ÿWPCL ûÿ2BJ|xÕ!Ð x ÐÐÐüð ä ØÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÐЊ‚ÐÈÐÁ`ÁSOURCES © page !ÕÐ °x ÐÐа¤˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐЊ‚ÐÈÐ Ã ÃÁàÁSOURCES IN RECREATIONAL MATHEMATICS ÁàÁAN ANNOTATED BIBLIOGRAPHY Ä ÄÁàÁEIGHTH PRELIMINARY EDITION ÁàÁà ÃDAVID SINGMASTERÄ Ä ÁàÁ87 Rodenhurst Road, London, SW4 8AF, UK ÁàÁTel/fax: 020©8674 3676; email: ZINGMAST @ LSBU.AC.UK ÁàÁLast updated on ØD1 3 4. DØ ÁÁThis is a copy of the current version from my source files. I had intended to reorganise the material before producing a Word version, but have decided to produce this version for G4G6 and to renumber it as the Eighth Preliminary Edition. ÁÁA version from early 2000 was converted into HTML by Bill Kalush and is available on www.geocities.com/mathrecsources/ and another version is at http://members.it.tripod.de/catur/singmast/intro.htm . ÁÁIf 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. [Fibonacci, translated by Grimm.]) ÁÁÁÁÁÁà ÃINTRODUCTIONÄ Ä ÁÁÁÁà ÃNATURE OF THIS WORKÄ Ä ÁÁRecreational mathematics is as old as mathematics itself. Recreational problems already occur in the oldest extant sources ©© the Rhind Papyrus and Old Babylonian tablets. The Rhind Papyrus has an example of a purely recreational problem ©© Problem 79 is like the "As I was going to St. Ives" nursery rhyme. The Babylonians give fairly standard practical problems with a recreational context ©© a man knows the area plus the difference of the length and width of his field, a measurement which no surveyor would ever make! There is even some prehistoric mathematics which could not have been practical ©© numerous 'carved stone balls' have been found in eastern Scotland, dating from the Neolithic period and they include rounded forms of all the regular polyhedra and some less regular ones. Since these early times, recreations have been a feature of mathematics, both as pure recreations and as pedagogic tools. In this work, I use recreational in a fairly broad sense, but I tend to omit the more straightforward problems and concentrate on those which 'stimulate the curiosity' (as Montucla says). ÁÁIn addition, recreational mathematics is certainly as diffuse as mathematics. Every main culture and many minor ones have contributed to the history. A glance at the Common References below, or at almost any topic in the text, will reveal the diversity of sources which are relevant to this study. Much information arises from material outside the purview of the ordinary historian of mathematics ©© e.g. patents; articles in newspapers, popular magazines and minor journals; instruction leaflets; actual artifacts and even oral tradition. ÁÁConsequently, it is very difficult to determine the history of any recreational topic and the history given in popular books is often extremely dubious or even simply fanciful. For example, Nim, Tangrams, and Magic Squares are often traced back to China of about 2000 BC. The oldest known reference to Nim is in America in 1903. Tangrams appear in China and Europe at essentially the same time, about 1800, though there are related puzzles in 18C Japan and in the Hellenistic world. Magic Squares seem to be genuinely a Chinese invention, but go back to perhaps a few centuries BC and are not clearly described until about 80AD. Because of the lack of a history of the field, results are frequently rediscovered. ÁÁWhen I began this bibliography in 1982, I had the the idea of producing a book (or books) of the original sources, translated into English, so people could read the original material. This bibliography began as the table of contents of such a book. I thought that this would be an easy project, but it has become increasingly apparent that the history of most recreations is hardly known. I have recently realised that mathematical recreations are really the folklore of mathematics and that the historical problems are similar to those of folklore. One might even say that mathematical recreations are the urban myths or the jokes or the campfire stories of mathematics. Consequently I decided that an annotated bibliography was the first necessity to make the history clearer. This bibliography alone has grown into a book, something like Dickson's à ÃHistory of the Theory of NumbersÄ Ä. Like that work, the present work divides the subject into a number of topics and treats them chronologically. ÁÁI have printed six preliminary editions of this work, with slightly varying titles. The first version of 4 Jul 1986 had 224 topics and was spaced out so entries would not be spread over two pages and to give room for page numbers. This stretched the text from 110pp to 129pp and was printed for the Strens Memorial Conference at the Univ. of Calgary in Jul/Aug 1986. I no longer worry about page breaks. The following editions had: 250 topics on 152 pages; 290 topics on 192 pages; 307 topics on 223 pages; 357 topics on 311 pages and 392 topics on 456 pages. The seventh edition was never printed, but was a continually changing computer file. It had about 419 topics (as of 20 Oct 95) and 587 pages, as of 20 Oct 1995. I then carried out the conversion to proportional spacing and this reduced the total length from 587 to 488 pages, a reduction of 16.87% which is conveniently estimated as 1/6. This reduction was fairly consistent throughout the conversion process. ÁÁThis eighth edition is being prepared for the Gathering for Gardner 6 in March 2004. The text is 818 pages as of 18 Mar 2004. There are about 457 topics as of 18 Mar 2004. ÁÁA fuller description of this project in 1984©1985 is given in my article ÃÃSome early sources in recreational mathematicsÄÄ, in: C. Hay et al., eds.; à ÃMathematics from Manuscript to PrintÄ Ä; Oxford Univ. Press, 1988, pp. 195-208. A more recent description is in my article: ÃÃRecreational mathematicsÄÄ; in: à ÃEncyclopedia of the History and Philosophy of the Mathematical SciencesÄ Ä; ed. by I. Grattan©Guinness; Routledge & Kegan Paul, 1993; pp. 1568©1575. ÁÁBelow I compare this work with Dickson and similar works and discuss the coverage of this work. ÁÁÁÁà ÃSIMILAR WORKSÄ Ä ÁÁAs already mentioned, the work which the present most resembles is Dickson's à ÃHistory of the Theory of NumbersÄ Ä. ÂÂÁÁThe history of science can be made entirely impartial, and perhaps that is what it should be, by merely recording who did what, and leaving all "evaluations" to those who like them. To my knowledge there is only one history of a scientific subject (Dickson's, of the Theory of Numbers) which has been written in this coldblooded, scientific way. The complete success of that unique example ©© admitted by all who ever have occasion to use such a history in their work ©© seems to indicate that historians who draw morals should have their own morals drawn. ÂÂÂÂE. T. Bell. à ÃThe Search for Truth.Ä Ä George Allen & Unwin, London, 1935, p. 131. ÁÁDickson attempted to be exhaustive and certainly is pretty much so. Since his time, many older sources have been published, but their number©theoretic content is limited and most of Dickson's topics do not go back that far, so it remains the authoritative work in its field. ÁÁThe best previous book covering the history of recreational mathematics is the second edition of Wilhelm Ahrens's à ÃMathematische Unterhaltungen und SpieleÄ Ä in two volumes. Although it is a book on recreations, it includes extensive histories of most of the topics covered, far more than in any other recreational book. He also gives a good index and a bibliography of 762 items, often with some bibliographical notes. I will indicate the appropriate pages at the beginning of any topic that Ahrens covers. This has been out of print for many years but Teubner has some plans to reissue it. ÁÁAnother similar book is the 4th edition of J. Tropfke's à ÃGeschichte der ElementarmathematikÄ Ä, revised by Vogel, Reich and Gericke. This is quite exhaustive, but is concerned with older problems and sources. It presents the material on a topic as a history with references to the sources, but it doesn't detail what is in each of the sources. Sadly, only one volume, on arithmetic and algebra, appeared before Vogel's death. A second volume, on geometry, is being prepared. For any topic covered in Tropfke, it should be consulted for further references to early material which I have not seen, particularly material not available in any western language. I cite the appropriate pages of Tropfke at the beginning of any topic covered by Tropfke. ÁÁAnother book in the field is W. L. Schaaf's à ÃBibliography of Recreational MathematicsÄ Ä, in four volumes. This is a quite exhaustive bibliography of recent articles, but it is not chronological, is without annotation and is somewhat less classified than the present work. Nonetheless it is a valuable guide to recent material. ÁÁCollecting books on magic has been popular for many years and quite notable collections and bibliographies have been made. Magic overlaps recreational mathematics, particularly in older books, and I have now added references to items listed in the bibliographies of Christopher, Clarke & Blind, Hall, Heyl, Toole Stott and Volkmann & Tummers ©© details of these works are given in the list of Common References below. There is a notable collection of Harry Price at Senate House, University of London, and a catalogue was printed in 1929 & 1935 ©© see HPL in Common References. ÁÁAnother related bibliography is Santi's à ÃBibliografia della EnigmisticaÄ Ä, which is primarily about word puzzles, riddles, etc., but has some overlap with recreational mathematics ©© again see the entry in the list of Common References. I have not finished working through this. ÁÁOther relevant bibliographies are listed in Section 3.B. ÁÁÁÁà ÃCOVERAGEÄ Ä ÁÁIn selecting topics, I tend to avoid classical number theory and classical geometry. These are both pretty well known. Dickson's à ÃHistory of the Theory of NumbersÄ Ä and Leveque's and Guy's à ÃReviews in Number TheoryÄ Ä cover number theory quite well. I also tend to avoid simple exercises, e.g. in the rule of three, in 'aha' or 'heap' problems, in the Pythagorean theorem (though I have now included 6.BF) or in two linear equations in two unknowns, though these often have fanciful settings which are intended to make them amusing and some of these are included ©© see 7.R, 7.X, 7.AX. I also leave out most divination (or 'think of a number') techniques (but a little is covered in 7.M.4.b) and most arithmetic fallacies. I also leave out Conway's approach to mathematical games ©© this is extensively covered by à ÃWinning WaysÄ Ä and Frankel's à ÃBibliographyÄ Ä. ÁÁThe classification of topics is still ad-hoc and will eventually get rationalised ©© but it is hard to sort things until you know what they are! At present I have only grouped them under the general headings: Biography, General, History & Bibliography, Games, Combinatorics, Geometry, Arithmetic, Probability, Logic, Physics, Topology. Even the order of these should be amended. The General section should be subsumed under the History & Bibliography. Geometry and Arithmetic need to be subdivided. ÁÁI have recently realised that some general topics are spread over several sections in different parts. E.g. fallacies are covered in 6.P, 6.R, 6.AD, 6.AW.1, 6.AY, 7.F, 7.Y, 7.Z, 7.AD, 7.AI, 7.AL, 7.AN, most of 8, 10.D, 10.E, 10.O. Perhaps I will produce an index to such topics. I try to make appropriate cross©references. ÁÁSome topics are so extensive that I include introductory or classifactory material at the beginning. I often give a notation for the problems being considered. I give brief explanations of those problems which are not well known or are not described in the notation or the early references. There may be a section index. I have started to include references to comprehensive surveys of a given topic ©© these are sometimes given at the beginning. ÁÁRecreational problems are repeated so often that it is impossible to include all their occurrences. I try to be exhaustive with early material, but once a problem passes into mathematical and general circulation, I only include references which show new aspects of the problem or show how the problem is transmitted in time and/or space. However, the point at which I start leaving out items may vary with time and generally slowly increases as I learn more about a topic. I include numerous variants and developments on problems, especially when the actual