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 Sp