle, asking for a route along its edges. Dudeney. Perplexities: No. 147: An old three-line puzzle. Strand Magazine 46 (Jul 1913) 110 & (Aug 1913) 221. c= AM, prob. 239: A juvenile puzzle, pp. 68-69 & 197. Five-brick form to be drawn or rubbed out on a board in three strokes. Either way requires doing two lines at once, either by folding the paper as you draw or using two fingers to rub out two lines at once. "I believe Houdin, the conjurer, was fond of showing this to his child friends, but it was invented before his time ©© perhaps in the Stone Age." Loyd. Problem of the bridges. Cyclopedia, 1914, pp. 155 & 359-360. = MPSL1, prob. 28, pp. 26-27 & 130-131. Eight bridges. Asks for number of routes. Loyd. Puzzle of the letter carrier's route. Cyclopedia, 1914, pp 243 & 372. Asks for a circuit on a 3 x 4 array with a minimal length of repeated path. Dudeney. AM. 1917. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐProb. 242: The tube inspector's puzzle, pp. 69 & 198. Minimal route on a 3 x 4 array. Prob. 261: The monk and the bridges, pp. 75©76 & 202©203. River with one island. Four bridges from island, two to each side of the river, and another bridge over the river. How many Euler paths from a given side of the river to the other? Answer:  16. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐCollins. Book of Puzzles. 1927. The fly on the octahedron, pp. 105©108. Asserts there are 1488 Euler circuits on the edges of an octahedron. He counts the reverse as a separate circuit. Harry Houdini [pseud. of Ehrich Weiss] Houdini's Book of Magic. 1927 (??NYS); Pinnacle Books, NY, 1976, p. 19: Can you draw this? Take a square inscribed in a circle and draw both diagonals. "The idea is to draw the figure without taking your pencil off the paper and without retracing or crossing a line. There is a trick to it, but it can be done. The trick in drawing the figure is to fold the paper once and draw a straight line between the folded halves; then, not removing your pencil, unfold the paper. You will find that you have drawn two straight lines with one stroke. The rest is simple." This perplexed me for some time, but I believe the idea is that holding the pencil between the two parts of the folded sheet and moving the pencil parallel to the fold, one can draw a line, parallel to the fold, on each part. Loyd Jr. SLAHP. 1928. Pp. 7-8. Discusses what he calls the "Five-brick puzzle", the common pattern of five rectangles in a rectangle. He says that the object was to draw the lines in four strokes ©© which is easily done ©© but that it was commonly misprinted as three strokes, which he managed to do by folding the paper. He says "a similar puzzle ... some ten or fifteen years ago" asked for a path crossing each of the 16 walls once, which is also impossible. The Bile Beans Puzzle Book. 1933. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐNo. 32. Draw the triangular array of three on an edge without crossing. No. 36. Draw the five©brick pattern in three lines. Folds paper and draws two lines at once. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐR. Ripley. Believe It Or Not! Book 2. (Simon & Schuster, 1931); Pocket Books, NY, 1948, pp. 70-71. = Omnibus Believe It Or Not! Stanley Paul, London, nd [c1935?], p. 270. Gives the five-brick problem of drawing a path crossing each wall once, with the trick solution having the path going along a wall. Asserts "This unicursal problem was solved thus by the great Euler himself." and cites the Euler paper above!! Meyer. Big Fun Book. 1940. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐTryangle, pp. 98 & 731. Triangle subdivided into triangles, with three small triangles along each edge. Draw an Euler circuit without crossings. Cutting the walls, pp. 637 & 794. Five©brick problem. Solution has line crossing through a vertex. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐErn Shaw. The Pocket Brains Trust © No. 2. W. H. Allen, London, nd but inscribed 1944. Prob. 29: Five bricks teaser, pp. 10 & 39. Leeming. 1946. Chap. 6, prob. 2: Through the walls, pp. 70 & 184. Five-brick puzzle, with trick solution having the path go through an intersection. John Paul Adams. We Dare You to Solve This!. Op. cit. in 5.C. 1957? Prob. 49: In just one line, pp. 30 & 48©49. Five©brick puzzle, with answer having the path going along a wall, as in Ripley. Asserts Euler invented this solution. Gibson. Op. cit. in 4.A.1.a. 1963. Pp. 70 & 75: The "impossible" diagram. Same as Tom Tit. Gardner. SA (Apr 1964) = 6th Book, chap. 10. Says Carroll knew that a planar Eulerian graph could be drawn without crossings. Gives a method of O'Beirne for doing this ©© two colour the regions and then make a path which separates the colours into simply connected regions. Ripley's Puzzles and Games. 1966. P. 39. Euler paths on the 'envelope', i.e. a rectangle with its diagonals drawn and an extra connection between the top corners, looking like an unfolded envelope. Asserts the envelope has 50 solutions, but it is not clear if the central crossing is a further vertex. I did this by hand but did not get 50, so I wrote a program to count Euler paths. If the central crossing is not a vertex, then I find 44 paths from one of the odd vertices to the other, and of course 44 going the other way ©© and I had found this number by hand. However, if the central crossing is a vertex, then my hand solution omitted some cases and the computer found 120 paths from one odd vertex to the other. ÁÁÁÁPp. 40©43 give many problems of drawing non©crossing Euler paths or circuits. W. Wynne Willson. How to abolish cross-roads. MTg 42 (Spring 1968) 56-59. Euler circuit of a planar graph can be made without crossings. [Henry] Joseph & Lenore Scott. Master Mind Brain Teasers. Tempo (Grosset & Dunlap), NY, 1973 (& 1978?? ©© both dates are given ©© I'm presuming the 1978 is a 2nd ptg or a reissue under a different imprint??). One line/no crossing, pp. 85©86. Non©crossing Euler circuits on the triangular array of side 3 and non©crossing Euler paths on the 'envelope' ©© cf under Ripley's, above. Asserts the envelope has 50 solutions. I adapted the program mentioned above to count the number of non©crossing Euler paths ©© one must rearrange the first case as a planar graph ©© and there are 16 in the first case and 26 in the second case. Taking the reversals doubles these numbers so it is possible that the Scotts meant the second case and missed one path and its reversal. David Singmaster, proposer; Jerrold W. Grossman & E. M. Reingold, solvers. Problem E2897 ©© An Eulerian circuit with no crossings. AMM 88:7 (Aug 1981) 537©538 & 90:4 (Apr 1983) 287©288. A planar Eulerian graph can be drawn with no crossings. Solution cites some previous work. Anon. [probably Will Shortz ??check with Shortz]. The impossible file. No. 2: In just one line. Games (Apr 1986) 34 & 64 & (Jul 1986) 64. Five brick pattern ©© draw a line crossing each wall once. Says it appeared in a 1921 newspaper [perhaps by Loyd Jr??]. Gives the 1921 solution where the path crosses a corner, hence two walls at once. Also gives a solution with the path going along a wall. In the July issue, Mark Kantrowitz gives a solution by folding over a corner and also a solution on a torus. Marcia Ascher. Graphs in cultures: A study in ethnomathematics. HM 15 (1988) 201-227. Discusses the history of Eulerian circuits and non©crossing versions and then exposits many forms of the idea in many cultures. Marcia Ascher. Ethnomathematics. Op. cit. in 4.B.10. 1991. Chapter Two: Tracing graphs in the sand, pp. 30©65. Sketches the history of Eulerian graphs with some interesting references ©© ??NYS. Describes graph tracing in three cultures: the Bushoong and the Tshokwe of central Africa and the Malekula of Vanuatu (ex©New Hebrides). Extensive references to the ethnographic literature. ÁÁà Ã5.E.1.ÁÁMAZESÄ Ä Ðа¤˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐÁÁThis section is mainly concerned with the theory. The history of mazes is sketched first, with references to more detailed sources. There is even a journal, Caerdroia (53 Thundersley Grove, Thundersley, Essex, SS7 3EB, England), devoted to mazes and labyrinths, mostly concentrating on the history. It is an annual, began in 1980 and issue 31 appeared in 2000. ÁÁMazes are considered under Euler Circuits, since the method of Euler Circuits is often used to find an algorithm. However, some mazes are better treated as Hamiltonian Circuits ©© see 5.F.2. ÁÁA maze can be considered as a graph formed by the nodes and paths ©© the path graph. For the usual planar maze, one can also look at the graph formed by the walls ©© the wall graph, which is a kind of dual to the path graph. In later mazes, the walls do not form a connected whole, and an isolated part of the wall appears as a region or 'face' in the path graph. Such isolated bits of walling are sometimes called islands, but they are the same as the components of the wall graph, with the outer wall being one component, so the number of components is one more than the number of islands. The 'hand©on©wall' method will solve a maze if and only if the goals are adjacent to walls in the component of the outer wall. ÁÁA 'ring maze' is a plate with holes and raised areas with an open ring which must be removed by moving it from hole to hole. I have put these in 11.K.5 as they are a kind of mechanical or topological puzzle, though there are versions with a simple two legged spacer. ÁÁÁÁHISTORICAL SOURCES ÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐW. H. Matthews. Mazes & Labyrinths: A General Account of Their History and Developments. Longmans, Green and Co., London, 1922. = Mazes and Labyrinths: Their History and Development. Dover, 1970. (21 pages of references.) [For more about the book and the author, see: Zeta Estes; My Father, W. H. Matthews; Caerdroia (1990) 6©8.] Walter Shepherd. For Amazement Only. Penguin, 1942; Let's go amazing, pp. 5©12. Revised as: Mazes and Labyrinths ©© A Book of Puzzles. Dover, 1961; Let's go a-mazing, pp. v-xi. (Only a few minor changes are made in the text.) Sketch of the history. Sven Bergling invented the rolling ball labyrinth puzzle/game and they began being produced in 1946. [Kenneth Wells; Wooden Puzzles and Games; David & Charles, Newton Abbot, 1983, p. 114.] Walter Shepherd. Big Book of Mazes and Labyrinths. Dover, 1973, More amazement, pp. vii©x. Extends the historical sketch in his previous book, arguing that mazes with multiple choices perhaps derive from Iron Age hill forts whose entrances were designed to confuse an enemy. Janet Bord. Mazes and Labyrinths of the World. Latimer, London, 1976. (Extensively illustrated.) Nigel Pennick. Mazes and Labyrinths. Robert Hale, London, 1990. Adrian Fisher [& Georg Gerster (photographer)]. The Art of the Maze. Weidenfeld and Nicolson, London, 1990. (Also as: Labyrinth; Solving the Riddle of the Maze; Harmony (Crown Publishers), NY, 1990.) Origins and History occupies pp. 11©56, but he also describes many recent developments and innovations. He has convenient tables of early examples. Adrian Fisher & Diana Kingham. Mazes. Shire Album 264. Shire, Aylesbury, 1991. Adrian Fisher & Jeff Saward. The British Maze Guide. Minotaur Designs, St. Alban's, 1991. Ðа¤˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐÙÙ ÁÁÁÁÀ ÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀ À ÁÁÁÁÀ À À À ÁÁÁÁÀ À À ÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀ À À À ÁÁÁÁÀ À À À À À À À ÁÁÁÁÀ À À À À ÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀ À À À À À ÁÁÁÁÀ À À À À À À À À À À À ÁÁÁÁÀ À À À À À À ÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀ À À À À À À À ÁÁÁÁÀ À À À À À À À À À À À À À À À ÁÁÁÁÀ À À À À À À À À ÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀ À À À À À À À À À ÁÁÁÁÀ À À À À À À À À À À À À À À À À À À À ÁÁÁÁÀ À À À À À À À À À À ÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀ À À À À À À À À À À À ÁÁÁÁÀ À À À À À À À À À À À À À À À À À À À À À À À ÁÁÁÁÀ À À À À À À À À À À À À ÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀ À À À À À À À À À À À À À ÁÁÁÁÀ À À À À À À À À À À À À À À À À À À À À À À À À À À À ÁÁÁÁÀ À À À À À À À À À À À À À À ÀÀÀÀÀÀ À À À À À À À À À À À À À À À ÁÁÁÁÀ À À À À À À À À À À À À À À À À À À À À À À À À À À À À À À À ÁÁÁÁÀ À À À À À À À À À À À À ÀÀÀÀ À À À À À À À À À À À À À ÁÁÁÁÀ À À À À À À À À ÀÀÀÀÀÀÀÀÀÀÀÀ À À À À ÀÀÀÀÀÀÀÀÀÀÀÀ À À À À À À À À À À À ÁÁÁÁÀ À À À À À À À À À À À À À À À À À À À ÁÁÁÁÀ À À À À À À ÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀ À À À À À À À À À ÁÁÁÁÀ À À À À À À À À À À À À À À À ÁÁÁÁÀ À À À À ÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀ À À À À ÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀ À À À À À À À ÁÁÁÁÀ À À À À À À À À À À À À À À À ÁÁÁÁÀ À À ÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀ0À À À À À À À À2ÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀ À À À À À ÁÁÁÁÀ À À À À À À À À À À À ÁÁÁÁÀ ÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀ À À À ÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀ À À À ÁÁÁÁ À À À À À À ÁÁÁÁ À À ÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀÀ À ÁÁÁÁ À:À ÁÁÁÁ ÁÁÁÁHISTORICAL SKETCH ÁÁUp to about the 16C, all mazes were unicursal, i.e. with no decision points. The word labyrinth is sometimes used to distinguish unicursal mazes from others, but this distinction is not made consistently. Until about 1000, all mazes were of the classical 'Cretan' seven©ring type shown above. (However, see Shepherd's point in his 1973 book, above.) The oldest examples are rock carvings, the earliest being perhaps that in the Tomba del Labirinto at Luzzanas, Sardinia, c-2000 [Fisher, pp. 12, 25, 26, with photo on p. 12]. (In fact, Luzzanas is a local name for an uninhabited area of fields, so does not appear on any ordinary map. It is near Benetutti. See my A Mathematical Gazetteer or Mazing in Sardinia (Caerdroia 30 (1999) 17©21). Jeff Saward writes that current archaeological feeling is that the maze is Roman, though the cave is probably c©2000.) On pottery, there are labyrinths on fragments, c-1300, from Tell Rif'at, Syria [the first photos of this appeared in [Caerdroia 30 (2000) 54©55]), and on tablets, c-1200, from Pylos. Fisher [p. 26] lists the early