origin is obscure. ÁÁWhen I began, I made minimal annotations, often nothing at all. In rereading sections, particularly when adding more material, I have often added annotations, but I have not done this for all the early entries yet. ÁÁRecently added topics often may exist in standard sources that I have not reread recently, so the references for such topics often have gaps ©© I constantly discover that Loyd or Dudeney or Ahrens or Lucas or Fibonacci has covered such a topic but I have forgotten this ©© e.g. looking through Dudeney recently, I added about 15 entries. New sections are often so noted to indicate that they may not be as complete as other sections. ÁÁSome of the sources cited are lengthy and I originally added notes as to which parts might be usable in a book of readings ©© these notes have now been mostly deleted, but I may have missed a few. ÁÁÁÁà ÃSTATUS OF THE PROJECTÄ Ä ÁÁI would like to think that I am about 75% of the way through the relevant material. However, I recently did a rough measurement of the material in my study ©© there is about 8 feet of read but unprocessed material and about 35 feet of unread material, not counting several boxes of unread Rubik Cube material and several feet of semi©read material on my desk and table. I recently bought two bookshelves just to hold unread material. Perhaps half of this material is relevant to this work. ÁÁIn particular, the unread material includes several works of Folkerts and Sesiano on medieval MSS, a substantial amount of photocopies from Schott, Schwenter and Dudeney (400 columns), some 2000 pages of photocopies recently made at Keele, some 500 pages of photocopies from Martin Gardner's files, as well as a number of letters. Marcel Gillen has made extracts of all US, German and EURO patents and German registered designs on puzzles ©© 26 volumes, occupying about two feet on my shelves. I have recently acquired an almost complete set of ÃÃScripta MathematicaÄÄ (but I have previously read about half of it), SchwenterªHarsdÀ?Àrffer's à ÃDeliciÀ%À Physico-MathematicaeÄ Ä, Schott's à ÃJoco©SeriorumÄ Ä and Murray's à ÃHistory of Board Games Other Than ChessÄ Ä. I have recently acquired the early issues of ÃÃEurekaÄÄ, but there are later issues that I have not yet read and they persist in not sending the current copies I have paid for! ÁÁI have not yet seen some of the earlier 19C material which I have seen referred to and I suspect there is much more to be found. I have examined some 18C & 19C arithmetic and algebra books looking for problem sections ©© these are often given the pleasant name of Promiscuous Problems. There are so many of these that a reference to one of them probably indicates that the problem appears in many other similar books that I have not examined. My examination is primarily based on those books which I happen to have acquired. There are a few 15©17C books which I have not yet examined, notably those included at the end of the last paragraph. ÁÁIn working on this material, it has become clear that there were two particularly interesting and productive eras in the 19C. In the fifteen years from 1857, there appeared about a dozen books in the US and the UK: à ÃThe Magician's Own BookÄ Ä (1857); à ÃParlour PastimeÄ Ä, by "Uncle George" (1857); à ÃThe SociableÄ Ä (1858); à ÃThe Boy's Own ToymakerÄ Ä, by Landells (1858); à ÃThe Book of 500 Curious PuzzlesÄ Ä (1859); à ÃThe Secret OutÄ Ä (1859); à ÃIndoor and Outdoor Games for Boys and GirlsÄ Ä (c1859); à ÃThe Boy's Own Conjuring BookÄ Ä (1860); à ÃThe Illustrated Boy's Own TreasuryÄ Ä (1860, but see below); à ÃThe Parlor MagicianÄ Ä (1863); à ÃThe Art of AmusingÄ Ä, by Bellew (1866); à ÃParlour Pastimes (1868); Hanky PankyÄ Ä (1872); à ÃWithin DoorsÄ Ä, by Elliott (1872); à ÃMagic No MysteryÄ Ä (1876), just to name those that I know. Most of these are of uncertain authorship and went through several editions and versions. à ÃThe Magician's Own BookÄ Ä, à ÃThe Book of 500 Curious PuzzlesÄ Ä, à ÃThe Secret OutÄ Ä, à ÃThe SociableÄ Ä, à ÃThe Parlor MagicianÄ Ä,à à Hanky PankyÄ Ä, and à ÃMagic No MysteryÄ Ä seem to be by the same author(s). I have recently had a chance to look at a number of previously unseen versions at Sotheby's and at Edward Hordern's and I find that sometimes two editions of the same title are essentially completely different! This is particularly true for US and UK editions. Many of the later UK editions say 'By the author of Magician's Own Book etc., translated and edited by W. H. Cremer Jr.' From the TPs, it appears that they were written by Wiljalba Frikell (1818-1903) and then translated into English. However, BMC and NUC generally attribute the earlier US editions to George Arnold (1834©1865), and some catalogue entries explicitly say the Frikell versions are later editions, so it may be that Frikell produced later editions in some other language (French or German ??) and these were translated by Cremer. On the other hand, the UK ed of à ÃThe Secret OutÄ Ä says it is based on à ÃLe Magicien des SalonsÄ Ä. This is probably à ÃLe Magicien des Salons ou le Diable Couleur de RoseÄ Ä, for which I have several references, with different authors! ©© J. M. Gassier, 1814; M. [Louis Apollinarie Christien Emmanuel] Comte, 1829; Richard (pseud. of A. O. Delarue), 1857 and earlier. There was a German translation of this. Some of these are at HPL but ??NYS. Items with similar names are: à ÃLe Magicien de SociÀ)ÀtÀ)ÀÄ Ä, Delarue, Paris, c1860? and à ÃLe Manuel des SorciersÄ Ä (various Paris editions from 178?©1825, cf in Common References). It seems that this era was inspired by these earlier French books. To add to the confusion, an advertisement for the UK ed. of à ÃMagician's Own BookÄ Ä (1871?) says it is translated from à ÃLe Magicien des SalonsÄ Ä which has long been a standard in France and Germany. Toole Stott opines that Frikell had nothing to do with these books ©© as a celebrated conjuror of the times, his name was simply attached to the books. Toole Stott also doubts whether à ÃLe Magicien des SalonsÄ Ä exists ©© but it now seems pretty clear that it does, though it may not have been the direct source for any of these works, but see below. ÁÁChristopher 242 cites the following article on this series. ÁÁCharles L. Rulfs. Origins of some conjuring works. ÃÃMagicolÄÄ 24 (May 1971) 3©5. He discusses the various books, saying that Cremer essentially pirated the Dick & Fitzgerald productions. He says à ÃThe Magician's Own BookÄ Ä draws on à ÃWyman's HandbookÄ Ä (1850, ??NYS), à ÃEndless AmusementÄ Ä, à ÃParlour MagicÄ Ä (by W. Clarke?, 1830s, ??NYS), Brewster's à ÃNatural MagicÄ Ä (??NYS). He says à ÃThe Secret OutÄ Ä is largely taken, illustrations and all, from Blismon de Douai's à ÃManuel du MagicienÄ Ä (1849, ??NYS) and Richard & Delion's à ÃMagicien des salons ou le diable couleur de roseÄ Ä (1857 and earlier, ??NYS). ÁÁChristopher 622 says Harold Adrian Smith [Dick and Fitzgerald Publishers; ÃÃBooks at BrownÄÄ 34 (1987) 108©114] has studied this series and concludes that Williams was the author of à ÃMagician's Own BookÄ Ä, assisted by Wyman. Actually Smith simply asserts: "The book was undoubtedly [sic] written by H. L. Williams, a "hack writer" of the period, assisted by John Wyman in the technical details." He gives no explanation for his assertion. He later says he doubts whether Cremer ever wrote anything. He suggests à ÃThe Secret OutÄ Ä book is taken from DeLion. He states that à ÃThe Boy's Own Conjuring BookÄ Ä is a London pirate edition. ÁÁSeveral of the other items are anonymous and there was a tremendous amount of copying going on ©© problems are often reproduced verbatim with the same diagram or sometimes with minor changes. In some cases, the same error is repeated in five different books! I have just discovered some earlier appearances of the same material in ÃÃThe Family FriendÄÄ, a periodical which ran in six series from 1849 to 1921 and which I have not yet tracked down further. However, vol. 1©3 of 1849-1850 and the volume for Jul-Dec 1859 contain a number of the problems which appear repeatedly and identically in the above cited books. Toole Stott 407 is an edition of à ÃThe Illustrated Boy's Own TreasuryÄ Ä of c1847 but the BM copy was destroyed in the war and the other two copies cited are in the US. If this date is correct, then this book is a forerunner of all the others and a major connection between à ÃBoy's Own BookÄ Ä and à ÃMagician's Own BookÄ Ä. I would be most grateful to anyone who can help sort out this material ©© e.g. with photocopies of these or similar books or magazines. ÁÁThe other interesting era was about 1900. In English, this was largely created or inspired by Sam Loyd and Henry Dudeney. Much of this material first appeared in magazines and newspapers. I have seen much less than half of Loyd's and Dudeney's work and very little of similar earlier material (but see below). Consequently problems due to Loyd or Dudeney may seem to first appear in the works of Ball (1892, et seq.), Hoffmann (1893) and Pearson (1907). Further examination of Loyd's and Dudeney's material will be needed to clarify the origin and development of many problems. Though both started puzzle columns about 1896, they must have been producing material for a decade or more previously which does not seem to be known. I have just obtained photocopies of 401 columns by Dudeney in the ÃÃWeekly DispatchÄÄ of 1897©1903, but have not had time to study them. Will Shortz and Angela Newing have been studying Loyd and Dudeney respectively and turning up their material. ÁÁThe works of Lucas (1882-1895), Schubert (1890s) and Ahrens (1900-1918) were the main items on the Continent and they interacted with the English language writers. Ahrens was the most historical of these and his book is one of the foundations of the present work. All of these also wrote in newspapers and magazines and I have not seen all their material. ÁÁI would be happy to hear from anyone with ideas or suggestions for this bibliography. I would be delighted to hear from anyone who can locate missing information or who can provide copies of awkward material. I am particularly short of information about recreations in the Arabic period. I prepared a separate file, 'Queries and Problems in the History of Recreational Mathematics', which is about 23 pages, and has recently been updated. I have also prepared three smaller letters of queries about Middle Eastern, Oriental and Russian sources and these are generally more up©to©date. ÙÙÁÁÁÁà ÃTECHNICAL NOTESÄ Ä ÁÁI have prepared a CD containing this and much else of my material. I divided à ÃSourcesÄ Ä into four files when I used floppy discs as it was too big to fit on one disc, and I have not yet changed this. The files are: 1: Introductory material and list of abbreviations/references; 2: Sections 1 © 6; 3: Section 7; 4: Sections 8 © 11. It is convenient to have the first file separate from the main material, but I might combine the other three files. (I have tried to send it by email in the past, but this document is very large (currently c4.1MB and the Word version will be longer) and most people who requested it by email found that it overflowed their mailbox and created chaos in their system ©© this situation has changed a bit with larger memories and improved transmission speeds.) ÁÁThis file started on a DEC©10, then was transferred to a VAX. It is now on my PC using Script Professional, the development of LocoScript on the Amstrad. Even in its earliest forms, this provided an easy and comprehensive set of diacritical marks, which are still not all available nor easy to use in WordPerfect or Word (except perhaps by using macros and/or overstriking??). It also provides multiple cut and paste buffers and easy formatting, though I have learned how to overcome these deficiencies in Word. ÁÁScript provides an ASCII output, but this uses IBM extended ASCII which has 8©bit codes. Not all computers will accept or print such characters and sometimes they are converted into printer control codes causing considerable confusion. I have a program that converts these codes to 7©bits ©© e.g. accents and umlauts are removed. However, ASCII loses a great deal of the information, such as sub© and superscripts, so this is not a terribly useful format. ÁÁScript also provides WordStar and "Revisable©Form©Text DCA" output, but neither of these seems to be very successful (DCA is better than WordStar). Script later added a WordPerfect exporting facility. This works well, though some (fairly rare) characters and diacritical marks are lost and the output requires some reformatting. (Nob Yoshigahara reports that Japanese WordPerfect turns all the extended ASCII characters into Kanji characters!) ÁÁReading the WordPerfect output in Word (you may need to install this facility) gives a good approximation to my text, but in Courier 10pt. Selecting All and changing to Times New Roman 12pt gives an better approximation. (Some files use a smaller font of 10pt and I may have done some into 9pt.) You have to change this in the Header separately, using View Header and Footer. The page layout is awkward as my page numbering header gets put into the text, leaving a large gap at the top. I go into Page Setup and set the Paper Size to A4 and the Top, Bottom and Header Margins to 15mm and the Left and Right Margins to 25mm. (It has taken me some time to work this out and some earlier files may have other settings.) However, I find that lines are a bit too close together and underlines and some diacritical marks are lost, so one needs to also go into Format Paragraph Spacing ©© Line Spacing and choose At least and 12pt (or 10pt). I use hanging indentation in most of the main material and this feature is not preserved in this conversion. By selecting a relevant section and going into Format Paragraph Indentation ©© Special and selecting Hanging, it should automatically select 10.6mm which corresponds to my automatic spacing of five characters in 12pt. Further, I use second level hanging indentation in quite a number of places. You need to create a style which is the basic style with the left hand margin at 10.6mm (or 10 or 11 mm). When second level indenting is needed, select the desired section and apply this style to it. ÁÁHowever, this still leaves some problems. I use em dashes a bit, i.e. ÀMÀ, which gets converted into an underline, _. In Word, this is obtained by use of CTRL and the © sign on the numeric keypad. One can use the find and replace feature, EXCEPT that a number of other characters are also converted into underlines. In particular, Cyrillic characters are all converted into underlines. This is not insuperable as I always(?) give a transliteration of Cyrillic (using the current ÃÃMathematical ReviewsÄÄ system) and one can reconstruct the original Cyrillic from it. I notice that the Cyrillic characters are larger than roman characters and hence may overlap. One can amend this by selecting the Cyrillic text and going into Format Font Character Spacing Spacing and choosing Expanded By 2 pt (or thereabout). But a number of characters with unusual diacritical marks are also converted to underlines or converted to the unmarked character and not all of these are available in Word. E.g. i, which is the transliteration of À À becomes just i. I am slowly forming a Word file containing the Word versions of entries having the Cyrillic or other odd characters, and I will include this file on my CD, named CYRILLIC.DOC. For diacritical marks not supported by Word, I use an approximation and/or an explanation. ÁÁIt is very tedious to convert the underlines back to em dashes, so I will convert every em dash to a double hyphen ©©. ÁÁFinally, I have made a number of diagrams by simple typing without proportional spacing and Word does not permit changing font spacing in mid©line and ignores spaces before a right©alignment instruction. The latter problem can be overcome by using hard spaces and the former problem is less of a problem, and I think it can be overcome. ÁÁLater versions of Script support Hewlett©Packard DeskJets and I am now on my second generation of these, so the 7th and future editions will be better printed (if they ever are!). However, this required considerable reformatting as the text looks best in proportional spacing (PS) and I found I had to check every table and every mathematical formula and diagram. Also, to set off letters used as mathematical symbols within text, I find PS requires two spaces on each side of the letter ©© i.e. I refer to x rather than to x. (I find this easier to do than to convert to italics.) I also sometimes set off numbers with two spaces, though I wasn't consistent in doing this at the beginning of my reformatting. The conversion to proportional spacing reduced the total length from 587 to 488 pages, a reduction of 16.87% which is conveniently estimated as 1/6. The percentage of reduction was fairly consistent throughout the conversion process. ÁÁThe printing of Greek characters went amiss in the second part of the 6th Preliminary Edition, apparently due to the printer setting having been changed without my noticing ©© this happens if an odd character gets sent to the printer, usually in DOS when trying to use or print a corrupted file, and there is no indication of it. I was never able to reproduce the effect! ÁÁThe conversion to (Loco)Script provided many improved features compared to my earlier DEC versions. I am using an A4 page (8ÀÀ by 11ÀAÀ inches) rather than an 8ÀÀ  by 11 inch page, which gives 60 lines of text per page, four more or 7% more than when using the DEC or VAX. ÁÁÁÁà ÃNEW SECTIONS IN THIS EDITION.Ä Ä [SIXTH EDITION: 1: Fibonacci, 1: Montucla; 3.B; 4.A.1.a, 4.B.9, 4.B.10, 4.B.11, 4.B.12; 5.R.1.a, 5.W.1, 5.AA, 5.AB; 6.AS.1.b, 6.AS.2.a, 6.AS.5, 6.AW.4, 6.BP, 6.BQ, 6.BR; 7.I.1, 7.Y.2, 7.AY, 7.AZ; 7.BA; 8.I, 8.J; 9.E.2, 9.K; 10.A.4, 10.A.5, 10.U, 10.V, 10.W; 11.K.6, 11.K.7, 11.K.8.] In the last edition, I had 8.K instead of 8.J in the list of New Sections and in the Contents. 1: Pacioli, Carroll, Perelman; 4.B.13, 4.B.14, 4.B.15; 5.B.2, 5.H.3 (the previous 5.H.3 has been renumbered 5.H.4), 5.K.3, 5.R.1.b, 5.X.4, 5.AC, 5.AD, 5.AE, 5.AF, 5.AG.1, 5.AG.2; 6.AJ.4, 6.AJ.5, 6.AS.3.a, 6.AT.8, 6.AT.9, 6.AY.2, 6.BF.4, 6.BF.5, 6.BS, 6.BT, 6.BU, 6.BV, 6.BW; 7.H.6, 7.H.7 (formerly part of 7.H.5), 7.M.4.a, 7.M.4.b, 7.M.6, 7.R.4, 7.AC.3.a, 7.AC.7, 7.AH.1, 7.AJ.1, 7.BB, 7.BC; 8.K, 8.L; 10.A.6, 10.A.7, 10.A.8, 10.D has become 10.D.1, 10.D.2, 10.D.3, 10.E.4, 10.X, 10.Y, 10.Z, 10.AA, 10.AB, 10.AC, 10.AD, 10.AE; 11.N, 11.O, 11.P, 11.Q, 11.R, 11.S. (65 new sections) ÙÙ ÁÁÁÁà ÃACKNOWLEDGMENTSÄ Ä ÁÁI am immensely indebted to many mathematicians, historians, puzzlers, bookdealers and others who have studied particular topics, as will be apparent. ÁÁI have had assistance from so many sources that I have probably forgotten some, but I would like to give thanks here to the following, and beg forgiveness from anyone inadvertently omitted ©© if you remind me, I will make amendment. In some cases, I simply haven't got to your letter yet! Also I have had letters from people whose only identification is an undecipherable signature and phone messages from people whose name and phone number are unintelligible. ÁÁSadly, a few of these have died since I corresponded with them and I have indicated those known to me with À'À. AndrÀ)À Allard, Eric J. AitonÀ'À, Sue Andrew, Hugh Ap Simon, Gino Arrighi, Marcia Ascher, Mohammad Bagheri, Banca Commerciale Italiana, Gerd Baron, Chris Base, Rainier [Ray] Bathke, John Beasley, Michael Behrend, JÀ?Àrg Bewersdorff, Norman L. Biggs, C. [Chris] J. Bouwkamp, Jean Brette, John Brillhart, Paul J. Campbell, Cassa di Risparmio di Firenze, Henry Cattan, Marianna Clark, Stewart Coffin, Alan & Philippa Collins, John H. Conway, H. S. M. Coxeter, James Dalgety, Ann E. L. Davis, Yvonne Dold, Underwood Dudley, Anthony W. F. Edwards, John Ergatoudis, John FauvelÀ'À, Sandro Ferace, Judith V. Field, Irving Finkel, Graham Flegg, Menso Folkerts, David Fowler, Aviezri S. Fraenkel, Raffaella Franci, Gregory N. Frederickson, Michael Freude, Walter W. Funkenbusch, Nora GÀÀdeke, Martin Gardner, Marcel Gillen, Leonard J. Gordon, Ron Gow, Ivor Grattan-Guinness, Christine Insley Green, Jennifer Greenleaves (Manco), Tom Greeves, H. [Rik] J. M. van Grol, Branko GrÀGÀnbaum, Richard K. Guy, John Hadley, Peter Hajek, Diana Hall, Joan Hammontree, Anton HanegraafÀ'À, Martin Hansen, Jacques Haubrich, Cynthia Hay, Takao Hayashi, Robert L. Helmbold, Hanno Hentrich, Richard I. Hess, Christopher Holtom, Edward HordernÀ'À, Peter Hosek, Konrad Jacobs, Anatoli Kalinin, Bill Kalush, Michael Keller, Edward S. Kennedy, Sarah Key (The Haunted Bookshop), Eberhard Knobloch, Don Knuth, Bob Koeppel, Joseph D. E. Konhauser, David E. Kullman, Mogens Esrom Larsen, Jim Lavis (Doxa (Oxford)), John  LeechÀ'À, Elisabeth Lefevre, C. Legel, Derrick [Dick] H. LehmerÀ'À, Emma Lehmer, Leisure Dynamics, Hendrik W. Lenstra, Alan L. Mackay, Andrzej Makowski, John Malkevitch, Giovanni Manco, Tatiana Matveeva, Ann Maury, Max Maven, Jim McArdle, Patricia McCulloch, Peter McMullen, Leroy F. MeyersÀ'À, D. P. Miles, Marvin Miller, Nobuo Miura, William O. J. Moser, Barbara Moss, Angela Newing, Jennie Newman, Tom and Greta O'BeirneÀ'ÀÀ'À, Owen O'Shea, Parker Brothers, Alan Parr, Jean J. Pedersen, Luigi Pepe, William Poundstone, Helen Powlesland, Oliver Pretzel, Walter Purkert, Robert A. RankinÀ'À, Eleanor Robson, David J. A. Ross, Lee Sallows, Christopher Sansbury, Sol Saul, William L. Schaaf, Doris Schattschneider, Jaap Scherphuis, Heribert Schmitz, À°À. Schwabik, Eileen ScottÀ'À, Al Seckel, Jacques Sesiano, Claude E. ShannonÀ'À, John Sheehan, A. Sherratt, Will Shortz, Kripa Shankar Shukla, George L. Sicherman, Deborah Singmaster, Man-Kit Siu, Gerald [Jerry] K. Slocum, Cedric A. B. SmithÀ'À (and Sue Povey & Jim Mallet at the Galton Laboratory for letting me have some of Cedric's books), Jurgen Stigter, Arthur H. Stone, Mel StoverÀ'À, Michael Stueben, Shigeo TakagiÀ'À, Michael Tanoff, Gary J. Tee, Andrew Topsfield, George TysonÀ'À, Dario Uri, Warren Van Egmond, Carlo Viola, Kurt VogelÀ'À, Anthony Watkinson, Chris Weeks, Maurice Wilkes, John Winterbottom, John Withers, Nob. Yoshigahara, Claudia Zaslavsky. ÁÁI would also like to thank the following libraries and museums which I have used: University of Aberdeen; University of Bristol; Buckleys Shop Museum, Battle, East Sussex; University of Calgary; University of Cambridge; Marsh's Library, Dublin; FLORENCE: ÂÂBiblioteca Nazionale; Biblioteca Riccardiana; University of Keele ©© The Turner Collection(À'À) and its librarian Martin Phillips; Karl-Marx-UniversitÀÀt, Leipzig: UniversitÀÀt Bibliothek and Sektion Mathematik Bibliothek, ÁÁespecially Frau Letzel at the latter; LONDON: ÂÂBirkbeck College; British Library (at Bloomsbury and then at St. Pancras; also at Colindale); The London Library; School of Oriental and African Studies, especially Miss Y. Yasumara, the Art Librarian; Senate House, particularly the Harry Price Library; South Bank University; Southwark Public Library; University College London, especially the Graves Collection and the Rare Book Librarians Jill Furlong, Susan Stead and their staff; Warburg Institute; MUNICH: ÂÂDeutsches Museum; Institut fÀGÀr Geschichte der Naturwissenschaften; NEW YORK: ÂÂBrooklyn Public Library; City College of New York; Columbia University; Newark Public Library, Newark, New Jersey; University of Newcastle upon Tyne ©© The Wallis Collection and its librarian Lesley Gordon; OXFORD: ÂÂAshmolean Museum; The Bodleian Library; Museum of the History of Science, and its librarian Tony Simcock; University of Reading; University of St. Andrews; SIENA: ÂÂBiblioteca Comunale degli Intronati; Dipartimento di Matematica, UniversitÀ!À di Siena; University of Southampton; Mathematical Institute, Warsaw. ÁÁI would like especially to thank the following. ÁÁInterlibrary Loans (especially Brenda Spooner) at South Bank University and the British Library Lending Division for obtaining many strange items for me. ÁÁRichard Guy, Bill Sands and the Strens bequest for a most useful week at the Strens/Guy Collection at Calgary in early 1986 and for organizing the Strens Memorial Meeting in summer 1986 and for printing the first preliminary edition of these à ÃSourcesÄ Ä. ÁÁGerd Lassner, Uwe Quasthoff and the Naturwissenschaftlich-Theoretisches Zentrum of the Karl-Marx-UniversitÀÀt, for a very useful visit to Leipzig in 1988. ÁÁSouth Bank University Computer Centre for the computer resources for the early stages of this project, and especially Ann Keen for finding this file when it was lost. ÁÁMy School for printing these preliminary editions. ÁÁMartin Gardner for kindly allowing me to excavate through his library and files. ÁÁJames Dalgety, Edward Hordern, Bill Kalush, Chris Lewin, Tom Rodgers and Will Shortz for allowing me to rummage through their libraries. ÁÁJohn Beasley, Edward Hordern, Bill Kalush, Will Shortz and Jerry Slocum for numerous photocopies and copies from their collections. ÁÁMenso Folkerts, Richard Lorch, Michael Segre and the Institut fÀGÀr Geschichte der Naturwissenschaft, Munich, for a most useful visit in Sep 1994 and for producing a copy of Catel. ÁÁRaffaella Franci and the Dipartimento di Matematica and the Centro Studi della Matematica Medioevale at UniversitÀ!