examples. Staffen LundÀÀn; The labyrinth in the Mediterranean; Caerdroia 27 (1996) 28©54, catalogues all known 'Cretan' labyrinths from prehistory to the end of antiquity, c250, excluding the Roman 'spoked' form. All these probably had some mystical significance about the difficulty of reaching a goal, often with substantial mythology ©© e.g. Theseus in the Labyrinth or, later, the Route to Jerusalem. ÁÁRoman mosaics were unicursal but essentially used the Cretan form four times over in the four corners. LundÀÀn, above, calls these 'spoked'. Most of the extant examples are 2C-4C, but some BC examples are known ©© the earliest seems to be c©110 at Selinunte, Sicily. Fisher [pp. 36©37] lists all surviving examples. Saward says the earliest Roman example is at Pompeii, so ÀÀ 79. ÁÁIn the medieval period, the Christians developed a quite different unicursal maze. See Fisher [pp. 60©67] for detailed comparison of this form with the Roman and Cretan forms. The earliest large Christian example is the Chemin de Jerusalem of 1235 on the floor of Chartres Cathedral. Fisher [pp. 41 & 48] lists early and later Christian examples. ÁÁThe legendary Rosamund's Bower was located in Woodstock Park, Oxfordshire, and its purported site is marked by a well and fountain. It was some sort of maze to conceal Rosamund Clifford, the mistress of Henry II (1133-1189), from the Queen, Eleanor of Aquitaine. Legend says that about 1176, Eleanor managed to solve the maze and confronted Rosamund with the choice of a dagger or poison ©© she drank the poison and Henry never smiled again. [Fisher, p. 105]. Historically, Henry had imprisoned Eleanor for fomenting rebellion by her sons and Rosamund was his acknowledged mistress. Rosamund probably spent her last days at a nunnery in Godstow, near Oxford. The legend of the bower dates from the 14C and her murder is a later addition [Collins, Book of Puzzles, 1927, p. 121.] In the 19C, many puzzle collections had a maze called Rosamund's Bower. ÁÁThe earliest record of a hedge maze is of one destroyed in a siege of Paris in 1431. ÁÁNon©unicursal mazes and islands in the wall graph start to appear in the late 16C. Matthews [p. 96] says that: "A simple "interrupted©circle" type of labyrinth was adopted as a heraldic device by Gonzalo Perez, a Spanish ecclesiastic ... and published ... in 1566 ..." in his translation of the Odyssey. Matthews doesn't show this, but he then [pp. 96©97] describes and illustrates a simple maze used as a device by Bois©dofin de Laval, Archbishop of Embrum. He copies it from Claude Paradin; Devises HÀ)Àroiques et EmblÀ/Àmes of the early 17C. It has four entrances and possibly three goals, with walls having 8 components, two being part of the outer wall. The central goals is accessible from two of the entrances, but the two minor goals are each accessible from just one of the other entrances. Presumably this sort of thing is what Matthews meant as an "interrupted circle". ÁÁHowever, Saward has found a mid 15C anonymous English poem, ÃÃThe Assembly of LadiesÄÄ, which describes the efforts of a group of ladies to reach the centre of a maze, which, as he observes, implies there must be some choices involved. ÁÁ[Matthews, p. 114] has three examples from a book by Androuet du Cerceau; Les Plus Excellents Bastiments de France of 1576. Fig. 82 was in the gardens at Charleval and has four entrances, only one of which goes to the central goal. There are four minor goals. The N entrance connects to the NE and SE goals, with several dead ends. The E entrance is a dead end. The S entrance goes to the SW goal. The W entrance goes to the central goal, but the NW goal is on an island, though 'left©hand©on©wall' goes past it. Figs. 83 and 84 are essentially identical and seem to be corruptions of unicursal examples so that most of the maze is bypassed. In fig. 84, one has to walk around to the back of the maze to find the correct entrance to get to the central goal, which is an interesting idea. A small internal change in both cases and moving the entrances converts them to a standard unicursal pattern. ÁÁMatthews' Chap. XIII [pp. 100©109] is on floral mazes and reproduces some from Jan Vredeman De Vries; Hortorum Viridariorumque Formae; Antwerp, 1583. Fig. 74 is one of these and has two components and a short dead©end, but the 'hand©on©wall' rule solves it. Fig. 73 is another of De Vries's, but it is not all shown. It appears to have two entrances and there is certainly a decision point by the far gate, but one route goes to the apparent exit at the bottom of the page. There is a small dead end near the central goal. Fig. 78 shows a maze from a 17C manuscript book in the Harley Manuscripts at the BL, identified on p. 224 as BM Harl. 5308 (71, a, 12). This has two components with the central goal in the inner component, so the 'hand©on©wall' rule fails, but it brings you within sight of the centre and Matthews describes it as unicursal! Fig. 79 is from Adam Islip; The Orchard and the Garden, compiled from continental sources and published in 1602. It has 5 components, but four of these are small enclosures which could be considered as minor goals, especially if they had seats in them. The 'hand©on©wall' rule gets to the central goal. There is a lengthy dead end which goes to two of the inner islands. Fig. 80 is from a Dutch book: J. Commelyn; Nederlantze Hesperides of 1676. It has two components, a central goal and four minor goals. The 'hand©on©wall' gets you to the centre and passes two minor goals. One minor goal is on a dead end so 'left©hand©on©wall' gets to it, but 'right©hand©on©wall' does not. The fourth minor goal is on the island. ÁÁAt Versailles, c1675, AndrÀ)À Le NÀ=Àtre built a Garden Maze, but the objective was to visit, in correct order, 40 fountains based on Aesop's Fables. Each node of the maze had at least one fountain. Some fountains were not at path junctions, but one can consider these as nodes of degree two. This is an early example of a Hamiltonian problem, except that one fountain was located at the end of a short dead end. [Fisher, pp. 49, 79, 130 & 144©145, with contemporary map on p. 144. Fisher says there are 39 fountains, and the map has 40. Close examination shows that the map counts two statues at the entrance but omits to count a fountain between numbers 37 and 38. Matthews, pp. 117-121, says it was built by J. Hardouin©Mansart and his map has 39 fountains.] It has a main entrance and exit but there is another exit, so the perimeter wall already has three components, and there are 14 other components. Sadly, it was destroyed in 1775. ÁÁSeveral other mazes, of increasing complexity, occur in the second half of the 17C [Matthews, figs. 93©109, opp. p. 120 © p. 127]. Several of these could be from 20C maze books. Fig. 94, designed for Chantilly by Le NÀ=Àtre, is surprisingly modern in that there are eight paths spiralling to the centre. The entrance path takes you directly to the centre, so the real problem is getting back out! One of the mazes presently at Longleat has this same feature. ÁÁThe Hampton Court Maze, planted c1690, is the oldest extant hedge maze and one of the earliest puzzle mazes. ([Christopher Turner; Hampton Court, Richmond and Kew Step by Step; (As part of: Outer London Step by Step, Faber, 1986); Revised and published in sections, Faber, 1987, p. 16] says the present shape was laid out in 1714, replacing an earlier circular shape, but I haven't seen this stated elsewhere.) Matthews [p. 128] says it probably replaced an older maze. It has dead ends and one island, i.e. the graph has two components, though the 'hand on wall' rule will solve it. ÁÁThe second Earl Stanhope (1714©1786) is believed to be the first to design mazes with the goal (at the centre) surrounded by an island, so that the 'hand on wall' rule will not solve it. It has seven components and only a few short dead ends.. The fourth Earl planted one of these at Chevening, Kent, in c1820 and it is extant though not open to the public. [Fisher, p. 71, with photo on p. 72 and diagram on p. 73.] However, investigation in Matthews revealed the earlier examples above. Further Bernhard Wiezorke (below at 2001) has found a hedge maze in Germany, dating from c1730, which is not solved by the 'hand on wall' rule. This maze has 12 components. ÁÁIn 1973, Stuart Landsborough, an Englishman settled at Wanaka, South Island, New Zealand, began building his Great Maze. This was the first of the board mazes designed by Landsborough which were immensely popular in Japan. Over 200 were built in 1984©1987, with 20 designed by Landsborough. Many of these were three dimensional ©© see below. About 60 have been demolished since then. [Fisher, pp. 78-79 & 118©121 has 6 colour photos, pp. 156©157 lists Landsborough's designs.] ÁÁIf Minos' labyrinth ever really existed, it may have been three dimensional and there may have been garden examples with overbridges, but I don't know of any evidence for such early three dimensional mazes. Lewis Carroll drew mazes which had paths that crossed over others making a simple three dimensional maze, in his Mischmasch of c1860, see below. John Fisher [The Magic of Lewis Carroll; (Nelson, 1973), Penguin, 1975, pp. 19©20] gives this and another example. Are there earlier examples? Boothroyd & Conway, 1959, seems to be the earliest cubical maze. Much more complex versions were developed by Larry Evans from about 1970 and published in a series of books, starting with 3©Dimensional Mazes (Troubador Press, San Francisco, 1976). His 3-Dimensional Maze Art (Troubador, 1980) sketches some general history of the maze and describes his development of pictures of three dimensional mazes. The first actual three dimensional maze seems to be Greg Bright's 1978 maze at Longleat House, Warminster. [Fisher, pp. 74, 76, 94©95 & 152©153, with colour photos on pp. 94©95.] Since then, Greg Bright, Adrian Fisher, Randoll Coate, Stuart Landsborough and others have made many innovations. Bright seems to have originated the use of colour in mazes c1980 and Fisher has extensively developed the idea. [Fisher, pp. 73©79.] ÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐ AbuÀÀl©Rayhan Al-Biruni (= ÀÀAbÀEÀ©alraihÀÀn Muhammad ibn ÀÀAhmad AlbÀ+ÀrÀEÀnÀ3À). India. c1030. Chapter XXX. IN: Al-Beruni's India, trans. by E. C. Sachau, 2 vols., London, 1888, vol. 1, pp. 306©307 (= p. 158 of the Arabic ed., ??NYS). In describing a story from the fifth and sixth books of the Ramayana, he says that the demon Ravana made a labyrinthine fortress, which in Muslim countries "is called ÃÃYÀÀvana©kotiÄÄ, which has been frequently explained as Rome." He then gives "the plan of the labyrinthine fortress", which is the classical Cretan seven©ring form. Sachau's notes do not indicate whether this plan is actually in the Ramayana, which dates from perhaps ©300. Pliny. Natural History. c77. Book 36, chap. 19. This gives a brief description of boys playing on a pavement where a thousand steps are contained in a small space. This has generally been interpreted as referring to a maze, but it is obviously pretty vague. See: Michael Behrend; Julian and Troy names; Caerdroia 27 (1996) 18©22, esp. note 5 on p. 22. Pacioli. De Viribus. c1500. Part II: Cap. (C)XVII. Do(cumento). de saper fare illa berinto con diligentia secondo Vergilio, f. 223v = Peirani 307©308. A sheet (or page) of the MS has been lost. Cites Vergil, À$Àneid, part six, for the story of PasiphÀ%À and the Minotaur, but the rest is then lost. Sebastiano Serlio. Architettura, 5 books, 1537©1547. The separate books had several editions before they were first published together in 1584. The material of interest is in Book IV which shows two unicursal mazes for gardens. I have seen the following. ÁÁÁÁTutte l'Opere d'Architetture et Prospetiva, .... Giacomo de'Franceschi, Venice, 1619; facsimile by Gregg Press, Ridgewood, New Jersey, 1964. F. 199r shows the designs and f. 197v has some text, partly illegible in my photocopy. [Cf Caerdroia 30 (1999) 15.] ÁÁÁÁSebastiano Serlio on Architecture Volume One Books I©V of 'Tutte l'Opere d'Architettura et Prospetiva'. Translated and edited by Vaughan Hart and Peter Hicks. Yale Univ. Press, New Haven, 1996. P. 388 shows the designs and p. 389 has the text, saying these 'are for the compartition of gardens'. The sidenotes state that these pages are ff. LXXVr and LXXIIIIr of the 3rd ed. of 1544 and ff. 198v©199r and 197v©198r of the 1618/19 ed. William Shakespeare. A Midsummer Night's Dream. c1610. Act II, scene I, lines 98©100: "The nine men's morris is fill'd up with mud, And the quaint mazes in the wanton green For lack of tread are undistinguishable." Fiske 126 opines that the latter two lines may indicate that the board was made in the turf, though he admits that they may refer just to dancers' tracks, but to me it clearly refers to turf mazes. John Cooke. Greene's Tu Quoque; or the Cittie Gallant; a Play of Much Humour. 1614. ??NYS ©© quoted by Matthews, p. 135. A challenge to a duel is given by Spendall to Staines. ÁÁÁÁÃÃStainesÄÄ. I accept it ; the meeting place? ÁÁÁÁÃÃSpendallÄÄ. Beyond the maze in Tuttle. ÁÁThis refers to a maze in Tothill Fields, close to Westminster Abbey. Lewis Carroll. Untitled maze. In: Mischmasch, the last of his youthful MS magazines, with entries from 1855 to 1862. Transcribed version in: The Rectory Umbrella and Mischmasch; Cassell, 1932; Dover, 1971; p. 165 of the Dover ed. John Fisher [The Magic of Lewis Carroll; (Nelson, 1973), Penguin, 1975, pp. 19©20] gives this and another example. Cf Carroll©Wakeling, prob. 35: An amazing maze, pp. 46©47 & 75 and Carroll©Gardner, pp. 80©81 for the Mischmasch example. I don't find the other example elsewhere, but it was for Georgina "Ina" Watson, so probably c1870. Mittenzwey. 