À di Siena for a most useful visit in Sep 1994. ÁÁTakao Hayashi for much material from Japan and India. ÁÁMy wife for organizing a joint trip to Newcastle in Sep 1997 where I made use of the Wallis Collection. ÁÁFinally, I would like to thank a large number of publishers, distributors, bookdealers and even authors who have provided copies of the books and documents upon which much of this work is based. Bookdealers have often let me examine books in their shops. Their help is greatly appreciated. There are too many of these to record here, but I would like to mention Fred Whitehart (À'À1999), England's leading dealer in secondhand scientific books for many years who had a real interest in mathematics. ÁÁÁÁÁÁà ÃCONTENTSÄ Ä Ã ÃINTRODUCTIONÄ Ä 1 ÁÁNature of This Work 1 ÁÁSimilar Works 2 ÁÁCoverage 3 ÁÁStatus of the Project 4 ÁÁTechnical Notes 6 ÁÁNew Sections in This Edition 7 ÁÁAcknowledgements 7 à ÃCONTENTSÄ Ä 10 à ÃABBREVIATIONSÄ Ä 20 ÁÁDiacritical Marks and Notation 20 ÁÁAbbreviations of Journals and Series 21 ÁÁAbbreviations of Publishers 21 ÁÁAbbreviations of Months 21 ÁÁPublishers' Locations 21 à ÃCOMMON REFERENCESÄ Ä 22 à ÃSOME OTHER RECURRING REFERENCESÄ Ä 79 Ðаlð ä ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐà Ã1.ÁÁBIOGRAPHICAL MATERIALÄ Ä in Chronological Order 82 Alcuin, Fibonacci, Pacioli, Bachet, Leurechon/van Etten, Ozanam, Montucla, Carroll, Hoffmann, Loyd & Loyd Jr, Lucas, Schubert, Ball, Dudeney, Ahrens, Perelman, Phillips. à Ã2.ÁÁGENERAL PUZZLE COLLECTIONS AND SURVEYSÄ Ä 90 à Ã3.ÁÁGENERAL HISTORICAL AND BIBLIOGRAPHICAL MATERIALÄ Ä 91 3.A.ÁÁGeneral Historical Material 91 3.B.ÁÁBibliographical Material 91 à Ã4.ÁÁMATHEMATICAL GAMESÄ Ä 97 4.A.ÁÁGeneral Theory and Nim-like Games 97 ÁÁ4.A.1.ÁÁOne Pile Game 97 ÁÁ4.A.1.a.ÁÁÁÁThe 31 Game 100 ÁÁ4.A.2.ÁÁSymmetry Arguments 102 ÁÁ4.A.3.ÁÁKayles 102 ÁÁ4.A.4.ÁÁNim 103 ÁÁ4.A.5.ÁÁGeneral Theory 105 4.B.ÁÁParticular Games 106 ÁÁ4.B.1.ÁÁTic-Tac-Toe = Noughts and Crosses 106 ÁÁ4.B.1.a.ÁÁÁÁIn Higher Dimensions 114 ÁÁ4.B.2.ÁÁHex 115 ÁÁ4.B.3.ÁÁDots and Boxes 116 ÁÁ4.B.4.ÁÁSprouts 117 ÁÁ4.B.5.ÁÁOvid's Game and Nine Men's Morris 118 ÁÁ4.B.6.ÁÁPhutball 124 ÁÁ4.B.7.ÁÁBridg-It 124 ÁÁ4.B.8.ÁÁChomp 124 ÁÁ4.B.9.ÁÁSnakes and Ladders 125 ÁÁ4.B.10.ÁÁMu Torere 127 ÁÁ4.B.11.ÁÁMastermind, etc. 127 ÁÁ4.B.12.ÁÁRithmomachia = The Philosophers' Game 128 ÁÁ4.B.13.ÁÁMancala Games 129 ÁÁ4.B.14.ÁÁDominoes, etc. 130 ÁÁ4.B.15.ÁÁSvoyi Kosiri 130 à Ã5.ÁÁCOMBINATORIAL RECREATIONSÄ Ä 131 5.A.ÁÁThe 15 Puzzle, etc. 131 ÁÁ General 131 ÁÁ Early Alphabetic Versions 131 ÁÁ Loyd 132 ÁÁ The 15 Puzzle 132 ÁÁ5.A.1.ÁÁNon-square Pieces 139 ÁÁ5.A.2.ÁÁThree Dimensional Versions 140 ÁÁ5.A.3.ÁÁRolling Piece Puzzles 141 ÁÁ5.A.4.ÁÁPanex Puzzle 142 5.B.ÁÁCrossing Problems 142 ÁÁ5.B.1.ÁÁLowering from Tower Problem 151 ÁÁ5.B.2.ÁÁCrossing a Bridge with a Torch 152 5.C.ÁÁFalse Coins with a Balance 152 ÁÁ5.C.1.ÁÁRanking Coins with a Balance 155 5.D.ÁÁMeasuring Problems 156 ÁÁ5.D.1.ÁÁJugs & Bottles 156 ÁÁ5.D.2.ÁÁRuler with Minimal Number of Marks 161 ÁÁ5.D.3.ÁÁFalse Coins with a Weighing Scale 162 ÁÁ5.D.4.ÁÁTiming with Hourglasses 162 ÁÁ5.D.5.ÁÁMeasure Half a Barrel 162 5.E.ÁÁEuler Circuits and Mazes 163 ÁÁ5.E.1.ÁÁMazes 167 ÁÁ5.E.2.ÁÁMemory Wheels = Chain Codes 172 ÁÁ5.E.2.aÁÁÁÁPantactic Squares 173 5.F.ÁÁHamiltonian Circuits 174 ÁÁ5.F.1.ÁÁKnight's Tours and Paths 174 ÁÁ5.F.2.ÁÁOther Hamiltonian Circuits 181 ÁÁ5.F.3.ÁÁKnight's Tours in Higher Dimensions 182 ÁÁ5.F.4.ÁÁOther Circuits In and On a Cube 183 5.G.ÁÁConnection Problems 183 ÁÁ5.G.1.ÁÁGas, Water and Electricity 183 5.H.ÁÁColoured Squares and Cubes, etc. 184 ÁÁ5.H.1.ÁÁInstant Insanity = The Tantalizer 184 ÁÁ5.H.2.ÁÁMacMahon Pieces 185 ÁÁ5.H.3.ÁÁPath Forming Puzzles 187 ÁÁ5.H.4.ÁÁOther and General 187 5.I.ÁÁLatin Squares and Euler Squares 190 ÁÁ5.I.1.ÁÁEight Queens Problem 192 ÁÁ5.I.2.ÁÁColouring Chessboard with No Repeats in a Line 195 5.J.ÁÁSquared Squares, etc. 195 ÁÁ5.J.1.ÁÁMrs Perkins's Quilt 197 ÁÁ5.J.2.ÁÁCubing the Cube 198 ÁÁ5.J.3.ÁÁTiling a Square of Side 70 with Squares of Sides 1, 2, ..., 24 198 5.K.ÁÁDerangements 198 ÁÁ5.K.1ÁÁDeranged Boxes of A, B and A & B 199 ÁÁ5.K.2ÁÁOther Logic Puzzles Based on Derangements 199 ÁÁ5.K.3ÁÁCayley's Mousetrap 200 5.L.ÁÁMÀ)Ànage Problem 200 5.M.ÁÁSix People at a Party ©© Ramsey Theory 201 5.N.ÁÁJeep or Explorer's Problem 201 5.O.ÁÁTait's Counter Puzzle: BBBBWWWW to WBWBWBWB 204 5.P.ÁÁGeneral Moving Piece Puzzles 207 ÁÁ5.P.1.ÁÁShunting Puzzles 207 ÁÁ5.P.2.ÁÁTaquin 209 5.Q.ÁÁNumber of Regions Determined by N Lines or Planes 209 ÁÁ5.Q.1.ÁÁNumber of Intersections Determined by N Lines 210 5.R.ÁÁJumping Piece Games 210 ÁÁ5.R.1.ÁÁPeg Solitaire 210 ÁÁ5.R.1.a.ÁÁÁÁTriangular Version 213 ÁÁ5.R.1.b.ÁÁÁÁOther shapes 214 ÁÁ5.R.2.ÁÁFrogs and Toads: BBB_WWW to WWW_BBB 215 ÁÁ5.R.3.ÁÁFore and Aft ©© 3 by 3 Squares Meeting at a Corner 217 ÁÁ5.R.4.ÁÁReversing Frogs and Toads: _12...n to _n...21 218 ÁÁ5.R.5.ÁÁFox and Geese, etc. 219 ÁÁ5.R.6.ÁÁOctagram Puzzle 222 ÁÁ5.R.7.ÁÁPassing Over Counters 223 5.S.ÁÁChain Cutting and Rejoining 226 ÁÁ5.S.1.ÁÁUsing Chain Links to Pay for a Room 227 5.T.ÁÁDividing a Cake Fairly 227 5.U.ÁÁPigeonhole Recreations 228 5.V.ÁÁThink-A-Dot, etc. 229 5.WÁÁMaking Three Pieces of Toast 230 ÁÁ5.W.1.ÁÁBoiling Eggs 230 5.XÁÁCounting Figures in a Pattern 231 ÁÁ5.X.1.ÁÁCounting Triangles 231 ÁÁ5.X.2.ÁÁCounting Rectangles or Squares 233 ÁÁ5.X.3.ÁÁCounting Hexagons 235 ÁÁ5.X.4.ÁÁCounting Circles 235 5.Y.ÁÁNumber of Routes in a Lattice 235 5.Z.ÁÁChessboard Placing Problems 238 ÁÁ5.Z.1.ÁÁKings 239 ÁÁ5.Z.2.ÁÁQueens 239 ÁÁ5.Z.3.ÁÁBishops 240 ÁÁ5.Z.4.ÁÁKnights 241 ÁÁ5.Z.5.ÁÁRooks 241 ÁÁ5.Z.6.ÁÁMixtures 241 5.AA.ÁÁCard Shuffling 242 5.AB.ÁÁFolding a Strip of Stamps 244 5.AC.ÁÁProperties of the Seven Bar Digital Display 244 5.AD.ÁÁStacking a Deck to Produce a Special Effect 245 5.AE.ÁÁReversing Cups 245 5.AF.ÁÁSpotting Dice 245 5.AG.ÁÁRubik's Cube and Similar Puzzles 246 ÁÁ5.AG.1.ÁÁRubik's Cube 246 ÁÁ5.AG.2.ÁÁHungarian Rings, etc. 246 à Ã6.ÁÁGEOMETRIC RECREATIONSÄ Ä 248 6.A.ÁÁPi 248 6.B.ÁÁStraight Line Linkages 249 6.C.ÁÁCurves of Constant Width 250 6.D.ÁÁFlexagons 251 6.E.ÁÁFlexatube 253 6.F.ÁÁPolyominoes, etc. 253 ÁÁ6.F.1.ÁÁOther Chessboard Dissections 262 ÁÁ6.F.2.ÁÁCovering Deleted Chessboard with Dominoes 264 ÁÁ6.F.3.ÁÁDissecting a Cross into Zs and Ls 264 ÁÁ6.F.4.ÁÁQuadrisecting an L-Tromino, etc. 266 ÁÁ6.F.5.ÁÁOther Dissections into Polyominoes 268 6.G.ÁÁSoma Cube 269 ÁÁ6.G.1.ÁÁOther Cube Dissections 270 ÁÁ6.G.2.ÁÁDissection of 6ÃÃ3ÄÄ into 3ÃÃ3ÄÄ, 4ÃÃ3ÄÄ and 5ÃÃ3ÄÄ, etc. 271 ÁÁ6.G.3.ÁÁDissection of a Die into Nine 1 x 1 x 3 271 ÁÁ6.G.4.ÁÁUse of Other Polyhedral Pieces 272 6.H.ÁÁPick's Theorem 272 6.I.ÁÁSylvester's Problem of Collinear Points 273 6.J.ÁÁFour Bugs and Other Pursuit Problems 273 6.K. ÁÁDudeney's Square to Triangle Dissection 275 6.L.ÁÁCrossed Ladders 275 ÁÁ6.L.1.ÁÁLadder Over Box 277 6.M.ÁÁSpider & Fly Problems 278 6.N.ÁÁDissection of a 1 x 1 x 2 Block to a Cube 279 6.O.ÁÁPassing a Cube Through an Equal or Smaller Cube ©© Prince Rupert's Problem 279 6.P.ÁÁGeometrical Vanishing 280 ÁÁ6.P.1.ÁÁParadoxical Dissections of the Chessboard Based on ÁÁÁÁÁÁFibonacci Numbers 280 ÁÁ6.P.2.ÁÁOther Types 282 6.Q.ÁÁKnotting a Strip to Make a Regular Pentagon 285 6.R.ÁÁGeometric Fallacies 285 ÁÁ6.R.1.ÁÁEvery Triangle is Isosceles 286 ÁÁ6.R.2.ÁÁA Right Angle is Obtuse 287 ÁÁ6.R.3.ÁÁLines Approaching but not Meeting 287 ÁÁ6.R.4.ÁÁOthers 287 6.S.ÁÁTangrams, et al. 287 ÁÁGeneral Histories 287 ÁÁSpecific Items 289 ÁÁ6.S.1.ÁÁLoculus of Archimedes 299 ÁÁ6.S.2.ÁÁOther Sets of Pieces 300 6.T. ÁÁNo Three in a Line Problem 301 6.U.ÁÁTiling 302 ÁÁ6.U.1.ÁÁPenrose Pieces 302 ÁÁ6.U.2.ÁÁPacking Bricks in Boxes 302 6.V.ÁÁSilhouette and Viewing Puzzles 303 6.W.ÁÁBurr Puzzles 306 ÁÁ6.W.1.ÁÁThree Piece Burr 306 ÁÁ6.W.2.ÁÁSix Piece Burr = Chinese Cross 307 ÁÁ6.W.3.ÁÁThree Piece Burr with Identical Pieces 309 ÁÁ6.W.4.ÁÁDiagonal Six Piece Burr = Trick Star 309 ÁÁ6.W.5.ÁÁSix Piece Burr with Identical Pieces 310 ÁÁ6.W.6.ÁÁAltekruse Puzzle 310 ÁÁ6.W.7.ÁÁOther Burrs 310 6.X.ÁÁRotating Rings of Polyhedra 311 6.Y.ÁÁRope Round the Earth 312 6.Z.ÁÁLangley's Adventitious Angles 314 6.AA.ÁÁNets of Polyhedra 314 6.AB.ÁÁSelf-Rising Polyhedra 316 6.AC.ÁÁConway's Life 316 6.AD.ÁÁIsoperimetric Problems 316 ÁÁ6.AD.1.ÁÁLargest Parcel One Can Post 318 6.AE.ÁÁ6" Hole Through Sphere Leaves Constant Volume 319 6.AF.ÁÁWhat Colour Was The Bear? 319 6.AG.ÁÁMoving Around a Corner 322 6.AH.ÁÁTethered Goat 323 6.AI.ÁÁTrick Joints 324 6.AJ.ÁÁGeometric Illusions 325 ÁÁ6.AJ.1.ÁÁTwo Pronged Trident 328 ÁÁ6.AJ.2.ÁÁTribar and Impossible Staircase 329 ÁÁ6.AJ.3.ÁÁCafÀ)À Wall Illusion 330 ÁÁ6.AJ.4.ÁÁStereograms 331 ÁÁ6.AJ.5.ÁÁImpossible Crate 331 6.AK.ÁÁPolygonal Path Covering N x N Lattice of Points, Queen's Tours, etc. 331 6.AL.ÁÁSteiner-Lehmus Theorem 334 6.AM.ÁÁMorley's Theorem 335 6.AN.ÁÁVolume of the Intersection of Two Cylinders 335 6.AO.ÁÁConfiguration Problems 336 ÁÁ6.AO.1.ÁÁPlace Four Points Equidistantly = Make Four Triangles with Six ÁÁÁÁÁÁMatchsticks 343 ÁÁ6.AO.2.ÁÁPlace an Even Number in Each Line 345 6.AP.ÁÁDissections of a Tetrahedron 346 ÁÁ6.AP.1.ÁÁTwo Pieces 346 ÁÁ6.AP.2.ÁÁFour Pieces 346 6.AQ.ÁÁDissections of a Cross, T or H 347 6.AR.ÁÁQuadrisected Square Puzzle 348 6.AS.ÁÁDissection of Squares into a Square 349 ÁÁ6.AS.1.ÁÁTwenty 1, 2, ÀÀ5 Triangles Make a Square, i.e. Five Equal Squares to a ÁÁÁÁÁÁSquare 349 ÁÁ6.AS.1.a.ÁÁÁÁGreek Cross to a Square 352 ÁÁ6.AS.1.b.ÁÁÁÁOther Greek Cross Dissections 353 ÁÁ6.AS.2.ÁÁTwo (Adjacent) Squares to a Square 353 ÁÁ6.AS.2.a.ÁÁÁÁTwo Equal Squares to a Square 356 ÁÁ6.AS.3.ÁÁThree Equal Squares to a Square 356 ÁÁ6.AS.3.a.ÁÁÁÁThree Equal 'Squares' to a Hexagon 356 ÁÁ6.AS.4.ÁÁEight Equal Squares to a Square 357 ÁÁ6.AS.5.ÁÁRectangle to a Square or Other Rectangle 357 6.AT.ÁÁPolyhedra and Tessellations 358 ÁÁ6.AT.1.ÁÁRegular Polyhedra 358 ÁÁ6.AT.2.ÁÁStar and Stellated Polyhedra 362 ÁÁ6.AT.3.ÁÁArchimedean Polyhedra 364 ÁÁ6.AT.4.ÁÁUniform Polyhedra 369 ÁÁ6.AT.5.ÁÁRegular-Faced Polyhedra 369 ÁÁ6.AT.6.ÁÁTessellations 369 ÁÁ6.AT.6.a.ÁÁÁÁTessellating with Congruent Figures 369 ÁÁ6.AT.7.ÁÁPlaiting of Polyhedra 371 ÁÁ6.AT.8.ÁÁDÀGÀrer's Octahedron 371 ÁÁ6.AT.9.ÁÁOther Polyhedra 371 6.AU.ÁÁThree Rabbits, Dead Dogs and Trick Ponies 372 ÁÁChina 373 ÁÁOther Asia 375 ÁÁPaderborn 378 ÁÁMedieval Europe 380 ÁÁModern Versions of the Three Rabbits Puzzle 388 ÁÁDead Dogs 389 ÁÁTrick Mules 394 6.AV.ÁÁCutting Up in Fewest Cuts 394 6.AW.ÁÁDivision into Congruent Pieces 394 ÁÁ6.AW.1.ÁÁMitre Puzzle 394 ÁÁ6.AW.2.ÁÁRep-Tiles 395 ÁÁ6.AW.3.ÁÁDividing a Square into Congruent Parts 396 ÁÁ6.AW.4.ÁÁDividing an L©Tromino into Congruent Parts 397 6.AX.ÁÁThe Packer's Secret 397 6.AY.ÁÁDissect 3A x 2B to Make 2A x 3B, etc. 398 ÁÁ6.AY.1.ÁÁO'Beirne's Steps 400 ÁÁ6.AY.2.ÁÁSwiss Flag Puzzle 400 6.AZ.ÁÁBall Pyramid Puzzles 401 6.BA.ÁÁCutting a Card so One Can Pass Through It 402 6.BB.ÁÁDoubling an Area Without Changing Its Height or Width 402 6.BC.ÁÁHoffmann's Cube 403 6.BD.ÁÁBridge a Moat with Planks 403 6.BE.ÁÁReverse a Triangular Array of Ten Circles 405 6.BF.ÁÁPythagorean Recreations 405 ÁÁ6.BF.1.ÁÁThe Broken Bamboo 406 ÁÁ6.BF.2.ÁÁSliding Spear = Leaning Reed 407 ÁÁ6.BF.3.ÁÁWell Between Two Towers 408 ÁÁ6.BF.4.ÁÁRail Buckling 3411 ÁÁ6.BF.5.ÁÁTravelling on Sides of a Right Triangle 412 6.BG.ÁÁQuadrisect a Paper Square with One Cut 412 6.BH.ÁÁMoirÀ)À Patterns 412 6.BI.ÁÁVenn Diagrams for n Sets 413 6.BJ.ÁÁ3D Dissection Puzzles 415 6.BK.ÁÁSuperellipse 415 6.BL.ÁÁTanÃé1ÄÄ À@À + TanÃé1ÄÄ ÀÀ = TanÃé1ÄÄ 1, etc. 416 6.BM.ÁÁDissect Circle into Two Hollow Ovals 417 6.BN.ÁÁRound Peg in Square Hole or Vice Versa 417 6.BO.ÁÁButterfly Problem 418 6.BP.ÁÁEarly Matchstick Puzzles 418 6.BQ.ÁÁCovering a Disc with Discs 419 6.BR.ÁÁWhat is a General Triangle? 420 6.BS.ÁÁForm Six Coins into a Hexagon 420 6.BT.ÁÁPlacing Objects in Contact 421 6.BU.ÁÁConstruction of n©gons 421 6.BV.ÁÁGeometric Constructions 423 6.BW.ÁÁDistances to Corners of a Square 424 ÙÙà Ã7.ÁÁARITHMETIC & NUMBER-THEORETIC RECREATIONSÄ Ä 426 7.A.ÁÁFibonacci Numbers 426 7.B.ÁÁJosephus or Survivor Problem 429 7.C.ÁÁEgyptian Fractions 443 7.D.ÁÁThe First Digit Problem 443 7.E.ÁÁMonkey and Coconuts Problems 444 ÁÁ7.E.1.ÁÁVersions with All Getting the Same 457 7.F.ÁÁIllegal Operations Giving Correct Result 458 7.G.ÁÁInheritance Problems 459 ÁÁ7.G.1.ÁÁHalf + Third + Ninth, etc. 459 ÁÁ7.G.2.ÁÁPosthumous Twins, etc. 464 7.H.ÁÁDivision and Sharing Problems ©© Cistern Problems 467 ÁÁ7.H.1.ÁÁWith Growth ©© Newton's Cattle Problem 484 ÁÁ7.H.2.ÁÁDivision of Casks 486 ÁÁ7.H.3.ÁÁSharing Unequal Resources ©© Problem of the Pandects 487 ÁÁ7.H.4.ÁÁEach Doubles Other's Money to Make All Equal, etc. 490 ÁÁ7.H.5.ÁÁSharing Cost of Stairs, etc. 492 ÁÁ7.H.6.ÁÁSharing a Grindstone 494 ÁÁ7.H.7.ÁÁDigging Part of a Well 494 7.I.ÁÁFour Fours, etc. 496 ÁÁ7.I.1.ÁÁLargest Number Using Four Ones, etc. 503 7.J.ÁÁSalary Puzzle 504 7.K.ÁÁCongruences 505 ÁÁ7.K.1.ÁÁCasting Out Nines 506 7.L.ÁÁGeometric Progressions 508 ÁÁ7.L.1.ÁÁ1 + 7 + 49 + ... & St. Ives 510 ÁÁ7.L.2.ÁÁ1 + 2 + 4 + .... 513 ÁÁ7.L.2.a.ÁÁÁÁChessboard Problem 513 ÁÁ7.L.2.b.ÁÁÁÁHorseshoe Nails Problem 517 ÁÁ7.L.2.c.ÁÁÁÁUse of 1, 2, 4, ... as Weights, etc. 518 ÁÁ7.L.3.ÁÁ1 + 3 + 9 + ... and Other Systems of Weights 519 7.M.ÁÁBinary System and Binary Recreations 521 ÁÁ7.M.1.ÁÁChinese Rings 523 ÁÁ7.M.2.ÁÁTower of Hanoi 527 ÁÁ7.M.2.a.ÁÁÁÁTower of Hanoi with More Pegs 530 ÁÁ7.M.3.ÁÁGray Code 531 ÁÁ7.M.4.ÁÁBinary Divination 531 ÁÁ7.M.4.a.ÁÁÁÁTernary Divination 533 ÁÁ7.M.4.b.ÁÁÁÁOther Divinations Using Binary or Ternary 533 ÁÁ7.M.5.ÁÁLoony Loop = Gordian Knot 537 ÁÁ7.M.6.ÁÁBinary Button Games 537 7.N.ÁÁMagic Squares 540 ÁÁSurveys 540 ÁÁPossible Early References 541 ÁÁ7.N.1.ÁÁMagic Cubes 554 ÁÁ7.N.2.ÁÁMagic Triangles 556 ÁÁ7.N.3.ÁÁAnti-Magic Squares and Triangles 557 ÁÁ7.N.4.ÁÁMagic Knight's Tour 558 ÁÁ7.N.5.ÁÁOther Magic Shapes 559 7.O.ÁÁMagic Hexagon 562 ÁÁ7.O.1ÁÁOther Magic Hexagons 563 7.P.ÁÁDiophantine Recreations 565 ÁÁ7.P.1.ÁÁHundred Fowls and Other Linear Problems 565 ÁÁ7.P.2.ÁÁChinese Remainder Theorem 582 ÁÁ7.P.3.ÁÁArchimedes' Cattle Problem 588 ÁÁ7.P.4.ÁÁPresent of Gems 589 ÁÁ7.P.5.ÁÁSelling Different Amounts 'At Same Prices' Yielding the Same 589 ÁÁ7.P.6.ÁÁConjunction of Planets, etc. 594 ÁÁ7.P.7.ÁÁRobbing and Restoring 595 7.Q.ÁÁBlind Abbess and her Nuns ©© Rearrangement Along Sides of a 3 x 3 Square ÁÁÁÁConserving Side Totals 597 ÁÁ7.Q.1.ÁÁRearrangement on a Cross 600 ÁÁ7.Q.2.ÁÁRearrange a Cross of Six to Make Two Lines of Four 601 7.R.ÁÁ"If I Had One From You, I'd Have Twice You" 602 ÁÁ7.R.1.ÁÁMen Find a Purse and 'Bloom of Thymaridas' 608 ÁÁ7.R.2.ÁÁ"If I Had 1/3 of Your Money, I Could Buy the Horse" 614 ÁÁ7.R.3.ÁÁSisters and Brothers 623 ÁÁ7.R.4.ÁÁ"If I Sold Your Eggs at my Price, I'd Get ...." 623 7.S.ÁÁDilution and Mixing Problems 624 ÁÁ7.S.1.ÁÁDishonest Butler Drinking Some and Replacing with Water 624 ÁÁ7.S.2.ÁÁWater in Wine Versus Wine in Water 625 7.T.ÁÁFour Number Game 626 7.U.ÁÁPostage Stamp Problem 627 7.V.ÁÁxÃÃyÄÄ = yÃÃxÄÄ and Iterated Exponentials 627 7.W.ÁÁCard Piling over a Cliff 628 7.X.ÁÁHow Old is Ann? and Other Age Problems 629 7.Y.ÁÁCombining Amounts and Prices Incoherently 638 ÁÁ7.Y.1.ÁÁReversal of Averages Paradox 640 ÁÁ7.Y.2.ÁÁUnfair Division 641 7.Z.ÁÁMissing Dollar and Other Erroneous Accounting 642 7.AA.ÁÁNegative Digits 643 ÁÁ7.AA.1.ÁÁNegative Bases, etc. 644 7.AB.ÁÁPerfect Numbers, etc. 645 7.AC.ÁÁCryptarithms, Alphametics and Skeleton Arithmetic 647 ÁÁ7.AC.1.ÁÁCryptarithms: SEND + MORE = MONEY, etc. 647 ÁÁ7.AC.2.ÁÁSkeleton Arithmetic: Solitary Seven, etc. 650 ÁÁ7.AC.3.ÁÁPan-Digital Sums 652 ÁÁ7.AC.3.aÁÁInsertion of Signs to Make 100, etc. 653 ÁÁ7.AC.4.ÁÁPan-Digital Products 654 ÁÁ7.AC.5.ÁÁPan-Digital Fractions 656 ÁÁ7.AC.6.ÁÁOther Pan-Digital and Similar Problems 657 ÁÁ7.AC.7.ÁÁSelf©descriptive Numbers, Pangrams, etc. 660 7.AD.ÁÁSelling, Buying and Selling Same Item 661 ÁÁ7.AD.1.ÁÁPawning Money 662 7.AE.ÁÁUse of Counterfeit Bill or Forged Cheque 662 7.AF.ÁÁArithmetic Progressions 664 ÁÁ7.AF.1.ÁÁCollecting Stones 664 ÁÁ7.AF.2.ÁÁClock Striking 666 7.AG.ÁÁ2592 667 7.AH.ÁÁMultiplying by Reversing 668 ÁÁ7.AH.1.ÁÁOther Reversal Problems 669 7.AI.ÁÁImpossible Exchange Rates 669 7.AJ.ÁÁMultiplying by Shifting 669 ÁÁ7.AJ.1.ÁÁMultiplying by Appending Digits 671 7.AK.ÁÁLazy Worker 671 7.AL.ÁÁIf A is B, What is C? 674 7.AM.ÁÁCrossnumber Puzzles 676 7.AN.ÁÁThree Odds Make an Even, etc. 678 7.AO.ÁÁDivination of a Permutation 680 7.AP.ÁÁKnowing Sum vs Knowing Product 684 7.AQ.ÁÁNumbers in Alphabetic Order 687 7.AR.ÁÁ1089 687 7.AS.ÁÁCigarette Butts 690 7.AT.ÁÁBookworm's Distance 690 7.AU.ÁÁNumber of Cuts to Make n Pieces 691 7.AV.ÁÁHow Long to Strike Twelve? 692 7.AW.ÁÁ28/7 = 13 692 7.AX.ÁÁSum = Product, etc. 692 7.AY.ÁÁSum of Powers of Digits 693 7.AZ.ÁÁDivination of a Pair of Cards from its Rows 694 7.BA.ÁÁCycle of Numbers with Each Closer to Ten than the Previous 696 7.BB.ÁÁIterated Functions of Integers 696 7.BC.ÁÁUnusual Difficulty Making Change 697 à Ã8.ÁÁPROBABILITY RECREATIONSÄ Ä 699 8.A.ÁÁBuffon's Needle Problem 699 8.B.ÁÁBirthday Problem 699 8.C.ÁÁProbability that a Triangle is Acute 701 8.D.ÁÁAttempts to Modify Boy-Girl Ratio 702 8.E.ÁÁSt. Petersburg Paradox 702 8.F.ÁÁProblem of Points 703 8.G.ÁÁProbability that Three Lengths Form a Triangle 704 8.H.ÁÁProbability Paradoxes 705 ÁÁ8.H.1.ÁÁBertrand's Box Paradox 705 ÁÁ8.H.2.ÁÁBertrand's Chord Paradox 705 8.I.ÁÁTaking the Next Train 706 8.J.ÁÁClock Patience or Solitaire 706 8.K.ÁÁSucker Bets 706 8.L.ÁÁNontransitive Games 707 à Ã9.ÁÁLOGICAL RECREATIONSÄ Ä 708 9.A.ÁÁAll Cretans are Liars, etc. 708 9.B.ÁÁSmith ©© Jones ©© Robinson Problem 712 9.C.ÁÁForty Unfaithful Wives 712 9.D.ÁÁSpots on Foreheads 712 9.E.ÁÁStrange Families 714 ÁÁGeneral Studies of Kinship Relations 714 ÁÁDeceased Wife's Sister, etc. 715 ÁÁGeneral Family Riddles 716 ÁÁ9.E.1.ÁÁThat Man's Father is My Father's Son, etc. 722 ÁÁ9.E.2.ÁÁIdentical Siblings who are not Twins 726 9.F.ÁÁThe Unexpected Hanging 726 9.G.ÁÁTruthtellers and Liars 726 9.H.ÁÁPrisoner's Dilemma 729 9.I.ÁÁHempel's Raven Paradox 729 9.J.ÁÁUse of a Fallen Signpost 729 9.K.ÁÁLewis Carroll's Barber Paradox 730 ÙÙ Ã Ã10. ÁÁPHYSICAL RECREATIONSÄ Ä 730 10.A.ÁÁOvertaking and Meeting Problems 731 ÁÁ10.A.1.ÁÁCircling an Army 743 ÁÁ10.A.2.ÁÁNumber of Buses Met 745 ÁÁ10.A.3.ÁÁTimes from Meeting to Finish Given 745 ÁÁ10.A.4.ÁÁThe Early Commuter 746 ÁÁ10.A.5.ÁÁHead Start Problems 747 ÁÁ10.A.6.ÁÁDouble Crossing Problems 748 ÁÁ10.A.7.ÁÁTrains Passing 748 ÁÁ10.A.8.ÁÁToo Slow, Too Fast 748 10.B.ÁÁFly Between Trains 748 10.C.ÁÁLewis Carroll's Monkey Problem 750 10.D.ÁÁMirror Problems 751 ÁÁ10.D.1.ÁÁMirror Reversal Paradox 751 ÁÁ10.D.2.ÁÁOther Mirror Problems 753 ÁÁ10.D.3.ÁÁMagic Mirrors 753 10.E.ÁÁWheel Paradoxes 754 ÁÁ10.E.1.ÁÁAristotle's Wheel Paradox 754 ÁÁ10.E.2.ÁÁOne Wheel Rolling Around Another 754 ÁÁ10.E.3.ÁÁHunter and Squirrel 755 ÁÁ10.E.4.ÁÁRailway Wheels Paradox 756 10.F.ÁÁFloating Body Problems 756 10.G.ÁÁMotion in a Current or Wind 758 10.H.ÁÁSnail Climbing out of Well 759 10.I.ÁÁLimited Means of Transport ©© Two Men and a Bike, etc. 763 10.J.ÁÁResistor Networks 764 10.K.ÁÁProblem of the Date Line 765 10.L.ÁÁFalling Down a Hole Through the Earth 767 10.M.ÁÁCelts = Rattlebacks 770 ÁÁ10.M.1.ÁÁTippee Tops 771 10.N.ÁÁShip's Ladder in Rising Tide 771 10.O.ÁÁErroneous Averaging of Velocities 772 10.P.ÁÁFalse Balance 772 10.Q.ÁÁPush a Bicycle Pedal 773 10.R.ÁÁClock Hand Problems 774 10.S.ÁÁWalking in the Rain 776 10.T.ÁÁCentrifugal Puzzles 777 10.U.ÁÁShortest Route Via a Wall 777 10.V.ÁÁPick Up Puzzles = Pluck It 777 10.W.ÁÁPuzzle Vessels 778 10.X.ÁÁHow Far Does a Phonograph Needle Travel? 784 10.Y.ÁÁDouble Cone Rolls Uphill 784 10.Z.ÁÁThe Wobbler 784 10.AA.ÁÁNon©Regular Dice 785 10.AB.ÁÁBicycle Track Problems 788 10.AC.ÁÁRoberval's Balance 790 10.AD.ÁÁPound of Feathers 790 10.AE.ÁÁJuggling over a Bridge 790 à Ã11.ÁÁTOPOLOGICAL RECREATIONSÄ Ä 791 11.A.ÁÁScissors on String 791 11.B.ÁÁTwo People Joined by Ropes at Wrists 792 11.C.ÁÁTwo Balls on String Through Leather Hole and Strap = Cherries Puzzle 793 11.D.ÁÁSolomon's Seal 795 11.E.ÁÁLoyd's Pencil Puzzle 797 11.F.ÁÁThe Imperial Scale 797 11.G.ÁÁTrick Purses 798 11.H.ÁÁRemoving Waistcoat Without Removing Coat 799 ÁÁ11.H.1.ÁÁRemoving Loop from Arm 799 11.I.ÁÁHeart and Ball Puzzle and Other Loop Puzzles 799 11.J.ÁÁMÀ?Àbius Strip 802 11.K.ÁÁWire Puzzles 805 ÁÁ11.K.1.ÁÁRing and Spring Puzzle 807 ÁÁ11.K.2.ÁÁString and Spring Puzzle 807 ÁÁ11.K.3.ÁÁMagic Chain = Tumble Rings 807 ÁÁ11.K.4.ÁÁPuzzle Rings 808 ÁÁ11.K.5.ÁÁRing Mazes 808 ÁÁ11.K.6.ÁÁInterlocked Nails, Hooks, Horns, etc. 809 ÁÁ11.K.7.ÁÁHorseshoes Puzzle 809 ÁÁ11.K.8.ÁÁThe Caught Heart 810 11.L.ÁÁJacob's Ladder and Other Hinging Devices 810 11.M.ÁÁPuzzle Boxes 811 11.N.ÁÁThree Knives Make a Support 813 11.O.ÁÁBorromean Rings 815 11.P.ÁÁThe Lonely Monk 816 11.Q.ÁÁTurning an Inner Tube Inside Out 817 11.RÁÁString Figures 817 11.S.ÁÁPuzzle Knives 818 Ðа¤˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐÁÁÁÁÁÁà ÃABBREVIATIONSÄ Ä ÁÁÁÁà ÃDIACRITICAL MARKS AND NOTATIONÄ Ä ÁÁBefore converting to LocoScript, I used various conventions, given below, to represent diacritical marks. Each symbol (except ') occurred after the letter it referred to. I have now converted these and all mathematical conventions into correct symbols, so far as possible, but I may have missed some, so I am keeping this information for the present. ÁÁCommon entries using such marks are given later in this section and only the abbreviated or simplified form is used later ©© e.g. I use Problemes for Bachet's work rather than ProblÀ/Àmes. (Though this may change??) ÁÁInitially, I did not record all diacritical marks, so some may be missing though I have checked almost all items. I may omit diacritical marks which are very peculiar. ÁÁTransliterations of Arabic, Sanskrit, Chinese, etc. are often given in very different forms. See Smith, History, vol. 1, pp. xvii©xxii for a discussion of the problems. The use of ^ and ÀÀ seems interchangeable and I have used ^ when different versions use both ^ and ÀÀ , except when quoting a title or passage when I use the author's form. [Smith, following Suter, uses ^  for Arabic, but uses ÀÀ for Indian. Murray uses ÀÀ for both. Wieber uses ÀÀ for Arabic. Van der Linde uses ÀÀ for Arabic. Datta & Singh use ^ for Indian.] ÁÁThere are two breathing marks in Arabic ©© ayn ÀÀ and alif/hamzah ÀÀ ©© but originally I didn't have two forms easily available, so both were represented by '. I have now converted almost all of these to ÀÀ and ÀÀ. These don't seem to be as distinct in the printing as on my screen. ÁÁFrench practice in accenting capitals is variable and titles are often in capitals, so expected marks may be missing. Also, older printing may differ from modern usage ©© e.g. I have seen: LiÀ/Àge and LiÀ)Àge; ProblÀ/Àmes, ProblÀ+Àmes and ProblÀ)Àmes. When available, I have transcribed the material as printed without trying to insert marks, but many places insert the marks according to modern French spelling. ÁÁGreek and Cyrillic titles are now given in the original with an English transliteration (using the Amer. Math. Soc. transliteration for Cyrillic). ÁÁI usually ignore the older usage of v for u and i for j, so that I give mathematiqve as mathematique and xiij as xiii. ÁÁI used a1, a2, ..., ai, etc. for subscripted variables, though I also sometimes used a(1), a(2), ..., a(i), etc. Superscripts or exponents were indicated by use of ^, e.g. 2^3 is 8. These have been converted to ordinary sub© and superscript usage, but ^ may be used when the superscript is complicated ©© e.g. for 2^aÃÃiÄÄ or 9^(9ÃÃ9ÄÄ). ÁÁGreek letters were generally spelled out in capitals or marked with square brackets, e.g. PI, [pi], PHI, but these have probably all been converted. ÁÁMy word processor does not produce binomial coefficients easily, so I use BC(n,k) for n!/k!(n-k)! ÁÁMany problems have solutions which are sets of fractions with the same denominator and I abbreviate a/z, b/z, c/z as (a, b, c)/z. Notations for particular problems are explained at the beginning of the topic. ÁÁRather than attempting to italicise letters used as symbols, I generally set them off by double©spaces on each side ©© see examples above. Other mathematical notations may be improvised as necessary and should be obvious. ÁÁRecall that the symbols below occurred after the letter they referred to, except for ' . ÐФ˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐ "ÁÁdenoted umlaut or diaeresis in general, e.g.: ÀÀ, À-À, À5À, À?