1880. Prob. 281, pp. 50 & 100; 1895?: 310, pp. 53©54 & 102; 1917: 310, pp. 49 & 97. The garden of a French place has a maze with 31 points to see. Find a path past all of them with no repeated edges and no crossings. The pattern is clearly based on the Versailles maze of c1675 mentioned in the Historical Sketch above, but I don't recall the additional feature of no crossings occurring before. C. Wiener. Ueber eine Aufgabe aus der Geometria situs. Math. Annalen 6 (1873) 29-30. An algorithm for solving a maze. BLW asserts this is very complicated, but it doesn't look too bad. M. TrÀ)Àmaux. Algorithm. Described in Lucas, RM1, 1891, pp. 47-51. ??check 1882 ed. BLW assert Lucas' description is faulty. Also described in MRE, 1st ed., 1892, pp. 130-131; 3rd ed., 1896, pp. 155©156; 4th ed., 1905, pp. 175©176 is vague; 5th©10th ed., 1911-1922, 183; 11th ed., 1939, pp. 255-256 (taken from Lucas); (12th ed. describes Tarry's algorithm instead) and in Dudeney, AM, p. 135 (= Mazes, and how to thread them, Strand Mag. 37 (No. 220) (Apr 1909) 442-448, esp. 446-447). G. Tarry. Le problÀ/Àme des labyrinthes. Nouv. Annales de Math. (3) 4 (1895) 187-190. ??NYR Collins. Book of Puzzles. 1927. How to thread any maze, pp. 122©124. Discusses right hand rule and its failure, then TrÀ)Àmaux's method. M. R. Boothroyd & J. H. Conway. Problems drive, 1959. Eureka 22 (Oct 1959) 15©17 & 22©23. No. 2. 5 x 5 x 5 cubical maze. Get from a corner to an antipodal corner in a minimal number of steps. Anneke Treep. Mazes... How to get out! (part I). CFF 37 (Jun 1995) 18©21. Based on her MSc thesis at Univ. of Twente. Notes that there has been very little systematic study. Surveys the algorithms of Tarry, TrÀ)Àmaux, Rosenstiehl. Rosenstiehl is greedy on new edges, TrÀ)Àmaux is greedy on new nodes and TrÀ)Àmaux is a hybrid of these. ??©oopsªcheck. Studies probabilities of various routes and the expected traversal time. When the maze graph is a tree, the methods are equivalent and the expected traversal time is the number of edges. Bernhard Wiezorke. Puzzles und Brainteasers. OR News, Ausgabe 13 (Nov 2001) 52©54. This reports his discovery of a hedge maze in Germany ©© the first he knew of. It is in Altjessnitz, near Dessau in Sachsen©Anhalt. (My atlas doesn't show such a place, but Jessnitz is about 10km south of Dessau.) This maze dates from 1720 and has 12 components, with the goal completely separated from the outside so that the 'hand on wall' rule does not solve it. Torsten Silke later told Wiezorke of two other hedge mazes in Germany. One, in Probststeierhagen, Schleswig©Holstein, about 12km NE of Kiel, is in the grounds of the restaurant Zum Irrgarten (At the Labyrinth) and is an early 20C copy of the Altjessnitz example. The other, in Kleinwelka, Sachsen, about 50km NE of Dresden, was made in 1992 and is private. Though it has 17 components, the 'hand on wall' method will solve it. He gives plans of both mazes. He discusses the Seven Bridges of KÀ?Ànigsberg, giving a B&W print of the 1641 plan of the city mentioned at the beginning of Section 5.E ©© he has sent me a colour version of it. He also describes Tremaux's solution method. ÁÁà Ã5.E.2.ÁÁMEMORY WHEELS = CHAIN CODESÄ Ä Ðа¤˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐÁÁThese are cycles of 2ÃÃnÄÄ 0s and 1s such that each n-tuple of 0s and 1s appears just once. They are sometimes called De Bruijn sequences, but they have now been traced back to the late 19C. An example for n = 3 is 00010111. ÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐÀ(Àmile Baudot. 1884. Used the code for 2ÃÃ5ÄÄ in telegraphy. ??NYS ©© mentioned by Stein. A. de RiviÀ/Àre, proposer; C. Flye Sainte©Marie, solver. Question no. 58. L'IntermÀ)Àdiare des MathÀ)Àmaticiens 1 (1894) 19©20 & 107©110. ??NYS ©© described in Ralston and Fredricksen (but he gives no. 48 at one point). Deals with the general problem of a cycle of kÃÃnÄÄ symbols such that every n-tuple of the k basic symbols occurs just once. Gives the graphical method and shows that such cycles always exist and there are k!ÃÃg(n)ÄÄ/ kÃÃnÄÄ of them, where g(n) = kÃÃn-1ÄÄ. This work was unknown to the following authors until about 1975. N. G. de Bruijn. A combinatorial problem. Nederl. Akad. Wetensch. Proc. 49 (1946) 758-764. ??NYS ©© described in Ralston and Fredricksen. Gives the graphical method for finding examples and finds there are 2ÃÃf(n)ÄÄ solutions, where f(n) = 2ÃÃn©1ÄÄ © n. I. J. Good. Normal recurring decimals. J. London Math. Soc. 21 (1946) 167©169. ??NYS ©© described in Ralston and Fredricksen. Shows there are solutions but doesn't get the number. R. L. Goodstein. Note 2590: A permutation problem. MG 40 (No. 331) (Feb 1956) 46-47. Obtains a kind of recurrence for consecutive n-tuples. Sherman K. Stein. Mathematics: The Man-made Universe. Freeman, 1963. Chap. 9: Memory wheels. c= The mathematician as explorer, SA (May 1961) 149-158. Surveys the topic. Cites the c1000 Sanskrit word: yamÀÀtÀÀrÀÀjabhÀÀnasalagÀÀm used as the mnemonic for 01110100(01) giving all triples of short and long beats in Sanskrit poetry and music. Describes the many reinventions, including Baudot (1882), ??NYS, and the work of Good (1946), ??NYS, and de Bruijn (1946), ??NYS. 15 references. R. L. Goodstein. A generalized permutation problem. MG 54 (No. 389) (Oct 1970) 266-267. Extends his 1956 note to find a cycle of aÃÃnÄÄ symbols such that the n-tuples are distinct. Anthony Ralston. De Bruijn sequences ©© A model example of the interaction of discrete mathematics and computer science. MM 55 (1982) 131-143 & cover. Deals with the general problem of cycles of kÃÃnÄÄ symbols such that every n-tuple of the k basic symbols occurs just once. Discusses the history and various proofs and algorithms which show that such cycles always exist. 27 references. Harold Fredricksen. A survey of full length nonlinear shift register cycle algorithms. SIAM Review 24:2 (Apr 1982) 195©221. Mostly about their properties and their generation, but includes a discussion of the door lock connection, a mention of using the 2ÃÃ3ÄÄ case as a switch for three lights, and gives a good history. The door lock connection is that certain push button door locks will open when a four digit code is entered, but they open if the last four buttons pressed are the correct code, so using a chain code reduces the number of button pushes required by a burglar to 1/4 of the number required if he tries all four digit combinations. 58 references. At G4G2, 1996, Persi Diaconis spoke about applications of the chain code in magic and mentioned uses in repeated measurement designs, random number generators, robot location, door locks, DNA comparison. ÁÁÁÁThey were first used in card tricks by Charles T. Jordan in 1910. Diaconis' example had a deck of cards which were cut and then five consecutive cards were dealt to five people in a row. He then said he would determine what cards they had, but first he needed some help so he asked those with red cards to step forward. The position of the red cards gives the location of the five cards in a cycle of 32 (which was the size of the deck)! Further, there are simple recurrences for the sequence so it is fairly easy to determine the location. One can code the binary quintuples to give the suit and value of the first card and then use the succeeding quintuples for the succeeding cards. ÁÁÁÁLong versions of the chain code are printed on factory floors so that a robot can read it and locate itself. In Jan 2000, I discussed the Sanskrit chain code with a Sanskrit scholar, Dominik Wujastyk, who said that there is no known Sanskrit source for it. He has asked numerous pandits who did not know of it and he said there is is a forthcoming paper on it, but that it did not locate any Sanskrit source. ÁÁà Ã5.E.2.a.ÁÁPANTACTIC SQUARESÄ Ä ÁÁHaubrich's 1995©1996 surveys, op. cit. in 5.H.4, include this. B. Astle. Pantactic squares. MG 49 (No. 368) (May 1965) 144-152. This is a two-dimensional version of the memory wheel. Take a 5 x 5 array of cells marked 0 or 1 (or Black or White). There are 16 ways to take a 2 x 2 subarray from the 5 x 5 array. If these give all 16 2 x 2 binary patterns, the array is called pantactic. The author shows a number of properties and some types of such squares. C. J. Bouwkamp, P. Janssen & A. Koene. Note on pantactic squares. MG 54 (No. 390) (Dec 1970) 348-351. They find 800 such squares, forming 50 classes of 16 forms. [Surprisingly, neither paper considers a 4 x 4 array viewed toroidally, which is the natural generalization of the memory wheel. Precisely two of the fifty classes, namely nos. 25 & 41, give such a solution and these are the same pattern on the torus. One can also look at the 4 x 4 subarrays of a 131 x 131 or a 128 x 128 array, etc., as well as 3 and higher dimensional arrays. I submitted the question of the existence and numbers of these as a problem for CM, but it was considered too technical.] Ivan Moscovich. US Patent 3,677,549 ©© Board Game Apparatus. Applied: 14 Jun 1971; patented: 18 Jul 1972. Front page, 1p diagrams, 2pp text. Reproduced in Haubrich, About ..., 1996, op. cit. in 5.H.4. 2pp + 2pp diagrams. This uses the 16 2 x 2 binary patterns as game pieces. He allows the pieces to be rotated, scoring different values according to the orientation. No mention of reversing pieces or of the use of the pieces as a puzzle. John Humphries. Review of Q©Bits. G&P 54 (Nov 1976) 28. This is Moscovich's game idea, produced by Orda. Though he mentions changing the rules to having non©matching, there is no mention of two©sidedness. Pieter van Delft & Jack Botermans. Creative Puzzles of the World. (As: Puzzels uit de hele wereld; Spectrum Hobby, 1978); Harry N. Abrams, NY, 1978. The colormatch square, p. 165. See Haubrich,1994, for description. Jacques Haubrich. Pantactic patterns and puzzles. CFF 34 (Oct 1994) 19©21. Notes the toroidal property just mentioned. Says Bouwkamp had the idea of making the 16 basic squares in coloured card and using them as a MacMahon©type puzzle, with the pieces double©sided and such that when one side had MacMahon matching, the other side had non©matching. There are two different bijections between matching patterns and nonªmatching patterns, so there are also 800 solutions in 50 classes for the non©matching problem. Bouwkamp's puzzle appeared in van Delft & Botermans, though they did not know about and hence did not mention the double©sidedness. [In an email of 22 Aug 2000, Haubrich says he believes Bouwkamp did tell van Delft and Botermans about this, but somehow it did not get into their book.] The idea was copied by two manufacturers (Set Squares by Peter Pan Playthings and Regev Magnetics) who did not understand Bouwkamp's ideas ©© i.e. they permitted pieces to rotate. Describes Verbakel's puzzle of 5.H.2. Jacques Haubrich. Letter: Pantactic Puzzles = ÃÃQ©BitsÄÄ. CFF 37 (Jun 1995) 4. Says that Ivan Moscovich has responded that he invented the version called "Q©Bits" in 1960©1964, having the same tiles as Bouwkamp's (but only one©sided [clarified by Haubrich in above mentioned email]). His US Patent 3,677,549 (see above) is for a game version of he idea. The version produced by Orda Ltd. was reviewed in G&P 54 (Nov 1976) (above). So it seems clear that Moscovich had the idea of the pieces before Bouwkamp's version was published, but Moscovich's application was to use them in a game where the orientations could be varied. ÁÁà Ã5.F.ÁÁHAMILTONIAN CIRCUITSÄ Ä Ðа¤˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐ ÁÁFor queen's, bishop's and rook's tours, see 6.AK. ÁÁA tour is a closed path or circuit. ÁÁA path has end points and is sometimes called an open tour. ÁÁà Ã5.F.1.ÁÁKNIGHT'S TOURS AND PATHSÄ Ä ÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐÁÁÁÁGENERAL REFERENCES Antonius van der Linde. Geschichte und Literatur des Schachspiels. (2 vols., Springer, Berlin, 1874); one vol. reprint, Olms, ZÀGÀrich, 1981. [There are two other van der Linde books: Quellenstudien zur Geschichte des Schachspiels, Berlin, 1881, ??NYS; and Das Erste Jartausend [sic] der Schachlitteratur (850-1880), (Berlin, 1880); reprinted with some notes and corrections, Caissa Limited Editions, Delaware, 1979, which is basically a bibliography of little use here.] Baron Tassilo von Heydebrand und von der Lasa. Zur Geschichte und Literatur des Schachspiels. Forschungen. Leipzig, 1897. ??NYS. Ahrens. MUS I. 1910. Pp. 319©398. Harold James Ruthven Murray. A History of Chess. OUP, 1913; reprinted by Benjamin Press, Northampton, Massachusetts, nd [c1986]. This has many references to the problem, which are detailed below. Reinhard Wieber. Das Schachspiel in der arabischen Literatur von den AnfÀÀngen bis zur zweiten HÀÀlfte des 16.Jahrhunderts. Verlag fÀGÀr Orientkunde Dr. H. Vorndran, Walldorf-Hessen, 1972. George P. Jelliss. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐSpecial Issue: Notes on the Knight's Tour. Chessics 22 (Summer 1985) 61-72. Further notes on the knight's tour. Chessics 25 (Spring 1986) 106-107. Notes on Chessics 22 continued. Chessics 29 & 30 (1987) 160. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐÁÁÁÁThis is a progress report on his forthcoming book on the knight's tour. I will record some of his comments at the appropriate points below. He also studies the 3 x n board extensively. Ðа¤˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐÁÁTwo problems with knights on a 3 x 3 board are generally treated here, but cf 5.R.6. ÁÁThe 4 knights problem has two W and two B knights at the corners (same colours at adjacent corners) and the problem is to exchange them in 16 moves. The graph of knight's connections is an 8©cycle with the pieces at alternate nodes. [Putting same colours at opposite corners allows a solution in 8 moves.] ÁÁThe 7 knights problem is to place 7 knights on a 3 x 3 board in the 4 corners and 3 of the sides so each is a knight's move from the previously placed one. This is equivalent to the octagram puzzle of 5.R.6. ÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐ4 knights problem ©© see: at-TilimsÀÀni, 1446; Civis Bononiae, c1475; 7 knights problem ©© see: King's Library MS.13, A.xviii, c1275; "Bonus Socius", c1275; at-TilimsÀÀni, 1446; Al-Adli (c840) and as-Suli (c880-946) are the first two great Arabic chess players. Although none of their works survive, they are referred to by many later writers who claim to have used their material. Rudrata. KÀ]ÀvyÀ]ÀlankÀ]Àra. c900. ??NYS ©© described in Murray 53-55, from an 1896 paper by Jacobi, ??NYS. The poet speaks of verses which have the shapes of "wheel, sword, club, bow, spear, trident, and plough, which are to be read according to the chessboard squares of the chariot [= rook], horse [= knight], elephant [c= bishop], &c." According to Jacobi, the poet placed syllables in the cells of a half chessboard so that it reads the same straight across as when following a piece's path. With help from the commentator Nami, of 1069, the rook's and knight's path's are reconstructed, and are given on Murray 54. Both are readily extended to full board paths, but not tours. The elephant's path is confused. KitÀÀb ash-shatranj mimmaÀÀl-lafahuÀÀl-ÀÀAdli was-SÀEÀlÀ3À wa ghair-huma [Book of the Chess; extracts from the works of al-'AdlÀ3À, as-SÀEÀlÀ3À and others]. Copied by AbÀEÀ IshÀÀq IbrÀÀhÀ3Àm ibn al-MubÀÀrak ibn ÀÀAlÀ3À al-Mudhahhab al-BaghdÀÀdÀ3À. Murray 171-172 says it is MS ÀÀAbd-al-Hamid I, no. 560, of 1140, and denotes it AH. Wieber 12-15 says it is now MS Lala Ismail Efendi 560, dates it July-August 1141, and denotes it L. Both cite van der Linde, Quellenstudien, no. xviii, p. 331+, ??NYS. The author is unknown. This MS was discovered in 1880. Catalogues in Istanbul listed it as RisÀÀla fiÀÀsh-shatranj by AbÀEÀÀÀl-ÀÀAbbÀÀs Ahmad al-ÀÀAdlÀ3À. It is sometimes attributed to al-LajlÀÀj who wrote one short section of this book. Murray, van der Linde and Wieber (p. 41) cite another version: MS Khedivial Lib., Cairo, Mustafa Pasha, no. 8201, copied c1370, which Murray denotes as C and Wieber lists as unseen. ÁÁÁÁMurray 336 gives two distinct tours: AH91 & AH92. The solution of AH91 is a numbered diagram, but AH92 is 'solved' four times by acrostic poems, where the initial letters of the lines give the tour in an algebraic notation. Wieber 479-480 gives 2 tours from ff. 74a-75b: L74a = AH91 and L74b = reflection of AH92. [Since the 'solutions' of AH92 are poetic, it is not unreasonable to consider the reflection as different.] Also AH94 = L75b is a knight/bishop tour, where moves of the two types alternate. These tours may be due to as-Suli. AH196 is a knight/queen tour. Arabic MS Atif Efendi 2234 (formerly Vefa (ÀÀAtÀ3Àq EfendÀ3À) 2234), Eyyub, Istanbul. Copied by Muhammad ibn HawÀÀ (or RahwÀÀr ©© the MS is obscure) ibn ÀÀOthmÀÀn al-MuÀÀaddib in 1221. Murray 174-175 describes it as mostly taken from the above book and denotes it V. A tour is shown on p. 336 as V93 = AH92. Wieber 20-24 denotes it A. On p. 479, he shows the tour from f. 68b which is the same as L74b, the reflection of AH92. King's Library MS.13, A.xviii, British Museum, in French, c1275. Described in van der Linde I 305-306. Described and transcribed in Murray 579-582 & 588-600, where it is denoted as K. Van der Linde discusses the knight's path on I 295, with diagram no. 244 on p. 245. Murray 589 gives the text and a numbered diagram of a knight's path as K1. The path splits into two half board paths: a1 to d1 and e3 to h1, so the first half and the whole are corner to corner. The first half is also shown as diagram K2 with the half board covered with pieces and the path described by taking of pieces. K3 is the 7 knights problem "Bonus Socius" [perhaps Nicolas de NicolaÀ5À]. This is the common name of a collection of chess problems, assembled c1275, which was copied and translated many times. See Murray 618-642 for about 11 MSS. Some of these are given below. Fiske 104 & 110-111 discusses some MSS of this collection. ÁÁÁÁMS Lat. 10286, Nat. Lib., Paris. c1350. Van der Linde I 293-295 describes this but gives the number as 10287 (formerly 7390). Murray 621 describes it and denotes it PL. Van der Linde describes a half board knight's path, with a diagram no. 243 shown on p. 245. The description indicates a gap in the path which can only be filled in one way. This is a path from a8 to h8 which cannot be extended to the full board. Murray 641 says that PL275 is the same as problems in two similar MSS and as CB244, diagrammed on p. 674. However, this is not the same as van der Linde's no. 243, though cells 1-19 and 31-32 are the same in both paths, so this is also an a8 to h8 path which does not extend to a full board. ÁÁÁÁMurray 620 mentions a path in a late Italian MS version of c1530 (Florence, Nat. Lib. XIX.7.51, which he denotes It) which may be the MS described by van der Linde I 284 as no. 4 and the half board path described on I 295 with diagram no. 245 on I 245. Fiske 210©211 describes this and says von der Lasa 163©165 (??NYS) describes it as early 16C, but Murray does not mention von der Lasa. Fiske says it contains a tour on f. 28b, which von der Lasa claims is "das ÀÀlteste beispiel eines vollkommenen rÀ?Àsselsprunges", but Murray does not detail the problems so I cannot compare these citations. Fiske also says it also contains the 7 knights problem. Dresden MS 0/59, in French, c1400. Murray describes this on pp. 607-613 and denotes it D. On p. 609, Murray describes D57 which asks for a knight's path on a 4 x 4 board. No solution is given ©© indeed this is impossible, cf Persian MS 211 in the RAS. Ibid. is D62 which asks for a half board tour, but no answer is provided. Persian MS 211 in Royal Asiatic Society. Early 15C. ??NYS. ÁÁÁÁExtensively described as MS 250 bequeathed by Major David Price in: N. Bland; On the Persian game of chess; J. Royal Asiatic Soc. 13 (1852) 1-70. He dates it as 'at least 500 years old' and doesn't mention the knight's tour. ÁÁÁÁDescribed, as MS No. 260, and partially translated in Duncan Forbes; The History of Chess; Wm. H. Allen, London, 1860. Forbes says Bland's description is "very detailed but unsatisfactory". On p. 82 is the end of the translation of the preface: '"Finally I will show you how to move a Knight from any individual square on the board, so that he may cover each of the remaining squares in as many moves and finally come to rest on that square whence he started. I will also show how the same thing may be done by limiting yourself only to one half, or even to one quarter (1) of the board." ©© Here the preface abruptly terminates, the following leaf being lost.' Forbes's footnote (1) correctly doubts that a knight's tour (or even a knight's path) is possible on the 4 x 4 board. ÁÁÁÁMurray 177 cites it as MS no. 211 and denotes it RAS. He says that it has been suggested that this MS may be the work of ÀÀAlÀÀ'addÀ3Àn TabrÀ3ÀzÀ3À = ÀÀAlÀ3À ash-ShatranjÀ3À, late 14C, described on Murray 171. Murray mentions the knight's tour passage on p. 335. This may be in van der Linde, ??NX. Wieber 45 mentions the MS. AbÀEÀ ZakarÀ3ÀyÀÀ Yahya ibn IbrÀÀhÀ3Àm al-HakÀ3Àm. Nuzhat al-arbÀÀb al-ÀÀaqÀEÀl fÀ3ÀÀÀsh-shatranj al-manqÀEÀl (The delight of the intelligent, a description of chess). Arabic MS 766, John Rylands Library, Manchester. ÁÁÁÁBland, loc. cit., pp. 27-28, describes this as no. 146 of Dr. Lee's catalogue and no. 76 of the new catalogue. Forbes, loc. cit., says that Dr. Lee had loaned his two MSS to someone who had not yet returned them, so Forbes copies Bland's descriptions (on pp. 27-31) as his Appendix C, with some clarifying notes. (The other of Dr. Lee's MSS is described below.) Van der Linde I 107ff (??NX) seems to copy Bland & Forbes. ÁÁÁÁMurray 175-176 describes it as Arab. 59 at John Rylands Library and denotes it H. He says it was Bland who had borrowed the MSS from Dr. Lee and Murray traces their route to Dr. Lee and to Manchester. Murray says it is late 15C, is based on al-Adli and as-Suli and he also describes a later version, denoted Z, late 18C. Wieber 32-35 cites it as MS 766(86) at John Rylands, dates it 1430 and denotes it Y1. ÁÁÁÁMurray 336 gives three paths. H73 = H75 are the same tour, but with different keys, one poetic as in Rudrata, one numeric. H74 is a path attributed to Ali Mani with similar poetic solution. Wieber 480 shows two diagrams. Y1-39a, Y1-39b, Y1-41b are the same tour as H73, but with different descriptions, the latter two being attributed to al-Adli. Y1-39a (second diagram) = H74 is attributed to ÀÀAli ibn Mani. ShihÀÀbaddÀ3Àn AbÀEÀÀÀl-ÀÀAbbÀÀs Ahmad ibn Yahya ibn AbÀ3À Hajala at-TilimsÀÀni alH-anbalÀ3À. KitÀÀb ÀÀanmÀEÀdhaj al-qitÀÀl fi laÀÀb ash-shatranj (Book of the examples of warfare in the game of chess). Copied by Muhammed ibn ÀÀAli ibn Muhammed al-ArzagÀ3À in 1446. ÁÁÁÁBland, loc. cit., pp. 28-31, describes this as the second of Dr. Lee's MSS, old no. 147, new no. 77. Forbes copies this and adds notes. Van der Linde I 105-107 seems to copy from Bland and Forbes. Murray 176-177 says the author died in 1375, so this might be c1370. He says it is Dr. Lee's on 175-176, that it is MS Arab. 93 at the John Rylands Library and denotes it Man. Wieber 29-32 cites it as MS 767(59) at the Rylands Library and denotes it H. On p. 481, he shows a half-board path which cannot be extended to the full board. ÁÁÁÁThis MS also gives the 4 knights and 7 knights problems. Murray 337, 673 (CB236) & 690 and Wieber 481 show these problems. RisÀÀlahi Shatranj. Persian poem of unknown date and authorship. A copy was sent to Bland by Dr. Sprenger of Delhi. See Bland, loc. cit., pp. 43-44. [Bland uses ÀÀ for ÀÀ.] Bland says it has the problem of the knight's tour or path. [I think this is the poem mentioned on Murray 182©183 and hence on Wieber 42.] Sifat mal ÀÀÀEÀb al-faras fÀ3À gamÀ3À abyÀÀt aÀ±À-À±ÀatranÀwÀ. MS Gotha 10, Teil 6; ar. 366; Stz. Hal. 408. Date unknown. Wieber 37 & 480 describes this and gives a path from h8 to e4 which occurs on ff. 70 & 68. Civis Bononiae [Citizen of Bologna]. Like Bonus Socius, this is a collection of chess problems, from c1475, which exists in several MSS and printings. All are in Latin, from Italy, and give essentially the same 288 problems. See Murray 643-703 for description of about 10 texts and transcription of the problems. Many of the texts are not in van der Linde. Murray 643 cites MS Lasa, in the library of Baron von der Lasa, c1475, as the most accurate and complete of the texts. Two other well known versions are described below. ÁÁÁÁPaulo Guarino (di Forli) (= Paulus Guarinus). No real title, but the end has 'Explicit liber de partitis scacorum' with the writer's name and the date 4 Jan 1512. This MS was in the Franz Collection and is now (1913) in the John G. White Collection in Cleveland, Ohio. This version only contains 76 problems. Van der Linde I 295-297 describes the MS and on p. 294 he describes a half board path and says Guarino's 74 is a reflection of his no. 243. Murray 645 describes the MS but doesn't list the individual problems. He implies that CB244, on p. 674, is the tour that appears in all of the Civis Bononiae texts, but this is not the same as van der Linde's no. 243. CB236, pp. 673 & 690, is the 4 knights problem, which is Guarino's 42 [according to Lucas, RM4, p. 207], but I don't have a copy of van der Linde's no. 215 to check this, ??NX. ÁÁÁÁAnon. Sensuit Jeux Partis des eschez: composez nouvellement Pour recreer tous nobles cueurs et pour eviter oysivete a ceulx qui ont voulente: desir et affection de le scavoir et apprendre et est appelle ce Livre le jeu des princes et damoiselles. Published by Denis Janot, Paris, c1535, 12 ff. ??NYS. (This is the item described by von der Lasa as 'bei Janot gedrucktes QuartbÀÀndchen' (MUS #195).) This a late text of 21 problems, mostly taken from Civis Bononiae. Only one copy is known, now (1913) in Vienna. See van der Linde I 306-307 and Murray 707-708 which identify no. 18 as van der Linde's no. 243 and with CB244, as with the Guarino work. I can't tell but van der Linde may identify no. 11 as the 4 knights problem (??NX). ÁÁÁÁMurray 730 gives another half board path, C92, of c1500 which goes from a8 to g5. Murray 732 notes that a small rearrangement makes it extendable to the whole board. Horatio Gianutio della Mantia. Libro nel quale si tratta della Maniera di giuocar' À!À Scacchi, Con alcuni sottilissimi Partiti. Antonio de' Bianchi, Torino, 1597. ??NYS. Gives half board tours which can be assembled into to a full tour. (Not in the English translation: The Works of Gianutio and Gustavus Selenus, on the game of Chess, Translated and arranged by J. H. Sarratt; J. Ebers, London, 1817, vol. 1. ©© though the copy I saw didn't say vol. 1. Van der Linde, Erste Jartausend ... says there are two volumes.) Bhatta NÀ…Àlakantha. BhagavantabhÀ]Àskara. 17C. End of 5th book. ??NYS, described by Murray 63-66. The author gives three tours, in the poetic form of Rudrata, which are the same tour starting at different points. The tour has 180 degree rotational symmetry. Ozanam. 1725. Prob. 52, 1725: 260-269. Gives solutions due to Pierre RÀ)Àmond de Montmort, Abraham de Moivre, Jean-Jacques d'Ortous de Mairan (1678©1771). Surprisingly, these are all distinct and different from the earlier examples. Ozanam says he had the problem and the solution from de Mairan in 1722. Says the de Moivre is the simplest. Kraitchik, Math. des Jeux, op. cit. in 4.A.2, p. 359, dates the de Montmort as 1708 and the de Moivre as 1722, but gives no source for these. Montmort died in 1719. Ozanam died in 1717 and this edition was edited by Grandin. Van der Linde and Ahrens say they can find no trace of these solutions prior to Ozanam (1725). See Ozanam©Montucla, 1778. ÁÁÁÁBall, MRE, 1st ed., 1892, p. 139, says the earliest examples he knows are the De Montmort & De Moivre of the late 17C, but he only cites them from Ozanam©Hutton, 1803, & Ozanam©Riddle, 1840. In the 5th ed., 1911, p. 123, he adds that "They were sent by their authors to Brook Taylor who seems to have previously suggested the problem." He gives no reference for the connection to Taylor and I have not seen it mentioned elsewhere. This note is never changed and may be the source of the common misconception that knight's tours originated c1700! Les Amusemens. 1749. Prob. 181, p. 354. Gives de Moivre's tour. Says one can imagine other methods, but this is the simplest and most interesting. L. Euler. Letter to C. Goldbach, 26 Apr 1757. In: P.-H. Fuss, ed.; Correspondance MathÀ)Àmatique et Physique de Quelques CÀ)ÀlÀ/Àbres GÀ)ÀomÀ/Àtres du XVIIIÀ/Àme SiÀ/Àcle; (Acad. Imp. des Sciences, St. PÀ)Àtersbourg, 1843) = Johnson Reprint, NY, 1968, vol. 1, pp. 654-655. Gives a 180ÃÃoÄÄ symmetric tour. L. Euler. Solution d'une question curieuse qui ne paroit soumise À!À aucune analyse. (MÀ)Àm. de l'AcadÀ)Àmie des Sciences de Berlin, 15 (1759 (1766)), 310-337.) = Opera Omnia (1) 7 (1923) 26-56. (= Comm. Arithm. Coll., 1849, vol. 1, pp. 337-355.) Produces many solutions; studies 180ÃÃoÄÄ symmetry, two halves, and other size boards. [Petronio dalla Volpe]. Corsa del Cavallo per tutt'i scacchi dello scacchiere. Lelio della Volpe, Bologna, 1766. 