À, ÀGÀ. /ÁÁwas used after a letter for accent acute, after l for À™À in Polish, and after o for ÀQÀ in Scandinavian. \ÁÁdenoted accent grave. ^ÁÁdenoted the circumflex, ^, in Czech, etc.; the overbar (macron) ÀÀ or ^ for a long vowel in Sanskrit, Hindu, etc.; and the overbar used to indicate omission in medieval MSS. @ÁÁdenoted the cedilla (French À'À and Arabic À³À) and the ogonek or Polish hook (Polish a). .ÁÁdenoted the underdot in h, k, n, r, s, t, in Sanskrit, Hindu, Arabic. These are sometimes written with a following h ©© e.g. k may also be written kh and I may sometimes have used this. (It is difficult to search for h. , etc., so not all of these may be converted.) This mark vanishes when converted to WordPerfect. *ÁÁdenoted the overdot for À}À, m, n, in Sanskrit, Hindu, Arabic. This vanishes over m and n in WordPerfect. ~ÁÁdenoted the Spanish tilde ~ and the caron or hachek ÀÀ, in ÀwÀ, À±À. The breve is a curved version, ÀÀ, of the same symbol and is essentially indistinguishable from the caron. It may occur in Russian, but vanishes in WordPerfect. _ÁÁdenoted the underbar in d, j, t. This mark vanishes in WordPerfect. 'ÁÁdenotes breathing marks in Arabic, etc. There are actually two forms of this ©© ayn ÀÀ and alif/hamzah ÀÀ ©© but I didn't have two forms easily available and originally entered both as apostrophe ' . These normally occur between letters and I placed the ' in the same space. I have converted most of these. Ðа¤˜Œ € tÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐ ÁÁCommonly occurring words with diacritical marks are: AcadÀ)Àmie, arithmÀ)Àtique, bibliothÀ/Àque, BirkhÀÀuser, cafÀ)À, carrÀ)À, À)Àcole, ErdÀ?Às, fÀGÀr, gÀ)ÀomÀ/Àtre, gÀ)ÀomÀ)Àtrie, GÀ?Àttingen, HanoÀ5À ©© in French only, -iÀ/Àme, littÀ)Àraire, mathÀ)Àmatique, mÀ)Àmoire, mÀ)Ànage, misÀ/Àre, MÀ?Àbius, moirÀ)À, numÀ)Àrique, PÀ)Àtersbourg, probabilitÀ)Às, problÀ/Àme (I have seen problÀ+Àmes??), RÀÀtsel, rÀ)ÀcrÀ)Àation, SÀÀndig, siÀ/Àcle, sociÀ)ÀtÀ)À, ThÀ)Àbault, thÀ)Àorie, ÀGÀber, umfÀGÀllung. ÁÁI have used ?? to indicate uncertainty and points where further work needs to be done. The following symbols after ?? indicate the action to be done. ÁÁ*ÁÁÁÁcheck for diacritical marks, etc. ÁÁNXÁÁÁÁno Xerox or other copy ÁÁNYSÁÁÁÁnot yet seen ÁÁNYRÁÁnot yet read ÁÁo/oÁÁÁÁon order ÁÁSPÁÁÁÁcheck spelling ÁÁOther comments may be given. ÁÁà ÃABBREVIATIONS OF JOURNALS AND SERIES.Ä Ä See: AMM, CFF, CM, CMJ, Family Friend, G&P, G&PJ, HM, JRM, MG, MiS, MM, MS, MTg, MTr, M500, OPM, RMM, SA, SM, SSM in Common References below. ÁÁà ÃABBREVIATIONS OF PUBLISHERS.Ä Ä See: AMS, C&W, CUP, Loeb Classical Library, MA, MAA, NCTM, OUP in Common References below. ÁÁà ÃABBREVIATIONS OF MONTHS.Ä Ä All months are given by their first three letters in English: Jan, Feb, .... ÁÁà ÃPUBLISHERS' LOCATIONS.Ä Ä The following publisher's locations will not be cited each time. Other examples may occur and can be found in the file PUBLOC. AMS (American Mathematical Society), Providence, Rhode Island, USA. Chelsea Publishing, NY, USA. CUP (Cambridge University Press), Cambridge, UK. Dover, NY, USA. Freeman, San Francisco, then NY, USA. Harvard University Press, Cambridge, Massachusetts, USA. MA (Mathematical Association), Leicester, UK. MAA (Mathematical Association of America), Washington, DC, USA. NCTM (National Council of Teachers of Mathematics), Reston, Virginia, USA. Nelson, London, UK. OUP (Oxford University Press), Oxford, UK (and also NY, USA). Penguin, Harmondsworth, UK. Simon & Schuster, NY, USA. Ù Ù ÁÁÁÁÁÁà ÃCOMMON REFERENCES.Ä Ä ÁÁNOTES. When referring to items below, I will usually include the earliest reasonable date, even though the citation may be to a much later edition. For example, I would say "Canterbury Puzzles, 1907", even though I am citing problem numbers or pages from the 1958 Dover reprint of the 1919 edition. Sometimes the earlier editions are hard to come by and I have sometimes found that the earlier edition has different pagination ©© in that case I will (eventually) make the necessary changes. ÁÁEdition information in parentheses indicates items or editions that I have not seen, though I don't always do this when the later version is a reprint or facsimile. ÐФ˜¸  ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ°ÐÐ Abbaco.ÁÁÁÁSee: Pseudo©dell'Abbaco. Abbot Albert.ÁÁAbbot Albert von Stade. Annales Stadenses. c1240. Ed. by J. M. Lappenberg. In: Monumenta Germaniae Historica, ed. G. H. Pertz, Scriptorum t. XVI, Imp. Bibliopolii Aulici Hahniani, Hannover, 1859 (= Hiersemann, Leipzig, 1925), pp. 271-359. (There are 13 recreational problems on pp. 332-335.) [Vogel, on p. 22 of his edition of the Columbia Algorism, dates this as 1179, but Tropfke gives 1240, which is more in line with Lappenberg's notes on variants of the text. The material of interest, and several other miscellaneous sections, is inserted at the year 1152 of the Annales, so perhaps Vogel was misled by this.] I have prepared an annotated translation of this: The problems of Abbot Albert (c1240). I have numbered the problems and will cite this problem number. Abraham.ÁÁÁÁR. M. Abraham. Diversions and Pastimes. Constable, London, 1933 = Dover, 1964 (slightly amended and with different pagination, later retitled: Tricks and Amusements with Coins, Cards, String, Paper and Matches). I will cite the Constable pages (and the Dover pages in parentheses). Ackermann.ÁÁAlfred S. E. Ackermann. Scientific Paradoxes and Problems and Their Solutions. The Old Westminster Press, London, 1925. D. Adams. New Arithmetic. 1835. ÁÁÁÁÁÁDaniel Adams (1773©1864). ADAMS NEW ARITHMETIC. Arithmetic, in which the principles of operating by numbers are analytically explained, and synthetically applied; thus combining the advantages to be derived both from the inductive and synthetic mode of instructing: The whole made familiar by a great variety of useful and interesting examples, calculated at once to engage the pupil in the study, and to give him a full knowledge of figures in their application to all the practical purposes of life. Designed for the use of schools and academies in the United States. J. Prentiss, Keene, New Hampshire, 1836, boarded. 1©262 pp + 2pp publisher's ads, apparently inserted backward. [Halwas 1©6 lists 1st ed as 1835, then has 1837, 1838, 1839, 1842, c1850.] This is a reworking of The Scholar's Arithmetic of 1801. D. Adams. Scholar's Arithmetic. 1801. ÁÁÁÁÁÁDaniel Adams (1773©1864). The Scholar's Arithmetic; or, Federal Accountant: Containing. I. Common arithmetic, .... II. Examples and Answers with Blank Spaces, .... III. To each Rule, a Supplement, comprehending, 1. Questions .... 2. Exercises. IV. Federal Money, .... V. Interest cast in Federal Money, .... VI. Demonstration by engravings .... VII. Forms of Notes, .... The Whole in a Form and Method altogether New, for the Ease of the Master and the greater Progress of the Scholar. Adams & Wilder, Leominster, Massachusetts, 1801; 2nd ed, 1802. 3rd ed ??. 4th ed, by Prentiss, 1807; 6th ed, 1810; 10th ed, 1816; Stereotype Edition, Revised and Corrected, with Additions, 1819, 1820, 1824; John Prentiss, Keene, New Hampshire, 1825. [Halwas 8©14.] I have the 1825, whose Preface is for the 10th ed of 1816, so is probably identical to that ed. The Preface says he has now made some revisions. The only change of interest to us is that he has added answers to some problems. So I will cite this as 1801 though I will be giving pages from the 1825 ed. The book was thoroughly reworked as Adams New Arithmetic, 1835. M. Adams. Indoor Games. 1912. ÁÁÁÁÁÁMorley Adams, ed. The Boy's Own Book of Indoor Games and Recreations. "The Boy's Own Paper" Office, London, 1912; 2nd ptg, The Religious Tract Society, London (same address), 1913. [This is a major revision of: G. A. Hutchison, ed.; Indoor Games and Recreations; The Boy's Own Bookshelf; New ed., Religious Tract Society, London, 1891 (possibly earlier) ©© see 5.A.] M. Adams. Puzzle Book. 1939. ÁÁÁÁÁÁMorley Adams. The Morley Adams Puzzle Book. Faber & Faber, London, 1939. M. Adams. Puzzles That Everyone Can Do. 1931. ÁÁÁÁÁÁMorley Adams. Puzzles that Everyone Can Do. Grant Richards, London, 1931, boarded. AGM.ÁÁÁÁAbhandlungen zur Geschichte der Mathematischen Wissenschaften mit Einschluss ihrer Anwendungen. BegrÀGÀndet von Moritz Cantor. Teubner, Leipzig. The first ten volumes were Supplements to Zeitschrift fÀGÀr Math. u. Physik, had a slightly different title and are often bound in with the journal volume. Ahrens, Wilhelm Ernst Martin Georg (1872©1927). See: A&N, MUS, 3.B, 7.N. al-Karkhi.ÁÁÁÁAboÀEÀ Beqr Mohammed Ben AlhaÀ'Àen AlkarkhÀ3À [= al-Karagi = al-KarajÀ…À]. Untitled MS called KitÀ]Àb al©FakhrÀ…À (or just AlfakhrÀ3À) (The Book Dedicated to Fakhr al©Din). c1010. MS 952, Supp. Arabe de la BibliothÀ/Àque ImpÀ)Àriale, Paris. Edited into French by Franz Woepcke as: Extrait du FakhrÀ3À. L'Imprimerie ImpÀ)Àriale, Paris, 1853; reprinted by Georg Olms Verlag, Hildesheim, 1982. My page citations will be to Woepcke. Woepcke often refers to Diophantos, but his numbering gets ahead of Heath's. Alberti. 1747.ÁÁGiuseppe Antonio (or Giusepp©Antonio) Alberti (1715©1768). I Giochi Numerici Fatti Arcani Palesati da Giuseppe Antonio Alberti. Bartolomeo Borghi, Bologna, 1747, 1749. Venice, 1780, 1788(?). 4th ed., adornata di figure, Giuseppe Orlandelli for Francesco di Niccolo' Pezzana, Venice, 1795 (reprinted: Arnaud, Florence, 1979), 1813. Adornata di 16 figure, Michele Morelli, Naples, 1814. As: Li Giuochi Numerici Manifestati, Edizione adorna di Figure in rame, Giuseppe Molinari, Venice, 1815. ÁÁÁÁThe editions have almost identical content, but different paginations. I have compared several editions and seen little difference. The 1747 ed. has a dedication which is dropped in the 2nd ed. which also omits the last paragraph of the Prefazione. I only saw one other point where a few words were changed. I will give pages of 1747 (followed by 1795 in parenthesis). Much of Alberti, including almost all the material of interest to us and many of the plates, is translated from vol. 4 of the 1723 ed. of Ozanam. ÁÁÁÁ(Serge Plantureux's 1993 catalogue describes a 1747©1749 ed. with Appendice al Trattato de' Giochi Numerici (1749, 72 pp) & Osservazioni all'Appendice de' Giochi Numerici (38 pp), ??NYS. The copy in the Honeyman Collection had the Appendice. Christopher 3 has the Osservazioni. The Appendice is described by Riccardi as a severe criticism of Alberti, attributed to Giovanni Antonio Andrea Castelvetri and published by Lelio dall Volpe, Bologna, 1749. The Osservazioni are Alberti's response.) Alcuin (c735©804). ÁÁÁÁÁÁPropositiones Alcuini doctoris Caroli Magni Imperatoris ad acuendos juvenes. 9C. ÁÁÁÁIN: B. Flacci Albini seu Alcuini, Abbatis et Caroli Magni Imperatoris Magistri. Opera Omnia: Operum pars octava: Opera dubia. Ed. D. Frobenius, Ratisbon, 1777, Tomus secundus, volumen secundum, pp. 440-448. ??NYS. Revised and republished by J.-P. Migne as: Patrologiae Cursus Completus: Patrologiae Latinae, Tomus 101, Paris, 1863, columns 1143-1160. ÁÁÁÁA different version appears in: Venerabilis Bedae, Anglo-Saxonis Presbyteri. Opera Omnia: Pars Prima, Sectio II ©© Dubia et Spuria: De Arithmeticus propositionibus. Tomus 1, Basel, 1563. (Rara, 131, says there were earlier editions: Paris, 1521 (part), 1544-1545 (all), 1554, all ??NYS.) Revised and republished by J.