12pp, of which 2 and 12 are blanks. [Lelio della Volpe is sometimes given as the author, but he died c1749 and was succeeded by his son Petronio.] Photographed and printed by Dario Uri from the example in the Libreria Comunale Archiginnasio di Bologna, no. 17 CAPS XVI 13. The booklet is briefly described in: Adriano Chicco; Note bibliografiche su gli studi di matematica applicata agli scacchi, publicati in Italia; Atti del Convegno Nazionale sui Giochi Creative, Siena, 11-14 Jun 1981, ed. by Roberto Magari; Tipografia Senese for GIOCREA (SocietÀ!À Italiana Giochi Creativi), 1981; p. 155. ÁÁÁÁThe Introduction by the publisher cites Ozanam as the originator of this 'most ingenious' idea and says he gives examples due to Montmort, Moivre and Mairan. He also says this material has 'come to hand' but doesn't give any source, so it is generally thought he was the author. He gives ten paths, starting from each of the 10 essentially distinct cells. He then gives the three cited paths from Ozanam. He then gives six tours. Each path is given as a numbered board and a line diagram of the path, which led Chicco to say there were 38 paths. The line drawing of the first tour is also reproduced on the cover/title page. Ozanam©Montucla. 1778. Prob. 23, 1778: 178©182; 1803: 177©180; 1814: 155©157. Prob. 22, 1840: 80-81. Drops the reference to de Mairan as the source of the problem and adds a fourth tour due to "M. de W***, capitaine au rÀ)Àgiment de Kinski". All of these have a misprint of 22 for 42 in the right hand column of De Moivre's solution. H. C. von Warnsdorff. Des RÀ?Àsselsprunges einfachste und allgemeinste LÀ?Àsung. Th. G. Fr. Varnhagenschen Buchhandlung, Schmalkalden, 1823, 68pp. ??NYS ©© details from Walker. Rule to make the next move to the cell with the fewest remaining neighbours. Lucas, L'ArithmÀ)Àtique Amusante, p. 241, gives the place of publication as Berlin. Boy's Own Book. Not in 1828. 1828©2: 318 states a knight's tour can be made. George Walker. The Art of Chess©Play: A New Treatise on the Game of Chess. (1832, 80pp. 2nd ed., Sherwood & Co, London, 1833, 160pp. 3rd ed., Sherwood & Co., London, 1841, 300pp. All ??NYS ©© details from 4th ed.) 4th ed., Sherwood, Gilbert & Piper, London, 1846, 375pp. Chap. V ©© section: On the knight, p. 37. "The problem respecting the Knight's covering each square of the board consecutively, has attracted, in all ages, the attention of the first mathematicians." States Warnsdorff's rule, without credit, but gives the book in his bibliography on p. 375, and asserts the rule will always give a tour. No diagram. Family Friend 2 (1850) 88 & 119, with note on 209. Practical Puzzle ©© No. III. Find a knight's path. Gives one answer. Note says it has been studied since 'an early period' and cites Hutton, who copies some from Montucla, an article by Walker in Frasers Magazine (??NYS) which gives Warnsdorff's rule and an article by Roget in Philosophical Magazine (??NYS) which shows one can start and end on any two squares of opposite colours. Describes using a pegged board and a string to make pretty patterns. Boy's Own Book. Moving the knight over all the squares alternately. 1855: 511©512; 1868: 573; 1881 (NY): 346©347. 1855 says the problem interested Euler, Ozanam, De Montmart [sic], De Moivre, De Majron [sic] and then gives Warnsdorff's rule, citing George Walker's 'Treatise on Chess' for it ©© presumably 'A New Treatise', London, 1832, with 2nd ed., 1833 & 3rd ed., 1841, ??NYS. Walker also wrote On Moving the Knight, London, 1840, ??NYS. 1868 drops all the names, but the NY ed. of 1881 is the same as the 1855. Gives a circuit due to Euler. Magician's Own Book. 1857. Art. 46: Moving the knight over all the squares alternately, pp. 283©287. Identical to Boy's Own Book, 1855, but adds Another Method. = Book of 500 Puzzles; 1859, art. 46, pp. 97©101. = Boy's Own Conjuring Book, 1860, prob. 45, pp. 246-251. Landells. Boy's Own Toy©Maker. 1858. Moving the knight over all the squares alternately, p. 143. This is the Another Method of Magician's Own Book, 1857. Cf Illustrated Boy's Own Treasury, 1860. Illustrated Boy's Own Treasury. 1860. Prob. 47: Practical chess puzzle, pp. 404 & 443. Knight's tour. This is the Another Method of Magician's Own Book. C. F. de Jaenisch. TraitÀ)À des Applications de l'Analyse MathÀ)Àmatiques au Jeu des À(Àchecs. 3 vols., no publisher, Saint©PÀ)Àtersbourg. 1862©1863. Vol. 1: Livre I: Section III: De la marche du cavalier, pp. 186©259 & Plate III. Vol. 2: Livre II: ProblÀ/Àme du Cavalier, pp. 1©296 & 31 plates (some parts ??NYS). Vol. 3: Addition au Livre II, pp. 239©243 (This Addition ??NYS). This contains a vast amount of miscellaneous material and I have not yet read it carefully. ??NYR Leske. Illustriertes Spielbuch fÀGÀr MÀÀdchen. 1864? Prob. 323, pp. 153©154 & 393: RÀ?Àsselsprung©Aufgaben. Three arrays of syllables and one must find a poetic riddle by following a knight's tour. Arrays are 8 x 8, 8 x 8, 6 x 4. C. Flye Sainte-Marie. Bull. Soc. Math. de France (1876) 144-150. ??NYS ©© described by Jelliss. Shows there is no tour on a 4 x n board and describes what a path must look like. Mittenzwey. 1880. Prob. 222©223, pp. 40 & 91; 1895?: 247©248, pp. 44 & 93; 1917: 247-248, pp. 40©41 & 89. First is a knight's path. Second is a board with word fragments and one has to make a poem, which uses the same path as in the first problem. Paul de Hijo [= AbbÀ)À Jolivald]. Le ProblÀ/Àme du Cavalier des À(Àchecs. Metz, 1882. ??NYS ©© described by Jelliss and quoted by Lucas. Jelliss notes the BL copy of de Hijo was destroyed in the war, but he has since told me there are copies in The Hague and Nijmegen. First determination of the five 6 x 6 tours with 4©fold rotational symmetry, the 150 ways to cover the 8 x 8 with two circuits of length 32 giving a pattern with 2-fold rotational symmetry, the 378 ways giving reflectional symmetry in a median, the 140 ways with four circuits giving 4©fold rotational symmetry and the 301 ways giving symmetry in both medians (quoted in Lucas, L'ArithmÀ)Àtique Amusante, pp. 238©241). Lucas. Nouveaux jeux scientifiques ..., 1889, op. cit. in 4.B.3. (Described on p. 302, figure on p. 301.) 'La Fasioulette' is an 8 x 8 board with 64 links of length ÀÀ5 to form knight's tours. Knight's move puzzles. The Boy's Own Paper 11 (Nos. 557 & 558) (14 & 21 Sep 1889) 799 & 814. Four Shakespearean quotations concealed as knight's tours on a 8 x 8 board. Beginnings not indicated! Hoffmann. 1893. Chap. X, no. 6: The knight's tour, pp. 335©336 & 367©373 = Hoffmann-Hordern, pp. 225©229. Gives knight's paths due to Euler and Du Malabare, a knight's tour due to Monneron, and four other unattributed tours. Gives Warnsdorff's rule, citing Walker's A New Treatise on Chess, 1832. Ahrens. Mathematische Spiele. Encyklopadie article, op. cit. in 3.B. 1904, pp. 1080-1093. Pp. 1084-1086 gives many references to 19C work, including estimates of the number of tours and results on 'semi-magic tours'. C. Planck. Chess Amateur (Dec 1908) 83. ??NYS ©© described by Jelliss. Shows there are 1728 paths on the 5 x 5 board. Jelliss notes that this counts each path in both directions and there are only 112 inequivalent tours. Ahrens. 1910. MUS I 325. Use of knight's tours as a secret code. Dudeney. AM. 1917. Prob. 339: The four knight's tours, pp. 103 & 229. Quadrisect the board into four congruent pieces such that there is a knight's tour on the piece. Jelliss asserts that the solution is unique and says this may be what Persian MS 260 (i.e. 211) intended. He notes that the four tours can be joined to give a tour with four fold rotational symmetry. W. H. Cozens. Cyclically symmetric knight's tours. MG 24 (No. 262) (Dec 1940) 315-323. Finds symmetric tours on various odd-shaped boards. H. J. R. Murray. The Knight's Tour. ??NYS. MS of 1942 described by G. P. Jelliss, G&PJ 2 (No. 17) (Oct 1999) 315. Observes that a knight can move from the (0, 0) cell to the (2, 1) and (1, 2) cella and that the angle between these lines is the smaller angle of a 3, 4, 5 triangle. One can see this by extending the lines to (8, 4) and (5, 10) and seeing these points form a 3, 4, 5 triangle with (0, 0). W. H. Cozens. Note 2761: On note 2592. MG 42 (No. 340) (May 1958) 124-125. Note 2592 tried to find the cyclically symmetric tours on the 6 x 6 board and found 4. Cozens notes two are reflections of the other two and that three such tours were omitted. He found all these in his 1940 paper. R. C. Read. Constructing open knight's tours blindfold! Eureka 22 (Oct 1959) 5©9. Describes how to construct easily a tour between given cells of opposite colours, correcting a method of Roget described by Ball (MRE 11th ed, p. 181). Says he can do it blindfold. W. H. Cozens. Note 2884: On note 2592. MG 44 (No. 348) (May 1960) 117. Estimates there are 200,000 cyclically symmetric tours on the 10 x 10 board. Roger F. Wheeler. Note 3059: The KNIGHT's tour on 4ÃÃ2ÄÄ and other boards. MG 47 (No. 360) (May 1963) 136-141. KNIGHT means a knight on a toroidal board. He finds 2688 tours of 19 types on the 4ÃÃ2ÄÄ toroid. (Cf Tylor, 1982??) J. J. Duby. Un algorithme graphique trouvant tous les circuits Hamiltoniens d'un graphe. Etude No. 8, IBM France, Paris, 22 Oct 1964. [In English with French title and summary.] Finds there are 9862 knight's tours on the 6 x 6 board, where the tours all start at a fixed corner and then go to a fixed one of the two cells reachable from the corner. He also finds 75,000 tours on the 8 x 8 board which have the same first 35 moves. He believes there may be over a million tours. Karl Fabel. Wanderungen von Schachfiguren. IN: Eero Bonsdorff, Karl Fabel & Olavi Riihimaa; Schach und Zahl; Walter Rau Verlag, DÀGÀsseldorf, 1966, pp. 40©50. On p. 50, he says that there are 122,802,512 tours where the knight does two joined half©board paths. He also says there are upper bounds, determined by several authors, and he gives 1.5 x 10ÃÃ26ÄÄ as an example. Gardner. SA (Oct 1967) = Magic Show, chap. 14. Surveys results of which boards have tours or paths. D. J. W. Stone. On the Knight's Tour Problem and Its Solution by Graph-Theoretic and Other Methods. M.Sc. Thesis, Dept. of Computing Science, Univ. of Glasgow, Jan. 1969. Confirms Duby's 9862 tours on the 6 x 6 board. David Singmaster. Enumerating unlabelled Hamiltonian circuits. International Series on Numerical Mathematics, No. 29. BirkhÀÀuser, Basel, 1975, pp. 117-130. Discusses the work of Duby and Stone and gives an estimate, which Stone endorses, that there are 10ÃÃ23ÀÀ3ÄÄ tours on the 8 x 8 board. C. M. B. Tylor. 2-by-2 tours. Chessics 14 (Jul-Dec 1982) 14. Says there are 17 knight's tours on a 2 x 2 torus and gives them. Doesn't mention Wheeler, 1963. Robert Cannon & Stan Dolan. The knight's tour. MG 70 (No. 452) (Jun 1986) 91-100. A rectangular board is tourable if it has a knight's path between any two cells of opposite colours. They prove that m x n is tourable if and only if mn is even and m ÀÀ 6, n ÀÀ 6. They also prove that m x n has a knight's tour if and only if mn is even and [(m ÀÀ 5, n ÀÀ 5) or (m = 3, n ÀÀ 10)] and that when mn is even, m x n has a knight's path if and only if m ÀÀ 3, n ÀÀ 3, except for the 3 x 6 and 4 x 4 boards. (These later results are well known ©© see Gardner. The authors only cite Ball's MRE.) George Jelliss. Figured tours. MS 25:1 (1992/93) 16©20. Exposition of paths and tours where certain stages of the path form an interesting geometric figure. E.g. Euler's first paper has a path on the 5 x 5 such that the points on one diagonal are in arithmetic progression: 1, 7, 13, 19, 25. Martin Loebbing & Ingo Wegener. The number of knight's tours equals 33,439,123,484,294 -©© Counting with binary decision diagrams. Electronic Journal of Combinatorics 3 (1996) article R5. A somewhat vague description of a method for counting knight's tours ©© they speak of directed knight's tours, but it is not clear if they have properly accounted for the symmetries of a tour or of the board. Several people immediately pointed out that the number is incorrect because it has to be divisible by four. Two comments have appeared, ibid. On 15 May 1996, the authors admitted this and said they would redo the problem, but they have submitted no further comment as of Jan 2001. On 18 Feb 1997, Brendan McKay announced that he had done the computation another way and found 13,267,364,410,532. ÁÁÁÁIn view of the difference between this and my 1975 estimate of 10ÃÃ23ÀÀ3ÄÄ tours, it might be worth explaining my reasoning. In 1964, Duby found 75,000 tours with the same first 35 moves. The average valence for a knight on an 8 x 8 board is 5.25, but one cannot exit from a cell in the same direction as one entered, so we might estimate the number of ways that the first 35 moves can be made as 4.25ÃÃ35ÄÄ = 9.9 x 10ÃÃ21ÄÄ. Multiplying by 75,000 then gives 7.4 x 10ÃÃ26ÄÄ. I think I assumed that some of the first moves had already been made, e.g. we only allow one move from the starting cell, and factored by 8 for the symmetries of the square, to get 2.2 x 10ÃÃ25ÄÄ. I can't find my original calculations, and I find the estimate 10ÃÃ25ÄÄ in later papers, so I suppose I tried to reduce the effect of the 4.25ÃÃ35ÄÄ some more. In retrospect, I had no knowledge of how many of these had already been tried. If about half of all moves from a cell had already been tried before any circuit was found, then the estimate would be more like 2.25ÃÃ34ÄÄ x 75,000 = 7.1 x 10ÃÃ16ÄÄ. If we divide the given number of circuits by 75,000 and take the 34th root, we get an average valence of 1.78 remaining, far less than I would have guessed. ÁÁÁÁI am grateful to Don Knuth for this reference. Neither he nor I expected to ever see this number calculated! ÁÁà Ã5.F.2.ÁÁOTHER HAMILTONIAN CIRCUITSÄ Ä ÁÁFor circuits on the n-cube, see also 5.F.4 and 7.M.1,2,3. ÁÁFor circuits on the chessboard, see also 6.AK. Le NÀ=Àtre. Le Labyrinte de Versailles, c1675. This was a hedge or garden maze, but the objective was to visit, in correct order, 40 fountains based on Aesop's à ÃFablesÄ Ä. Each node of the maze had at least one fountain. Some fountains were not at path junctions, but one can consider these as nodes of degree two. This is an early example of a Hamiltonian problem, except that one fountain was located at the end of a short dead end. [Fisher, op. cit. in 5.E.1, pp. 49, 79, 130 & 144©145, with contemporary diagram on p. 144. He says there are 39 fountains, but the diagram has 40.] T. P. Kirkman. On the partitions of the R-pyramid, being the first class of R-gonous X-edra. Philos. Trans. Roy. Soc. 148 (1858) 145-161. W. R. Hamilton. The Icosian Game. 4pp instructions for the board game. J. Jaques and Son, London, 1859. (Reproduced in BLW, pp. 32©35, with frontispiece photo of the board at the Royal Irish Academy.) For a long time, the only known example of the game, produced by Jaques, was at the Royal Irish Academy in Dublin. This example is inscribed on the back as a present from Hamilton to his friend, J. T. Graves. It is complete, with pegs and instructions. None of the obvious museums have an example. Diligent searching in the antique trade failed to turn up an example in twenty years, but in Feb 1996, James Dalgety found and acquired an example of the board ©© sadly the pegs and instructions were lacking. Dalgety obtained another board in 1998, again without the pegs and instructions, but in 1999 he obtained another example, with the pegs. Mittenzwey. 1880. Prob. 281, pp. 50 & 100; 1895?: 310, pp. 53©54 & 102; 1917: 310, pp. 49 & 97. The garden of a French palace has a maze with 31 points to see. Find a path past all of them with no repeated edges and no crossings. The pattern is clearly based on the Versailles maze of c1675 mentioned above, but I don't recall the additional feature of no crossings occurring before. T. P. Kirkman. Solution of problem 6610, proposed by himself in verse. Math. Quest. Educ. Times 35 (1881) 112-116. On p. 115, he says Hamilton told him, upon occasion of Hamilton presenting him 'with his handsomest copy of the puzzle', that Hamilton got the idea for the Icosian Game from p. 160 of Kirkman's 1858 article, Lucas. RM2, 1883, pp. 208-210. First? mention of the solid version. The 2nd ed., 1893, has a footnote referring to Kirkman, 1858. John Jaques & Son. The Traveller's Dodecahedron; or, A Voyage Round the World. A New Puzzle. "This amusing puzzle, exercising considerable skill in its solution, forms a popular illustration of Sir William Hamilton's Icosian Game. A wood dodecahedron with the base pentagon stretched so that when it sits on the base, all vertices are visible. With ivory? pegs at the vertices, a handle that screws into the base, a string with rings at the ends and one page of instructions, all in a box. No date. The only known example was obtained by James Dalgety in 2002. Pearson. 1907. Part III, no. 60: The open door, pp. 60 & 130. Prisoner in one corner of an 8 x 8 array is allowed to exit from from the other corner provided he visits every cell once. This requires him to enter and leave a cell by the same door. Ahrens. Mathematische Spiele. 2nd ed., Teubner, Leipzig, 1911. P. 44, note, says that a Dodekaederspiel is available from Firma Paul Joschkowitz ©© Magdeburg for .65 mark. This is not in the 1st ed. of 1907 and the whole Chapter is dropped in the 3rd ed. of 1916 and the later editions. Anonymous. The problems drive. Eureka 12 (Oct 1949) 7©8 & 15. No. 3. How many Hamiltonian circuits are there on a cube, starting from a given point? Reflections and reversals count as different tours. Answer is 12, but this assumes also that rotations are different. See Singmaster, 1975, for careful definitions of how to count. There are 96 labelled circuits, of which 12 start at a given vertex. But if one takes all the 48 symmetries of the cube as equivalences (six of which fix the given vertex), there are just 2 circuits from a given starting point. However, these are actually the same circuit started at different points. Presumably Kirkman and Hamilton knew of this. C. W. Ceram. Gods, Graves and Scholars. Knopf, New York, 1956, pp. 26©29. 2nd ed., Gollancz, London, 1971, pp. 24©25. Roman knobbed dodecahedra ©© an ancient solid version?? R. E. Ingram. Appendix 2: The Icosian Calculus. In: The Mathematical Papers of Sir William Rowan Hamilton. Vol. III: Algebra. Ed. by H. Halberstam & R. E. Ingram. CUP, 1967, pp. 645-647. [Halberstam told me that this Appendix is due to Ingram.] Discusses the method and asserts that the tetrahedron, cube and dodecahedron have only one unlabelled circuit, the octahedron has two and the icosahedron has 17. David Singmaster. Hamiltonian circuits on the regular polyhedra. Notices Amer. Math. Soc. 20 (1973) A-476, no. 73T-A199. Confirms Ingram's results and gives the number of labelled circuits. David Singmaster. Op. cit. in 5.F.1. 1975. Carefully defines labelled and unlabelled circuits. Discusses results on regular polyhedra in 3 and higher dimensions. David Singmaster. Hamiltonian circuits on the n-dimensional octahedron. J. Combinatorial Theory (B) 18 (1975) 1-4. Obtains an explicit formula for the number of labelled circuits on the n-dimensional octahedron and shows it is ÀsÀ (2n)!/e. Gives numbers for n ÀÀ 8. In unpublished work, it is shown that the number of unlabelled circuits is asymptotic to the above divided by n!2ÃÃnÄÄÀ À4n. Angus Lavery. The Puzzle Box. G&P 2 (May 1994) 34©35. Alternative solitaire, p. 34. Asks for a knight's tour on the 33©hole solitaire board. Says he hasn't been able to do it and offers a prize for a solution. In Solutions, G&P 3 (Jun 1994) 44, he says it cannot be done and the proof will be given in a future issue, but I never saw it. ÁÁà Ã5.F.3.ÁÁKNIGHT'S TOURS IN HIGHER DIMENSIONSÄ Ä A.-T. Vandermonde. Remarques sur les problÀ/Àmes de situation. Hist. de l'Acad. des Sci. avec les MÀ)Àmoires (Paris) (1771 (1774)) MÀ)Àmoires: pp. 566-574 & Plates I & II. ??NYS. First? mention of cubical problem. (Not given in BLW excerpt.) F. Maack. Mitt. ÀGÀber Raumschak. 1909, No. 2, p. 31. ??NYS ©© cited by Gibbins, below. Knight's tour on 4 x 4 x 4 board. Dudeney. AM. 1917. Prob. 340: The cubic knight's tour, pp. 103 & 229. Says Vandermonde asked for a tour on the faces of a 8 x 8 x 8 cube. He gives it as a problem with a solution. N. M. Gibbins. Chess in three and four dimensions. MG 28 (No. 279) (1944) 46-50. Gives knight's tour on 3 x 3 x 4 board ©© an unpublished result due to E. Hubar-Stockar of Geneva. This is the smallest 3-D board with a tour. Gives Maack's tour on 4 x 4 x 4 board. Ian Stewart. Solid knight's tours. JRM 4:1 (Jan 1971) 1. Cites Dudeney. Gives a tour through the entire 8 x 8 x 8 cube by stacking 8 knight's paths. T. W. Marlow. Closed knight tour of a 4 x 4 x 4 board. Chessics 29 & 30 (1987) 162. Inspired by Stewart. ÁÁà Ã5.F.4.ÁÁOTHER CIRCUITS IN AND ON A CUBEÄ Ä Ðа¤˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐÁÁThe number of Hamiltonian Circuits on the n©dimensional cube is the same as the number of Gray codes (see 7.M.3) and has been the subject of considerable research. I will not try to cover this in detail. ÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐD. W. Crowe. The n-dimensional cube and the Tower of Hanoi. AMM 63:1 (Jan 1956) 29-30. E. N. Gilbert. Gray codes and paths on the n©cube. Bell System Technical Journal 37 (1958) 815©826. Shows there are 9 inequivalent circuits on the 4©cube and 1 on the n©cube for n = 1, 2, 3. The latter cases are sufficiently easy that they may have been known before this. Allen F. Dreyer. US Patent 3,222,072 ©© Block Puzzle. Filed: 11 Jun 1965; patented: 7 Dec 1965. 4pp + 2pp diagrams. 27 cubes on an elastic. The holes are straight or diagonal so that three consecutive cubes are either in a line or form a right angle. A solution is a Hamiltonian path through the 27 cells. Such puzzles were made in Germany and I was given one about 1980 (see Singmaster and Haubrich & Bordewijk below). Dreyer gives two forms. Gardner. The binary Gray code. SA (Aug 1972) c= Knotted, chap. 2. Notes that the number of circuits on the n©cube, n > 4, is not known. SA (Apr 1973) reports that three (or four) groups had found the number of circuits on the 4©cube ©© this material is included in the Addendum in Knotted, chap. 2, but none of the groups ever seem to have published their results elsewhere. Unfortunately, none of these found the number of inequivalent circuits since they failed to take all the equivalences into account ©© e.g. for n = 1, 2, 3, 4, 5, their enumerations give: 2, 8, 96, 43008, 5 80189 28640 for the numbers of labelled circuits. Gardner's Addendum describes some further work including some statistical work which estimates the number on the 6©cube is about 2.4 x 10ÃÃ25ÄÄ. David Singmaster. A cubical path puzzle. Written in 1980 and submitted to JRM, but never published. For the 3 x 3 x 3 problem, the number, S, of straight through pieces (ignoring the ends) satisfies 2 ÀÀ S ÀÀ 11. Mel A. Scott. Computer output, Jun 1986, 66pp. Determines there are 3599 circuits through the 3 x 3 x 3 cube such that the resulting string of 27 cubes can be made into a cube in just one way. But cf the next article which gives a different number?? Jacques Haubrich & Nanco Bordewijk. Cube chains. CFF 34 (Oct 1994) 12-15. Erratum, CFF 35 (Dec 1994) 29. Says Dreyer is the first known reference to the idea and that they were sold 'from about 1970' Reproduces the first page of diagrams from Dreyer's patent. Says his first version has a unique solution, but the second has 38 solutions. They have redone previous work and get new numbers. First, they consider all possible strings of 27 cubes with at most three in a line (i.e. with at most a single 'straight' piece between two 'bend' pieces and they find there are 98,515 of these. Only 11,487 of these can be folded into a 3 x 3 x 3 cube. Of these, 3654 can be folded up in only one way. The chain with the most solutions had 142 different solutions. They refer to Mel Scott's tables and indicate that the results correspond ©© perhaps I miscounted Scott's solutions?? ÁÁà Ã5.G.ÁÁCONNECTION PROBLEMSÄ Ä ÁÁà Ã5.G.1.ÁÁGAS, WATER AND ELECTRICITYÄ Ä Dudeney. Problem 146 ©© Water, gas, and electricity. Strand Mag. 46 (No. 271) (Jul 1913) 110 & (No. 272) (Aug 1913) 221 (c= AM, prob. 251, pp. 73 & 200-201). Earlier version is slightly more interesting, saying the problem 'that I have called "Water, Gas, and Electricity" ... is as old as the hills'. Gives trick solution with pipe under one house. A. B. Nordmann. One Hundred More Parlour Tricks and Problems. Wells, Gardner, Darton & Co., London, nd [1927 ©© BMC]. No. 96: The "three houses" problem, pp. 89©90 & 114. "Were all the houses connected up with all three supplies or not?" Answer is no ©© one connection cannot be made. Loyd, Jr. SLAHP. 1928. The three houses and three wells, pp. 6 & 87-88. "A puzzle ... which I first brought out in 1900 ..." The drawing is much less polished than Dudeney's. Trick solution with a pipe under one house, a bit differently laid out than Dudeney. The Bile Beans Puzzle Book. 1933. No. 46: Water, gas & electric light. Trick solution almost identical to Dudeney. Philip Franklin. The four color problem. In: Galois Lectures; Scripta Mathematica Library No. 5; Scripta Mathematica, Yeshiva College, NY, 1941, pp. 49©85. On p. 74, he refers to the graph as "the basis of a familiar puzzle, to join each of three houses with each of three wells (or in a modern version to a gas, water, and electricity plant)". Leeming. 1946. Chap. 6, prob. 4: Water, gas and electricity, pp. 71 & 185. Dudeney's trick solution. H. ApSimon. Note 2312: All modern conveniences. MG 36 (No. 318) (Dec 1952) 287-288. Given m houses and n utilities, the maximum number of non-crossing connections is 2(m+n-2) and this occurs when all the resulting regions are 4-sided. He extends to p-partite graphs in general and a special case. John Paul Adams. We Dare You to Solve This! Op. cit. in 5.C. 1957? Prob. 50: Another enduring favorite appears below, pp. 30 & 49. Electricity, gas, water. Dudeney's trick solution. Young World. c1960. P. 4: Crossed lines. Electricity, TV and public address lines. Trick solution with a line passing under a house. T. H. O'Beirne. For boys, men and heroes. New Scientist 12 (No. 266) (21 Dec 1961) 751-753. Shows you can join 4 utilities to 4 houses on a torus without crossing. ÁÁà Ã5.H.ÁÁCOLOURED SQUARES AND CUBES, ETC.Ä Ä ÁÁà Ã5.H.1.ÁÁINSTANT INSANITY = THE TANTALIZERÄ Ä Ðа¤˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐÁÁNote. Often the diagrams do not show all sides of the pieces so I cannot tell if one version is the same as another. ÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐ Frederick A. Schossow. US Patent 646,463 ©© Puzzle. Applied: 19 May 1899; patented: 3 Apr 1900. 1p + 1p diagrams. Described in S&B, p. 38, which also says it is described in O'Beirne, but I don't find it there?? Four cubes with suit patterns. The net of each cube is shown. The fourth cube has three clubs. George Duncan Moffat. UK Patent 9810 ©© Improvements in or relating to Puzzle©apparatus. Applied: 28 May 1900; accepted: 30 Jun 1900. 2pp + 1p diagrams. For a six cube version with "letters R, K, B, W, F and B©P, the initials of the names of General Officers of the South African Field Force." Joseph Meek. UK Patent 2775 ©© Improved Puzzle Game. Applied: 5 Feb 1909; complete specification: 16 Jun 1909; accepted: 3 Feb 1910. 2pp + 1p diagrams. A four cube version with suit patterns. His discussion seems to describe the pieces drawn by Schossow. Slocum. Compendium. Shows: The Great Four Ace Puzzle (Gamage's, 1913); Allies Flag Puzzle (Gamage's, c1915); Katzenjammer Puzzle (Johnson Smith, 1919). Edwin F. Silkman. US Patent 2,024,541 ©© Puzzle. Applied: 9 Sep 1932; patented: 17 Dec 1935. 2pp + 1 p diagrams. Four cubes marked with suits. The net of each cube is shown. The third cube has three hearts. This is just a relabelling of Schossow's pattern, though two cubes have to be reflected which makes no difference to the solution process. E. M. Wyatt. The bewitching cubes. Puzzles in Wood. (Bruce Publishing, Co., Milwaukee, 1928) = Woodcraft Supply Corp., Woburn, Mass., 1980, p. 13. A six cube, six way version. Abraham. 1933. Prob. 303 ©© The four cubes, p. 141 (100). 4 cube version "sold ... in 1932". A. S. Filipiak. Four ace cube puzzle. 100 Puzzles, How do Make and How to Solve Them. A. S. Barnes, NY, (1942) = Mathematical Puzzles, and Other Brain Twisters; A. S. Barnes, NY, 1966; Bell, NY, 1978; p. 108. Leeming. 