-P. Migne as: Patrologiae Cursus Completus: Patrologiae Latinae, Tomus 90, Paris, 1904, columns 665-672. Incipiunt aliae propositiones ad acuendos juvenes is col. 667-672. A version of this occurs in Ens' Thaumaturgus Mathematicus of 1636 ©© cf under Etten. ÁÁÁÁThe Alcuin has 53 numbered problems with answers. The Bede has 3 extra problems, but the problems are not numbered, there are only 31 1/2 answers and there are several transcription errors. The editor has used the Bede to rectify the Alcuin. ÁÁÁÁThere is a recent critical edition of the text by Folkerts ©© Die ÀÀlteste mathematische Aufgabensammlung in lateinischer SprÀÀche: Die Alkuin zugeschriebenen Propositiones ad Acuendos Iuvenes; Denkschriften der À>Àsterreichischen Akademie der Wissenschaften, Mathematische-naturwissenschaftliche Klasse 116:6 (1978) 13-80. (Also separately published by Springer, Vienna, 1978. The critical part is somewhat revised as: Die Alkuin zugeschriebenen "Propositiones ad Acuendos Iuvenes"; IN: Science in Western and Eastern Civilization in Carolingian Times, ed. by P. L. Butzer & D. Lohrmann; BirkhÀÀuser, Basel, 1993, pp. 273©281.) He finds that the earliest text is late 9C and is quite close to the first edition cited above. He uses the same numbers for the problems as above and numbers the extra Bede problems as 11a, 11b, 33a. I use Folkerts for the numbering and the titles of problems. ÁÁÁÁJohn Hadley kindly translated Alcuin for me some years ago and made some amendments when Folkerts' edition appeared. I annotated it and it appeared as: Problems to Sharpen the Young, MG 76 (No. 475) (Mar 1992) 102©126. A slightly corrected and updated edition, containing some material omitted from the MG version, is available as Technical Report SBU©CISM©95©18, School of Computing, Information Systems, and Mathematics, South Bank University, Oct 1995, 28pp. ÁÁÁÁMenso Folkerts and Helmuth Gericke have produced a German edition: Die Alkuin zugeschriebenen Propositiones ad Acuendos Juvenes (Aufgabe zur SchÀÀrfung des Geistes der Jugend); IN: Science in Western and Eastern Civilization in Carolingian Times, ed. by P. L. Butzer & D. Lohrmann; BirkhÀÀuser, Basel, 1993, pp. 283©362. ÁÁÁÁSee also: David Singmaster. The history of some of Alcuin's ÃÃPropositionesÄÄ. IN: Charlemagne and his Heritage 1200 Years of Civilization and Science in Europe: Vol. 2 Mathematical Arts; ed. by P. L. Butzer, H. Th. Jongen & W. Oberschelp; Brepols, Turnhout, 1998, pp. 11-29. AM. 1917.ÁÁH. E. Dudeney. Amusements in Mathematics. Nelson, 1917. (There were reprintings in 1919, 1920, 1924, 1925, 1927, 1928, 1930, 1932, 1935, 1938, 1939, 1941, 1943, 1946, 1947, 1949, 1951, but it seems that the date wasn't given before 1941?) = Dover, 1958. AMM.ÁÁÁÁAmerican Mathematical Monthly. AMS.ÁÁÁÁAmerican Mathematical Society. Les Amusemens. 1749. ÁÁÁÁÁÁLes Amusemens MathÀ)Àmatiques PrecedÀ)Às Des ElÀ)Àmens d'ArithmÀ)Àtique, d'AlgÀ)Àbre & de GÀ)ÀomÀ)Àtrie nÀ)Àcessaires pour l'intelligence des ProblÀ+Àmes. AndrÀ)À-Joseph Panckoucke, Lille, 1749. Often listed with Panckoucke as author (e.g. by the NUC, the BNC and Poggendorff), but the book gives no such indication. Sometimes spelled Amusements. There were 1769 and 1799 editions. Apianus. Kauffmanss Rechnung. 1527. ÁÁÁÁÁÁPetrus Apianus (= Peter Apian or Bienewitz or Bennewitz) (1495-1552). Eyn Newe Unnd wolgegrÀGÀndte underweysung aller Kauffmanss Rechnung in dreyen BÀGÀchern / mit schÀ?Ànen Regeln uÀÙÀ fragstucken begriffen. Sunderlich was fortl unnd behendigkait in der WelschÀoÀ Practica uÀÙÀ Tolletn gebraucht wirdt / des gleychen fÀGÀrmalss wider in TeÀGÀtzscher noch in Welscher sprach nie gedrÀGÀckt. durch Petrum ApianÀÁÀ von Leyssnick / d Astronomei zÀÅÀ Ingolstat OrdinariÀÁÀ / verfertiget. Georgius Apianus, Ingolstadt, (1527), facsimile, with the TP of the 1544 ed. and 2pp of publication details added at the end, Polygon©Verlag, Buxheim©EichstÀÀtt, 1995, with 8pp commentary leaflet by Wolfgang Kaunzner. (The TP of this has the first known printed version of Pascal's Triangle.) Smith, Rara, pp. 155©157. (The d is an odd symbol, a bit like a À À or an 8, which is used regularly for der both as a single word and as the ending of a word, e.g. and for ander.) Smith notes that Apianus follows Rudolff (1526) very closely. AR. c1450.ÁÁFrater Friedrich Gerhart (attrib.). Latin & German MSS, c1450, known as Algorismus Ratisbonensis. Transcribed and edited from 6 MSS by Kurt Vogel as: Die Practica des Algorismus Ratisbonensis; C. H. Beck'sche Verlagsbuchhandlung, Munich, 1954. (Kindly sent by Prof. Vogel.) Vogel classifies the problems and gives general comments on the mathematics on pp. 155-189. He gives detailed historical notes on pp. 203-232. When appropriate, I will cite these pages before the specific problems. He says (on p. 206) that almost all of Munich 14684 (see below) is included in AR. Arnold, George. See: Book of 500 Puzzles, Boy's Own Conjuring Book, Hanky Panky. Arrighi, Gino. See: Benedetto da Firenze, Calandri, Pseudo©Dell'Abbaco, della Francesca, Gherardi, Lucca 1754, P. M. Calandri. Aryabhata.ÁÁÁÁÀ\Àryabhata (I) (476© ). À\ÀryabhatÀ…Àya. 499. Critically edited and translated into English by Kripa Shankar Shukla, with K. V. Sarma. Indian National Science Academy, New Delhi, 1976. (Volume 1 of a three volume series ©© the other two volumes are commentaries, of which Vol. 2 includes the commentary À\ÀryabhatÀ…ÀyaªBhÀ]Àsya, written by Bhaskara I in 629. Aryabhata rarely gives numerical examples, so Bhaskara I provided them and these were later used by other Indian writers such as Chaturveda, 860. The other commentaries are later and of less interest to us. Prof. Shukla has sent a photocopy of an introductory booklet, which is an abbreviated version of the introductory material of Vol. 1, with some extensions relating Aryabhata to other writers.) The material is organized into verses. There is an older translation by Walter Eugene Clark as: The ÀÀryabhatÀ3Àya of ÀÀryabhata; Univ. of Chicago Press, Chicago, 1930. (There was an Aryabhata II, c950, but he only occurs in 7.K.1.) A&N.ÁÁÁÁWilhelm Ahrens. Altes und Neues aus der Unterhaltungsmathematik. Springer, Berlin, 1918. Bachet, Claude-Gaspar (1581©1638). See: Problemes. Bachet©Labosne. See: Problemes. Badcock. Philosophical Recreations, or, Winter Amusements. [1820]. ÁÁÁÁÁÁPhilosophical Recreations, or, Winter Amusements. Thomas Hughes, London, nd [1820]. [BCB 18©19; OCB, pp. 180 & 197. Heyl 22©23. Toole Stott 75©77. Christopher 54©56. Wallis 34 BAD, 35 BAD. These give dates of 1820, 1822, 1828.] HPL [Badcock] RBC has three versions with slightly different imprints on the title pages, possibly the three dates mentioned. ÁÁÁÁWallis 34 BAD has this bound after the copy of: John Badcock; Domestic Amusements, or Philosophical Recreations ...; T. Hughes, London, nd [1823], and it is lacking its Frontispiece and TP ©© cf in 6.BH. HPL [Badcock] has both books, including the folding Frontispieces. The earlier does not give an author, but its Preface is signed J. B. and the later book does give his name and calls itself a sequel to the earlier. Toole Stott 75©80 clearly describes both works. Some of the material is used in Endless Amusement II. Baker. Well Spring of Sciences. 1562? ÁÁÁÁÁÁHumfrey Baker (fl. 1557©1587). The Well Sprynge of Sciences Which teacheth the perfect worke and practise of Arithmeticke both in whole numbers and fractions, with such easye and compendious instruction into the sayde arte, .... Rouland Hall for James Rowbotham, London, 1562. [Smith, Rara, p. 327, says it was written in 1562 but wasn't actually printed until 1568, but a dealer says the 1st ed. was 1564 and there was a 4th ed. in 1574, which I have examined.] Apparently much revised and extended, (1580). Reprinted, with title: The Wel [sic] Spring of Sciences: Which teacheth the perfect worke and practise of Arithmetike; Thomas Purfoote, London, 1591. I have seen Thomas Purfoot, London, 1612, which is essentially identical to 1591. I have also seen: Christopher Meredith, London, 1646; Christopher Meredith, London, 1650; R. & W. L. for Andrew Kemb, London, 1655; which are all the same, but differently paged than the 1591. I have also seen Baker's Arithmetick, ed. by Henry Phillippes, Edward Thomas, London, 1670, which has different pagination and some additional problems compared to the 1646/1655 ed. [Smith, Rara, 327©330 & 537, says it was rewritten in 1580, but there is little difference between the 1580 and the many later editions, so the 1591 ed. is probably close to the 1580 ed. The copy of the 1562 in the Graves collection ends on f. 160r, but an owner has written a query as to whether the book is complete. Neither Smith nor De Morgan seems to have seen a 1562 so they don't give a number of pages for it. (STC records no copies of the 1562, 1564, 1576, 1584, 1607 editions, but there was a 1576 by [T. Purfoote], apparently the 5th ed., of c500pp, in the Honeyman Collection.) Almost all the problems of interest occur on ff. 189r©198r of the 1591 ed. and hence are not in the Graves copy of the 1562 ed., but H&S 61 refers to one of these problems as being in Baker, 1568. The 1574 ends at fol. 200 (misprinted as 19?, where the ? is an undecipherable blob) and Chapter 16, which is headed: The 16 Chapter treateth of sportes and pastime, done by number, is on ff. 189r©200v, and contains just a few recreations, as in Recorde. So I will date the book as 1562?, but most of the later material as 1580?. The problems of 7.AF.1 and 10.A may be in Graves copy of the 1562 ed. ©© ??check. I will cite the 1580?, 1646 and 1670 editions, e.g. 1580?: ff. 192r 193r; 1646: pp. 302©304; 1670: pp. 344©345.] Bill Kalush has recently sent a CD with 1574, 1580, 1591, 1598, 1602, 1607, 1612, 1617, 1650, 1655 on it ©© ??NYR. Bakhshali MS.ÁÁThe BakhshÀ]ÀlÀ…À Manuscript, c7C. This MS was found in May 1881 near the village of BakhshÀ]ÀlÀ…À, in the YusufzÀ]ÀÀ…À district of the Peshawer division, then at the northwestern frontier of India, but apparently now in Pakistan. This is discussed in several places, such as the following, but a complete translation has only recently appeared. David Pingree says it is 10C, but his student Hayashi opts for 7C which seems pretty reasonable and I will adopt c7C. ÁÁÁÁ1.ÁÁA. F. Rudolf Hoernle. Extract of his report in some journal of the previous year. The Indian Antiquary 12 (Mar 1883) 89©90. A preliminary report, saying it was found near BakhshÀÀlÀ3À in the Yusufzai District in the PanjÀÀb. ÁÁÁÁ2.ÁÁA. F. Rudolf Hoernle. On the BakhshÀ]ÀlÀ…À Manuscript. Berichte des VII. Internationalen Orientalisten-Congresses, Wien, 1886. Alfred HÀ?Àlder, Vienna, 1889. Arische Section, p. 127©147 plus three folding plates. Cf next item. I will cite this as Hoernle, 1886. ÁÁÁÁ3.ÁÁA. F. Rudolf Hoernle. The Bakhshali manuscript. The Indian Antiquary 17 (Feb 1888) 33-48 & Plate I opp. p. 46; 275-279 & Plates II & III opp. pp. 276 & 277. This is essentially a reprint of the previous item, with a few changes or corrections, but considerable additional material. He dates it c4C. I will cite this as Hoernle, 1888. ÁÁÁÁ4.ÁÁG. R. Kaye. The BakhshÀ]ÀlÀ…À Manuscript ©© A Study in Medieval Mathematics. ArchÀ%Àeological Survey of India ©© New Imperial Series XLIII: I©III, with parts I & II as one volume, (1927-1933). (Facsimile reprint in two volumes, Cosmo Publications, New Delhi, 1981 ©© this is a rather poor facsimile, but all the text is preserved. I have a letter detailing the changes between the original and this 'facsimile'.) I will only cite Part I ©© Introduction, which includes a discussion of the text. Part II is a discussion of the script, transliteration of the text and pictures of the entire MS. Part III apparently was intended to deal with the language used, but Kaye died before completing this and the published Part III consists of only a rearranged version of the MS with footnotes explaining the mathematics. Gupta, below, cites part III, as Kaye III and I will reproduce these citations. He dates it c12C. ÁÁÁÁ5.ÁÁB. Datta. The BakhshÀÀlÀ3À mathematics. Bull. Calcutta Math. Soc. 21 (1929) 1-60. This is largely devoted to dating of the MS and of its contents. He asserts that the MS is a copy of a commentary on some lost work of 4C or 5C (?). ÁÁÁÁ6.ÁÁR. C. Gupta. Some equalization problems from the BakhshÀ]ÀlÀ…À manuscript. Indian Journal of the History of Science 21 (1986) 51©61. Notes that Hoernle gave the MS to the Bodleian Library in 1902, where it remains, with shelf mark MS. Sansk. d.14. He follows Datta in believing that this is a commentary on a early work, though the MS is 9C, as stated by Hoernle. He gives many problems from Kaye III, sometimes restoring them, and he discusses them in more detail than the previous works. ÁÁÁÁ7.ÁÁTakao Hayashi. The BakhshÀ]ÀlÀ…À Manuscript An ancient Indian mathematical treatise. Egbert Forsten, Groningen, Netherlands, 1995. (Based on his PhD Dissertation in History of Mathematics, Brown University, May 1985, 774pp.) A complete edition and translation with extensive discussion of the context of the problems. He dates it as 7C. Ball, Walter William Rouse (1850©1925). See: Ball-FitzPatrick; MRE. Ball-FitzPatrick. ÁÁÁÁÁÁFrench translation of MRE by J. Fitz-Patrick, published by Hermann, Paris. ÁÁÁÁ1st ed., RÀ)ÀcrÀ)Àations et ProblÀ/Àmes MathÀ)Àmatiques des Temps Anciens & Modernes. From the 3rd ed, 1896, of MRE, 'Revue et augmentÀ)Àe par l'auteur'. 1898. The Note says 'M. Ball ... a bien voulu apporter À!À la troisiÀ/Àme À)Àdition anglaise des additions et des modifications importantes.' 