1946. Chap. 10, prob. 9: The six cube puzzle, pp. 128-129 & 212. Identical to Wyatt. F. de Carteblanche [pseud. of Cedric A. B. Smith]. The coloured cubes problem. Eureka 9 (1947) 9-11. General graphical solution method, now the standard method. T. H. O'Beirne. Note 2736: Coloured cubes: A new "Tantalizer". MG 41 (No. 338) (Dec 1957) 292©293. Cites Carteblanche, but says the current version is different. Gives a nicer version. T. H. O'Beirne. Note 2787: Coloured cubes: a correction to Note 2736. MG 42 (No. 342) (Dec 1958) 284. Finds more solutions than he had previously stated. Norman T. Gridgeman. The 23 colored cubes. MM 44:5 (Nov 1971) 243©252. The 23 colored cubes are the equivalence classes of ways of coloring the faces with 1 to 6 colors. He cites and describes some later methods for attacking Instant Insanity problems. Jozsef BognÀÀr. UK Patent Application 2,076,663 A ©© Spatial Logical Puzzle. Filed 28 May 1981; published 9 Dec 1981. Cover page + 8pp + 3pp diagrams. Not clear if the patent was ever granted. Describes BognÀÀr's Planets, which is a four piece instant insanity where the pieces are spherical and held in a plastic tube. This was called Bolygok in Hungarian and there is a reference to an earlier Hungarian patent. Also describes his version with eight pieces held at the corners of a plastic cube. ÁÁà Ã5.H.2.ÁÁMACMAHON PIECESÄ Ä Ðа¤˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐÁÁHaubrich's 1995©1996 surveys, op. cit. in 5.H.4, include MacMahon puzzles as one class. ÁÁI have just added the Carroll result that there are 30 six©coloured cubes, but this must be older?? ÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐ Frank H. Richards. US Patent 331,652 ©© Domino. Applied: 13 Jun 1885; patented: 1 Dec 1885. 2pp + 2pp diagrams. Cited by Gardner in Magic Show, but with date 1895. Reproduced in Haubrich, About ..., 1996, op. cit. in 5.H.4. For triangular matching games, specifically showing the MacMahon 5©coloured triangles, but considering reflections as equivalences, so he has 35 pieces. [One of the colours is blank and hence Gardner said it was a 4©colouring.] Carroll©Wakeling. c1890? Prob. 15: Painting cubes, pp. 18©19 & 67. This is one of the problems on undated sheets of paper that Carroll sent to Bartholomew Price. How many ways can one six©colour a cube? Wakeling gives a solution, but this apparently is not on Carroll's MS. Percy Alexander MacMahon & Julian Robert John Jocelyn. UK Patent 3927 A.D. 1892 ©© Appliances to be used in Playing a New Class of Games. Applied: 29 Feb 1892; Complete Specification Left: 28 Nov 1892; Accepted: 28 Jan 1893. 5pp + 2pp diagrams. Reproduced in Haubrich, About ..., 1996, op. cit. in 5.H.4. Describes the 24 triangles with four types of edge and mentions other numbers of edge types. Describes various games and puzzles. Percy Alexander MacMahon & Julian Robert John Jocelyn. UK Patent 8275 A.D. 1892 ©© Appliances for New Games of Puzzles. Applied: 2 May 1892; Complete Specification Left: 31 Jan 1893; Accepted: 4 Mar 1893. 2pp. 27 cubes with three colours, opposite faces having the same colour. Similar sets of 8, 27, 64, etc. cubes. Various matching games suggested. Using six colours and all six on each cube gives 30 cubes ©© the MacMahon Cubes. Gives a complex matching problem of making two 2 x 2 x 2 cubes. Paul Garcia (email of 15 Nov 2002) commented: "8275 describes 2 different sets of blocks, using either three colours or six colours. The three colour blocks form a set of 27 that can be assembled into a large cube with single coloured faces and internal contact faces matching. For the six colour cubes, the puzzle suggested is to pick out two associated cubes, and find the sixteen cubes that can be assembled to make a copy of each. Not quite Mayblox, although using the same colouring system." James Dalgety. R. Journet & Company A Brief History of the Company & its Puzzles. Published by the author, North Barrow, Somerset, 1989. On p. 13, he says Mayblox was patented in 1892. In an email on 12 Nov 2002, he cited UK Patent 8275. Anon. Report: "Mathematical Society, February 9". Nature 47 (No. 1217) (23 Feb 1893) 406. Report of MacMahon's talk: The group of thirty cubes composed by six differently coloured squares. See: Au Bon MarchÀ)À, 1907, in 5.P.2, for a puzzle of hexagons with matching edges. Manson. 1911. Likoh, pp. 171©172. MacMahon's 24 four©coloured isosceles right triangles, attributed to MacMahon. "Toymaker". The Cubes of Mahomet Puzzle. Work, No. 1447 (9 Dec 1916) 168. 8 sixªcoloured cubes to be assembled into a cube with singly©coloured faces and internal faces to have matching colours. P. A. MacMahon. New Mathematical Pastimes. CUP, 1921. The whole book deals with variations of the problem and calculates the numbers of pieces of various types. In particular, he describes the 24 4©coloured triangles, the 24 3©coloured squares, the MacMahon cubes, some right©triangular and hexagonal sets and various subsets of these. With n colours, there are n(nÃÃ2ÄÄ+2)/3 triangles, n(n+1)(nÃÃ2ÄÄ-n+2)/4 squares and n(n+1)(nÃÃ4ÄÄ©nÃÃ3ÄÄ+nÃÃ2ÄÄ+2)/6 hexagons. [If one allows reflectional equivalence, one gets n(n+1)(n+2)/6 triangles, n(n+1)(nÃÃ2ÄÄ+n+2)/8 squares and n(n+1)(nÃÃ4ÄÄ©nÃÃ3ÄÄ+4nÃÃ2ÄÄ+2)/12 hexagons. Problem ©© is there an easy proof that the number of triangles is BC(n+2, 3)?] On p. 44, he says that Col. Julian R. Jocelyn told him some years ago that one could duplicate any cube with 8 other cubes such that the internal faces matched. Slocum. Compendium. Shows Mayblox made by R. Journet from Will Goldston's 1928 catalogue. F. Winter. Das Spiel der 30 bunten WÀGÀrfel MacMahon's Problem. Teubner, Leipzig, 1934, 128pp. ??NYR. Clifford Montrose. Games to play by Yourself. Universal Publications, London, nd [1930s?]. The coloured squares, pp. 78©80. Makes 16 squares with four©coloured edges, using five colours, but there is no pattern to the choice. Uses them to make a 4 x 4 array with matching edges, but seems to require the orientations to be fixed. M. R. Boothroyd & J. H. Conway. Problems drive, 1959. Eureka 22 (Oct 1959) 15©17 & 22©23. No. 6. There are twelve ways to colour the edges of a pentagon, when rotations and reflections are considered as equivalences. Can you colour the edges of a dodecahedron so each of these pentagonal colourings occurs once? [If one uses tiles, one has to have reversible tiles.] Solution says there are three distinct solutions and describes them by describing contacts between 10 pentagons forming a ring around the equator. Richard K. Guy. Some mathematical recreations I & II. Nabla [= Bull. Malayan Math. Soc.] 7 (Oct & Dec 1960) 97©106 & 144©153. Pp. 101©104 discusses MacMahon triangles, squares and hexagons. T. H. O'Beirne. Puzzles and paradoxes 5: MacMahon's three©colour set of squares. New Scientist 9 (No. 220) (2 Feb 1961) 288©289. Finds 18 of the 20 possible monochrome border patterns. Gardner. SA (Mar 1961) = New MD, Chap. 16. MacMahon's 3©coloured squares and his cubes. Addendum in New MD cites Feldman, below. Gary Feldman. Documentation of the MacMahon Squares Problem. Stanford Artificial Intelligence Project Memo No. 12, Stanford Computation Center, 16 Jan 1964. ??NYS Finds 12,261 solutions for the 6 x 4 rectangle with monochrome border ©© but see Philpott, 1982, for 13,328 solutions!! Gardner. SA (Oct 1968) = Magic Show, Chap. 16. MacMahon's four©coloured triangles and numerous variants. Wade E. Philpott. MacMahon's three©color squares. JRM 2:2 (1969) 67©78. Surveys the topic and repeats Feldman's result. N. T. Gridgeman, loc. cit. in 5.H.1, 1971, covers some ideas on the MacMahon cubes. J. J. M. Verbakel. Digitale tegels (Digital tiles). Niet piekeren maar puzzelen (name of a puzzle column). Trouw (a Dutch newspaper) (1 Feb 1975). ??NYS ©© described by Jacques Haubrich; Pantactic patterns and puzzles; CFF 34 (Oct 1994) 19©21. There are 16 ways to 2-colour the edges of a square if one does ÃÃnotÄÄ allow them to rotate. Assemble these into a 4 x 4 square with matching edges. There are 2,765,440 solutions in 172,840 classes of 16. One can add further constraints to yield fewer solutions ©© e.g. assume the 4 x 4 square is on a torus and make all internal lines have a single colour. Gardner. Puzzling over a problem-solving matrix, cubes of many colours and three-dimensional dominoes. SA 239:3 (Sep 1978) 20-30 & 242 c= Fractal, chap. 11. Good review of MacMahon (photo) and his coloured cubes. Bibliography cites recent work on Mayblox, etc. Wade E. Philpott. Instructions for Multimatch. Kadon Enterprises, Pasadena, Maryland, 1982. Multimatch is just the 24 MacMahon 3©coloured squares. This surveys the history, citing several articles ??NYS, up to the determination of the 13,328 solutions for the 6 x 4 rectangle with monochrome border, by Hilario FernÀÀndez Long (1977) and John W. Harris (1978). Torsten Sillke. Three 3 x 3 matching puzzles. CFF 34 (Oct 1994) 22©23. He has wanted an interesting 9 element subset of the MacMahon pieces and finds that of the 24 MacMahon 3©coloured squares, just 9 of them contain all three colours. He considers both the corner and the edge versions. The editor notes that a 3 x 3 puzzle has 36 x 32/2 = 576 possible edge contacts and that the number of these which match is a measure of the difficulty of the puzzle, with most 3 x 3 puzzles having 60 to 80 matches. The corner version of Sillke's puzzle has 78 matches and one solution. The edge version has 189 matches and many solutions, hence Sillke proposes various further conditions. ÁÁà Ã5.H.3.ÁÁPATH FORMING PUZZLESÄ Ä Ðа¤˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐ ÁÁHere we have a set of pieces and one has to join them so that some path is formed. This is often due to a chain or a snake, etc. New section. Again, Haubrich's 1995©1996 surveys, op. cit. in 5.H.4, include this as one class. ÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐ Hoffmann. 1893. Chap. III, No. 18: The endless chain, pp. 99©100 & 131 = Hoffmann-Hordern, pp. 91©92, with photo. 18 pieces, some with parts of a chain, to make into an 8 x 8 array with the chain going through 34 of the cells. All the pieces are rectangles of width one. Photo shows The Endless Chain, by The Reason Manufacturing Co., 1880©1895. Hordern Collection, p. 62, shows the same and La Chaine sans fin, 1880©1905. Loyd. Cyclopedia. 1914. Sam Loyd's endless chain puzzle, pp. 280 & 377. Chain through all 64 cells of a chessboard, cut into 13 pieces. The chessboard dissection is of type: 13: 02213 131. Hummerston. Fun, Mirth & Mystery. 1924. The dissected serpent, p. 131. Same pieces as Hoffmann, and almost the same pattern. Collins. Book of Puzzles. 1927. The dissected snake puzzle, pp. 126©127. 17 pieces forming an 8 x 8 square. All the piece are rectangular pieces of width one except for one L-hexomino ©© if this were cut into straight tetromino and domino, the pieces would be identical to Hoffmann. The pattern is identical to Hummerston. See Haubrich in 5.H.4. ÁÁà Ã5.H.4.ÁÁOTHER AND GENERALÄ Ä Ðа¤˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐ ÁÁThese all have coloured edges unless specified. See S&B, p. 36, for examples. ÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐEdwin L[ajette] Thurston. US Patent 487,797 ©© Puzzle. Applied: 30 Sep 1890; patented: 13 Dec 1892. 3pp + 3pp diagrams. Reproduced in Haubrich, About ..., 1996, op. cit. below. 4 x 4 puzzles with 6©coloured corners or edges, but assuming no colour is repeated on a piece ©© indeed he uses the 15 = BC(6,2) ways of choosing 4 out of 6 colours once only and then has a sixteenth with the same colours as another, but in different order. Also a star©shaped puzzle of six parallelograms. Edwin L. Thurston. US Patent 487,798 ©© Puzzle. Applied: 30 Sep 1890; patented: 13 Dec 1892. 2pp + 1p diagrams. Reproduced in Haubrich, About ..., 1996, op. cit. below. As far as I can see, this is the same as the 4 x 4 puzzle with 6-coloured edges given above, but he seems to be emphasising the 15 pieces. Edwin L. Thurston. US Patent 490,689 ©© Puzzle. Applied: 30 Sep 1890; patented: 31 Jun 1893. 2pp + 1p diagrams. Reproduced in Haubrich, About ..., 1996, op. cit. below. The patent is for 3 x 3 puzzles with 4-coloured corners or edges, but with pieces having no repeated colours and in a fixed orientation. He selects some 8 of these pieces for reasons not made clear and mentions moving them "after the manner of the old 13, 14, 15 puzzle." S&B, p. 36, describes the Calumet Puzzle, Calumet Baking Powder Co., Chicago, which is a 3 x 3 head to tail puzzle, claimed to be covered by this patent. Le Berger Malin. France, c1900. 3 x 3 head to tail puzzle, but the edges are numbered and the matching edges must add to 10. ??NYS ©© described by K. Takizawa, N. Takashima & N. Yoshigahara; Vess Puzzle and Its Family ©© A Compendium of 3 by 3 Card Puzzles; published by the authors, Tokyo, 1983. Slocum has this in two different boxes and dates it to c1900 ©© I had c1915 previously. Haubrich has one version, Produced by GB&O N.K. Atlas. Angus K. Rankin. US Patent 1,006,878 ©© Puzzle. Applied: 3 Feb 1911; patented: 24 Oct 1911. 2pp + 1p diagrams. Reproduced in Haubrich, About ..., 1996, op. cit. below. Described in S&B, p. 36. Grandpa's Wonder Puzzle. 3 x 3 square puzzle. Each piece has corners coloured, using four colours, and the colours meeting at a corner must differ. The patent doesn't show the advertiser's name ©© Grandpa's Wonder Soap ©© but is otherwise identical to S&B's photo. Daily Mail World Record Net Sale puzzle. 1920-1921. Instructions and picture of the pieces. Letter from Whitehouse to me describing its invention. 19 6©coloured hexagons without repeated colours. Daily Mail articles as follows. There may be others that I missed and sometimes the page number is a bit unclear. Note that 5 Dec was a Sunday. Ð ¤x ÐÐИŒ € thÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ¤ÐÐ 9 Nov 1920, p. 5. "Daily Mail" puzzle. To be issued on 7 Dec. 13 Nov 1920, p. 4. Hexagon mystery. 17 Nov 1920, p. 5. New mystery puzzle. Asserts the inventor does not know the solution ©© i.e. the solution has been locked up in a safe. 20 Nov 1920, p. 4. What is it? 23 Nov 1920, p. 5. Fascinating puzzle. The most fascinating puzzle since "Pigs in Clover". 25 Nov 1920, p. 5. Can you do it? 29 Nov 1920, p. 5. À À250 puzzle. 1 Dec 1920, p. 4. Mystery puzzle clues. 2 Dec 1920, p. 5. À À250 puzzle race. 3 Dec 1920, p. 5. The puzzle. 4 Dec 1920, p. 4. The puzzle. Amplifies on the inventor not knowing the solution ©© after the idea was approved, a new pattern was created by someone else and locked up. 6 Dec 1920, unnumbered back page. Photo with caption: À À250 for solving this. 7 Dec 1920, p. 7. "Daily Mail" Puzzle. Released today. À À100 for getting the locked up solution. À À100 for the first alternative solution and À À50 for the next alternative solution. "It is believed that more than one solution is possible." 8 Dec 1920, p. 5. "Daily Mail" puzzle. 9 Dec 1920, p. 5. Can you do it? 10 Dec 1920, p. 4. It can be done. 13 Dec 1920, p. 9. Most popular pastime. "More than 500,000 ÃÃDaily MailÄÄ Puzzles have been sold." 15 Dec 1920, p. 4. Puzzle king & the 19 hexagons. Dudeney says he does not think it can be solved "ÃÃexcept by trialÄÄ." 16 Dec 1920, p. 4. Tantalising 19 hexagons. 16 Dec 1920, unnumbered back page. Banner at top has: "The Daily Mail" puzzle. Middle of page has a cartoon of sailors trying to solve it. 17 Dec 1920, p. 5? The Xmas game. 18 Dec 1920, p. 7. Puzzle Xmas 'card'. 20 Dec 1920, p. 7. Hexagon fun. 22 Dec 1920, p. 3. 3,000,000 fascinated. It is assumed that about 5 people try each example and so this indicates that nearly 600,000 have been sold. 23 Dec 1920, p. 3. Too many cooks. 23 Dec 1920, unnumbered back page. Cartoon: The hexagonal dawn! 28 Dec 1920, p. 3? Puzzled millions. "On Christmas Eve the sales exceeded 600,000 ...." 29 Dec 1920, p. 3? "I will do it." 30 Dec 1920, p. 8. Puzzle fun. 3 Jan 1921, p. 3. The Daily Mail Puzzle. C. Lewis, aged 21, a postal clerk solved it within two hours of purchase and submitted his solution on 7 Dec. Hundreds of identical solutions were submitted, but no alternative solutions have yet appeared. There are two pairs of identical pieces: 1 & 12, 4 & 10. 3 Jan 1921, p. 10 = unnumbered back page. Hexagon Puzzle Solved, with photo of C. Lewis and diagram of solution. 10 Jan 1921, p. 4. Hexagon puzzle. Since no alternative hexagonal solutions were received, the other À À150 is awarded to those who submitted the most ingenious other solution ©© this was judged to be a butterfly shape, submitted by 11 persons, who shared the À À150. Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐHorace Hydes & Francis Reginald Beaman Whitehouse. UK Patent 173,588 ©© Improvements in Dominoes. Applied: 29 Sep 1920; complete application: 29 Jun 1921; accepted: 29 Dec 1921. Reproduced in Haubrich, About ..., 1996, op. cit. below. 3pp + 1p diagrams. This is the patent for the above puzzle, corresponding to provisional patent 27599/20 on the package. The illustration shows a solved puzzle based on 'A stitch in time saves nine'. George Henry Haswell. US Patent 1,558,165 ©© Puzzle. Applied: 3 Jul 1924; patented: 11 Sep 1925. Reproduced in Haubrich, About ..., 1996, op. cit. below. 2pp + 1p diagrams. For edge©matching hexagons with further internal markings which have to be aligned. [E.g. one could draw a diagonal and require all diagonals to be vertical ©© this greatly simplifies the puzzle!] If one numbers the vertices 1, 2, ..., 6, he gives an example formed by drawing the diagonals 13, 15, 42, 46 which produces six triangles along the edges and an internal rhombus. C. Dudley Langford. Note 2829: Dominoes numbered in the corners. MG 43 (No. 344) (May 1959) 120-122. Considers triangles, squares and hexagons with numbers at the corners. There are the same number of pieces as with numbers on the edges, but corner numbering gives many more kinds of edges. E.g. with four numbers, there are 24 triangles, but these have 16 edge patterns instead of 4. The editor (R. L. Goodstein) tells Langford that he has made cubical dominoes "presumably with faces numbered". Langford suggests cubes with numbers at the corners. [I find 23 cubes with two corner numbers and 333 with three corner numbers. ??check] Piet Hein. US Patent 4,005,868 ©© Puzzle. Applied: 23 Jun 1975; patented: 1 Feb 1977. Front page + 8pp diagrams + 5pp text. Basically non©matching puzzles using marks at the corners of faces of the regular polyhedra. He devises boards so the problems can be treated as planar. Kiyoshi Takizawa; Naoaki Takashima & Nob. Yoshigahara. Vess Puzzle and Its Family ©© A Compendium of 3 by 3 Card Puzzles. Published by the authors, Tokyo, Japan, 1983. Studies 32 types (in 48 versions) of 3 x 3 'head to tail' matching puzzles and 4 related types (in 4 versions). All solutions are shown and most puzzles are illustrated with colour photographs of one solution. (Haubrich counts 51 versions ©© check??) Melford D. Clark. US Patent 4,410,180 ©© Puzzle. Applied: 16 Nov 1981; patented: 18 Oct 1983. Reproduced in Haubrich, About ..., 1996, op. cit. in 5.H.4. 2pp + 2pp diagrams. Corner matching squares, but with the pieces marked 1, 2, ..., so that the pieces marked 1 form a 1 x 1 square, the pieces marked 2 allow this to be extended to a 2 x 2 square, etc. There are nÃÃ2ÄÄ © (n©1)ÃÃ2ÄÄ pieces marked n. Jacques Haubrich. Compendium of Card Matching Puzzles. Printed by the author, Aeneaslaan 21, NL©5631 LA Eindhoven, Netherlands, 1995. 2 vol., 325pp. describing over 1050 puzzles. He classifies them by the nine most common matching rules: Heads and Tails; Edge Matching (i.e. MacMahon); Path Matching; Corner Matching; Corner Dismatching; Jig©Saw©Like; Continuous Path; Edge Dismatching; Hybrid. He does not include Jig©Saw©Like puzzles here. Using the number of cards and their shape, then the matching rules, he has 136 types. 31 different numbers of cards occur: 4, 6ª16, 18©21, 23©25, 28, 30, 36, 40, 45, 48, 56, 64, 70, 80, 85, 100. There is an index of 961 puzzle names. He says Hoffmann is the earliest published example. He notes that most path puzzles have a global criterion that the result have a single circuit which slightly removes them from his matching criterion and he does not treat them as thoroughly. He has developed computer programs to solve each type of puzzle and has checked them all. Jacques Haubrich. About, Beyond and Behind Card Matching Puzzles. [= Vol. 3 of above]. Ibid, Apr 1996, 87pp. This is a general discussion of the different kinds of puzzles, how to solve them and their history, reproducing ten patents and two obituaries. ÁÁà Ã5.I.ÁÁLATIN SQUARES AND EULER SQUARESÄ Ä Ðа¤˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐ ÁÁThis topic ties in with certain tournament problems but I have not covered them. See also Hoffmann and Loughlin & Flood in 5.A.2 for examples of two orthogonal 3 x 3 Latin squares. The derangement problems in 5.K.2 give Latin rectangles. ÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐ Ahrens©1 & Ahrens©2. Opp. cit. in 7.N. 1917 & 1922. Ahrens©1 discusses and cites early examples of Latin squares, going back to medieval Islam (c1200), where they were used on amulets. Ahrens©2 particularly discusses work of al-Buni ©© see below. (Ahmed ibn ÀÀAlÀ3À ibn JÀEÀsuf) el-BÀEÀni, (AbÀEÀ'l-ÀÀAbbÀÀs, el-QoresÀ3À.) = Abu-lÀÀAbbas al-Buni. (??= Muhyi'l-DÀ3Àn AbÀEÀÀÀl©ÀÀAbbÀÀs al-BÀEÀnÀ3À ©© can't relocate my source of this form.) Sams al-maÀÀÀÀrif = Shams al-maÀÀÀÀrif al-kubrÀÀ = À°Àams al©maÀÀÀ]Àrif. c1200. ??NYS. Ahrens©1 describes this briefly and incorrectly. He expands and corrects this work in Ahrens©2. See 7.N for more details. Ahrens notes that a 4 x 4 magic square can be based on the pattern of two orthogonal Latin squares of order 4, and Al©Buni's work indicates knowledge of such a pattern, exemplified by the square ÁÁ 8, 11, 14, 1; 13, 2, 7, 12; 3, 16, 9, 6; 10, 5, 4, 15 considered (mod 4). He also has Latin squares of order 4 using letters from a name of God. He goes on to show 7 Latin squares of order 7, using the same 7 letters each time ©© though four are corrupted. (Throughout, the Latin squares also have 'Latin' diagonals, i.e. the diagonals contain all the values.) These are arranged so each has a different letter in the first place. It is conjectured that these are associated with the days of the week or the planets. Tagliente. Libro de Abaco. (1515). 1541. F. 18v. 7 x 7 Latin square with entries 1, 13, 2, 14, 3, 10, 4 cyclically shifted forward ©© i.e. the second row starts 13, 2, .... This is an elaborate plate which notes that the sum of each file is 47 and has a motto: Sola Virtu la Fama Volla, but I could find no text or other reason for its appearance! Inscription on memorial to Hannibal Bassett, d. 1708, in Meneage parish church, St. Mawgan, Cornwall. I first heard of this from Chris Abbess, who reported it in some newsletter in c1993. However, [Peter Haining; The Graveyard Wit; Frank Graham, Newcastle, 1973, p. 133] cites this as being at Cunwallow, near Helstone, Cornwall. [W. H. Howe; Everybody's Book of Epitaphs Being for the Most Part What the Living Think of the Dead; Saxon & Co., London, nd [c1895] (facsimile by Pryor Publications, Whitstable, 1995); p. 173] says it is in Gunwallow Churchyard. Spelling and punctuation vary a bit. The following gives a detailed account. Alfred Hayman Cummings. The Churches and Antiquities of Cury & Gunwalloe, in the Lizard District, including Local Traditions. E. Marlborough & Co., London & Truro, 1875, pp. 130©131. ??NX. "It has been said that there once existed ... the curious epitaph ©©" and gives a considerable rearrangement of the inscription below. He continues "But this is in all probability a mistake, as repeated search has been made for it, not only by the writer, but by a former Vicar of Gunwalloe, and it could nowhere be found, while there ÃÃisÄÄ a plate with an inscription in the church at Mawgan, the next parish, which might be very easily the one referred to." He gives the following inscription, saying it is to Hannibal Basset, d. 1708©9. Chris Weeks was kind enough to actually go to the church of St. Winwaloe, Gunwalloe, where he found nothing, and to St. Mawgan in Meneage, a few miles away. Chris Weeks sent pictures of Gunwallowe ª© the church is close to the cliff edge and it looks like there could once have been more churchyard on the other side of the church where the cliff has fallen away. In the church at St. Mawgan is the brass plate with 'the Acrostic Brass Inscription', but it is not clearly associated with a grave and I wonder if it may have been moved from Gunwallowe when a grave was eroded by the sea. It is on the left of the arch by the pulpit. I reproduce Chris Weeks' copy of the text. He has sent a photograph, but it was dark and the photo is not very clear, but one can make out the Latin square part. ÁÁÁÁÁÁHanniball BaÀ(ÀÀ(Àet here Inter'd doth lye ÁÁÁÁÁÁWho dying lives to all Eternitye ÁÁÁÁÁÁhee departed this life the 17ÃÃthÄÄ of Ian ÁÁÁÁÁÁ1709/8 in the 22ÃÃthÄÄ year of his age À À ÁÁÁÁÁÁÁÁA lover of learning Ð °Ü ÐÐФŒ H Àÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐÁÁÁÁShallÁÁweeÁÁallÁÁdye ÁÁÁÁWeeÁÁshallÁÁdyeÁÁall ÁÁÁÁallÁÁdyeÁÁshallÁÁwee ÁÁÁÁdyeÁÁallÁÁweeÁÁshall Ð °x ÐÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐ ÁÁÁÁThe À(ÀÀ(À are old style long esses. The superscript th is actually over the numeral. The 9 is over the 8 in the year and there is no stroke. This is because it was before England adopted the Gregorian calendar and so the year began on 25 Mar and was a year behind the continent between 1 Jan and 25 Mar. Correspondence of the time commonly would show 1708/9 at this time, and I have used this form for typographic convenience, but with the 9 over the 8 as on the tomb. ÁÁÁÁA word game book points out that this inscription is also palindromic!! Richard Breen. Funny Endings. Penny Publishing, UK, 1999, p. 35. Gives the following form: Shall we all die? / We shall die all. / All die shall we? / Die all we shall and notes that it is a word palindrome and says it comes from Gunwallam [sic], near Helstone. Joseph Sauveur. Construction gÀ)ÀnÀ)Àrale des quarrÀ)Às magiques. MÀ)Àmoires de l'AcadÀ)Àmie Royale des Sciences 1710(1711) 92-138. ??NYS ©© described in Cammann-4, p. 297, (see 7.N for details of Cammann) which says Sauveur invented Latin squares and describes some of his work. Ozanam. 1725. 1725: vol. IV, prob. 29, p. 434 & fig. 35, plate 10 (12). Two 4 x 4 orthogonal squares, using A, K, Q, J of the 4 suits, but it looks like: ÁÁÁÁJÀÀ, AÀÀ, KÀÀ, QÀÀ; QÀÀ, KÀÀ, AÀÀ, JÀÀ; AÀÀ, JÀÀ, QÀÀ, KÀÀ; KÀÀ, QÀÀ, JÀÀ, AÀÀ; but the ÀÀ and ÀÀ look very similar. From later versions of the same diagram, it is clear that the first row should have its ÀÀ and ÀÀ reversed. Note the diagonals also contain all four ranks and suits. (I have a reference for this to the 1723 edition.) Minguet. 1733. Pp. 146©148 (1864: 142©143; not noticed in other editions). Two 4 x 4 orthogonal squares, using A, K, Q, J (= As, Rey, Caballo (knight), Sota (knave)) of the 4 suits, but the Spanish suits, in descending order, are: Espadas, Bastos, Oros, Copas. The result is described but not drawn, as: ÁÁÁÁRO, AE, CC, SB; SC, CB, AO, RE; AB, RC, SE, CO; CE, SO, RB, AC; ÁÁwhich would translate into the more usual cards as: ÁÁÁÁKÀÀ, AÀÀ, QÀÀ, JÀÀ; JÀÀ, QÀÀ, AÀÀ, KÀÀ; AÀÀ, KÀÀ, JÀÀ, QÀÀ; QÀÀ, JÀÀ, KÀÀ, AÀÀ. ÁÁHowever, I'm not sure of the order of the Caballo and Sota; if they were reversed, which would interchange Q and J in the latter pattern, then both Ozanam and Minguet would have the property that each row is a cyclic shift or reversal of A, K, Q, J. Alberti. 1747. Art. 29, p. 203 (108) & fig. 36, plate IX, opp. p. 204 (108). Two 4 x 4 orthogonal squares, figure simplified from the correct form of Ozanam, 1725. L. Euler. Recherches sur une nouvelle espÀ/Àce de QuarrÀ)Às Magiques. (Verhandelingen uitgegeven door het zeeuwsch Genootschap der Wetenschappen te Vlissingen (= Flessingue) 9 (1782) 85-239.) = Opera Omnia (1) 7 (1923) 291-392. (= Comm. Arithm. 2 (1849) 302-361.) Manuel des Sorciers. 1825. Pp. 78©79, art. 39. ??NX Correct form of Ozanam. The Secret Out. 1859. How to Arrange the Twelve Picture Cards and the four Aces of a Pack in four Rows, so that there will be in Neither Row two Cards of the same Value nor two of the same Suit, whether counted Horizontally or Perpendicularly, pp. 90©92. Two 4 x 4 orthogonal Latin squares, not the same as in Ozanam. Bachet©Labosne. Problemes. 3rd ed., 1874. Supp. prob. XI, 1884: 200-202. Two 4 x 4 orthogonal squares. Berkeley & Rowland. Card Tricks and Puzzles. 1892. Card Puzzles, No. XVI, pp. 17©18. Similar to Oza