352pp. ÁÁÁÁ2nd ed., RÀ)ÀcrÀ)Àations et ProblÀ/Àmes MathÀ)Àmatiques des Temps Anciens et Modernes. From the 4th ed, 1905, of MRE, 'et enrichie de nombreuses additions'. ÁÁ As three volumes, 1907-09. [I have vol. 1, 1907, which is 356pp. Pp. 327-355 is a note by A. Hermann, ComptabilitÀ)À d'une persone qui dÀ)Àpense plus que son revenu. I have not yet seen the other volumes to compare with the 1926 reprint, but Strens's notes in his copy indicate that they are identical.] ÁÁ Reprinted in one vol., Gabay, Paris, 1992, 544pp. ÁÁ Reprinted, 1926©1927. The only copies that I have seen are bound as one volume, but with separate pagination. My copy has left out the title pages of vols. 2 & 3. The copy in the Strens Collection has these title pages, but its vol. II is 1908. The 1926 vol. 1 says Nouvelle À)Àdition franÀ'Àaise, but the 1927 vol. 3 says DeuxiÀ/Àme À)Àdition franÀ'Àaise. ÁÁÁÁ[Vol. 1 is 326pp, omitting the note by Hermann. Vol. 2 is 363pp (pp. 322-355 is a historical note on the cubic, based on Cossali (1797)). Vol. 3 is 363pp including: Notes diverses de M. Aubry, pp. 137-206 (or 340? ©© the Table des MatiÀ/Àres and the page set up do not make it clear if Aubry's Notes end on p. 206); Note de M. Fitz-Patrick, La gÀ)ÀomÀ)Àtrie par le pliage et dÀ)Àcoupage du papier, pp. 341-360; A. Margossian, De l'ordonnance des nombres dans les carrÀ)Às magiques impairs, pp. 1-60 (pp. 61©64 is a Note on the same subject, presumably part of Margossian's material); Capt. Reinhart, some geometric notes, pp. 130©136.] Barnard. 50 Observer Brain©Twisters. 1962. ÁÁÁÁÁÁDouglas St. Paul Barnard. Fifty Observer Brain-Twisters A Book of Mathematical and Reasoning Problems. Faber, 1962. US ed.: A Book of Mathematical and Reasoning Problems: Fifty Brain Twisters; Van Nostrand, 1962. The editions have identical pagination. Bartl. c1920.ÁÁJÀÀnos Bartl. Preis©Verzeichnis von Bartl's Akademie fÀGÀr moderne magische Kunst. Hamburg, c1920. Reprinted by Olms Verlag, ZÀGÀrich, 1983, as: Zauberkatalog Bartl. References are to the section: Vexier© und Geduldspiele, pp. 305-312. Bartoli. Memoriale. c1420. ÁÁÁÁÁÁFrancesco Bartoli ( ©1425). Memoriale (= Notebook) containing some 30 mathematical problems copied during 1400?©1425. Ms 1 F 54 of the Archives dÀ)Àpartementales du Vaucluse, France. ??NYS ©© described and quoted in: Jacques Sesiano; Les problÀ/Àmes mathÀ)Àmatiques du Memoriale de F. Bartoli; Physis 26:1 (1984) 129©150. BC.ÁÁÁÁÁÁBinomial Coefficient, i.e. BC(n, k) = n!/k!(n©k)!. BCB. See: Hall, BCB. BDM. See under DSB. Bede, The Venerable (c672©735). (Now St. Bede.) See: Alcuin. Benedetto da Firenze. c1465. ÁÁÁÁÁÁBenedetto da Firenze. Trattato d'Abacho. c1465. This was a popular treatise and Van Egmond's Catalog 356 lists 18 copies under Benedetto. Six show B as author, one has Benedetto, one has Benedetto da Firenze, one has PÃÃoÄÄ MÃÃaÄÄ and one has Filipo Chalandri, so it seems Benedetto is the most likely author. The MSS date from c1465 to c1525 and contain 9 to 25 chapters. ÁÁÁÁThe version in Cod. Acq. e doni 154, Biblioteca Medicea Laurenziana, Florence, c1480. has been transcribed and edited by Gino Arrighi as: Pier Maria Calandri; Tractato d'Abbacho; Domus Galilaeana, Pisa, 1974. The incipit names PÃÃoÄÄ MÃÃaÄÄ as author. Cf Van Egmond's Catalog 96. This version has 23 chapters. Benson. 1904.ÁÁJ. K. Benson. The Book of Indoor Games for Young People of All Ages. C. Arthur Pearson, London, 1904. [This copies a lot from Hoffmann (or a common ancestor?).] ÁÁÁÁMuch of the material of Indoor Games is repeated in: J. K. Benson, ed.; The Pearson Puzzle Book; C. Arthur Pearson, London, nd [1921 ©© BMC]. This is not in BMC or NUC under Benson ©© but I have seen an ad listing this as by Mr. X and it is listed under Mr. X in BMC. Puzzle Book pp. 1©96 =  Indoor Games pp. 189©257; Puzzle Book pp. 109©114 = Indoor Games pp. 258©262. The only different material in Puzzle Book is pp. 97©108. Neither book refers to the other. Cf Mr. X in Section 4.A.1 Berkeley & Rowland. Card Tricks & Puzzles. 1892. ÁÁÁÁÁÁ"Berkeley" [Peel, Walter H.] & Rowland, T. B. Card Tricks and Puzzles. The Club Series, George Bell & Sons, London, 1892 ©© according to BMC, but my copy is 1897. Card Puzzles, etc., pp. 1©74 is by Berkeley; Arithmetical Puzzles, pp. 75©120 is by Rowland. Berlekamp, Elwyn R. (1940© ) See: Winning Ways. Bestelmeier. 1801©1803. ÁÁÁÁÁÁG. H. [Georg Hieronimus] Bestelmeier. Magazin von verschiedenen Kunst- und andern nÀGÀtzlichen Sachen .... [Toy catalogues.] Nuremberg, 1801-1803. ÁÁÁÁEight issues and cumulative classified index reprinted by Olms, Zurich, 1979. Issue VII is 1801; the others are 'neue verbesserte Auflage', 1803. This includes items numbered 1 through 1111. ÁÁÁÁSelections, with English translations. Daniel S. Jacoby, ed. The Amazing Catalogue of the Esteemed Firm of George Hieronimus Bestelmeier. Selected Excerpts from Editions of 1793 and 1807. [A comment inside makes me wonder if 1793©1807 is meant??] Merrimack Publishing Corp., NY, 1971, 82pp. The numeration is the same as in the Olms edition, but the Jacoby continues to item 1321. Obviously these later items come from the 1807 edition, but we cannot tell if they might date from 1805, say, nor whether all the earlier items go back to 1793. Jerry Slocum uses Jacoby in his Compendium and has kindly provided photocopies of Jacoby's pp. 70©82 containing all the items after 1111 and some examples of the earlier items. Jacoby does not translate the texts, but just provides English labels for each picture and these labels are sometimes unconnected with the text. ÁÁÁÁMany of Bestelmeier's items are taken from Catel; Kunst©Cabinet; 1790. Sometimes the figure is identical (often reversed) or is a poor copy. Texts are often copied verbatim, or slightly modified, but often abbreviated. E.g. Catel often explains the puzzle and this part is frequently omitted in Bestelmeier. Bestelmeier was the successor to Catel, qv. The booklet by Slocum & Gebhardt (qv under Catel) gives precise datings for the various parts of these catalogues, but I have not yet entered these details. Bhaskara I. 629. ÁÁÁÁÁÁBhÀ]Àskara I. À\ÀryabhatÀ…Àya©BhÀ]Àsya. 629. Critically edited, including an English Appendix of the numerical examples used, by Kripa Shankar Shukla. Indian National Science Academy, New Delhi, 1976. (Volume 2 of a three volume series devoted to the À\ÀryabhatÀ…Àya (499) of Aryabhata (476© ).) Bhaskara I repeats and exposits Aryabhata verse by verse, but Aryabhata rarely gives numerical examples, so Bhaskara I provided them and these were later used by other Indian writers such as Chaturveda, 860. His earlier Maha©Bhaskariya (MahÀ]À-BhÀ]ÀskarÀ…Àya) of c629 is cited in 7.P.2. Shukla's Appendix is sometimes brief, but sometimes very detailed, e.g. on the 26 examples of Chinese remainder problems. Bhaskara II (1114©c1185). ÁÁÁÁÁÁBhÀÀskara II (1114©c1185, see Colebrooke). Biggs, Norman L. See: BLW. Bijaganita.ÁÁÁÁBÀ3Àjaganita of Bhaskara II, 1150 (see Colebrooke). The Bile Beans Puzzle Book. 1933. ÁÁÁÁÁÁBile Beans (C. E. Fulford, Ltd., Leeds, England). The Bile Beans Puzzle Book. 1933. Birtwistle. Math. Puzzles & Perplexities. 1971. ÁÁÁÁÁÁClaude Birtwistle. Mathematical Puzzles and Perplexities How to Make the Most of Them. George Allen & Unwin, London, 1971. Birtwistle. Calculator Puzzle Book. 1978. ÁÁÁÁÁÁClaude Birtwistle. The Calculator Puzzle Book. Paperfronts (Elliot Right Way Books), Kingswood, Surrey, 1978. (There is a US ed. by Bell, NY, 1978.) BL(LD).ÁÁÁÁBritish Library (Lending Division). Blasius. 1513.ÁÁJohannis (or Joannes) Martinus Blasius (later denoted Sileceus or Sciliceus). Liber Arithmetice Practice Astrologis Phisicis et Calculatioribus admodum utilis. Thomas Kees for Joannis Parui & Joannis Lambert (in colophon; TP has Jehanlambert), Paris, 1513. Facsimile by Heffer Scientific Reprint, Cambridge, 1960. See Smith, Rara, pp. 95©97. The Glaisher article in 7.P.5 [Messenger of Mathematics 53 (1923©24) 1-131] discusses this book and says he only knows one example of it, which he has in front of him, so I suspect this facsimile is from that copy. See Rara 95©97. The Honeyman Collection had a copy, saying it was printed for J. Petit and J. Lambert and that copy had Petit's device on the TP while the TP shown in Rara has Lambert's device, which is as in this facsimile. There was a reprinting in 1514 and extended editions in 1519 (ed. by Oronce FinÀ)À) and 1526 (ed. by T. Rhaetus) [Honeyman Collection, nos. 350©352]. BLC.ÁÁÁÁBritish Library Catalogue, replacing BMC, in progress since 1970s. BLC©ÀPÀÁÁÁÁIndicates that I could not find the item in the BLC. BLW. 1976.ÁÁNorman L. Biggs, E. Keith Lloyd & Robin J. Wilson. Graph Theory 1736-1936. OUP, 1976. Blyth. Match©Stick Magic. 1921. ÁÁÁÁÁÁWill Blyth. Match©Stick Magic. C. Arthur Pearson, London, 1921, reprinted 1923, 1939. BM(C).ÁÁÁÁBritish Museum (Catalogue (of books) to 1955. c1963). BMC65.ÁÁÁÁSupplement to the above Catalogue for 1956-1965. c1968. BN(C).ÁÁÁÁBibliothÀ/Àque National, Paris. (Catalogue, 1897©1981.) Bodleian.ÁÁÁÁThe Bodleian Library, University of Oxford, or its catalogue. Bonnycastle. Algebra. 1782 ÁÁÁÁÁÁJohn Bonnycastle (??©1821). An Introduction to Algebra, with Notes and Observations; designed for the Use of Schools, and Other Places of Public Education. 1782. The first nine editions appeared "without any material alterations". In 1815, he produced a 10th ed., "an entire revision of the work" which "may be considered as a concise abridgment" of his two volume Treatise on Algebra, 1813, (2nd ed. in 1820). The 1815 ed. had an Appendix: On the application of Algebra to Geometry. I have a copy of the 7th ed., 1805, printed for J. Johnson, London, and it is identical to the 2nd ed. of 1788 except for a problem in the final section of Miscellaneous Questions. However, the 9th ed. of 1812 has page numbers advanced by 10 except toward the end of the book. I also have the 13th ed. of 1824, printed for J. Nunn and 11 other publishers, London, 1824. This version has an Addenda: A New Method of resolving Numerical Equations, by his son Charles Bonnycastle (1797©1840), but is otherwise identical to the 10th ed. of 1815. The earlier text was expanded by about 10% in 1815, so a number of problems only occur in later editions. I will cite these later problems as 1815 and will cite the earlier problems as 1782. [Halwas 36©38 gives some US editions.] Book of 500 Puzzles. 1859. ÁÁÁÁÁÁThe Book of 500 Curious Puzzles: Containing a Large Collection of Entertaining Paradoxes, Perplexing Deceptions in Numbers, and Amusing Tricks in Geometry. By the author of "The Sociable," "The Secret Out," "The Magician's Own Book," "Parlor Games," and " Parlor Theatricals," etc. Illustrated with a great Variety of Engravings. Dick & Fitzgerald, NY, 1859. Compiled from The Sociable (qv) and Magician's Own Book. Pp. 1©2 are the TP and its reverse. Pp. 3-36, are identical to pp. 285©318 of The Sociable; pp. 37©54 are identical to pp. 199©216 of Magician's Own Book and pp. 55©116 are identical to pp. 241©302 of Magician's Own Book. [Toole Stott 103 lists it as anonymous. NUC, under Frikell, says to see title. NUC, under Book, also has an 1882 ed, compiled by William B. Dick. Christopher 129. C&B lists it under Cremer.] ÁÁÁÁThe authorship of this and the other books cited ©© The Sociable, The Secret Out, The Magician's Own Book, Parlor Games, and Parlor Theatricals, etc. ©© is confused. BMC & NUC generally assign them to George Arnold (1834©1865) or Wiljalba (or Gustave) Frikell (1818 (or 1816) © 1903), sometimes with Frikell as UK editor of Arnold's US version ©© but several UK versions say they are translated and edited by W. H. Cremer Jr, and one even cites an earlier French book (though the given title may not exist!, but cf Manuel des Sorciers, 1825) ©© see the discussion under Status of The Project, in the Introduction, above. The names of Frank Cahill, Henry Llewellyn Williams and Gustave Frikell (Jr.) are sometimes associated with versions of these as authors or coauthors. The Preface of The Sociable says that most of the Parlor Theatricals are by Frank Cahill and George Arnold ©© this may indicate they had little to do with the parts that interest us. Toole Stott 640 opines that this reference led Harry Price to ascribe these books to these authors. ÁÁÁÁA publisher's ad in the back says: "The above five books are compiled from the "Sociable" and "Magician's Own."", referring to: The Parlor Magician [Toole Stott 543, 544]; Book of Riddles and Five Hundred Home Amusements [Toole Stott 107, 951]; Book of Fireside Games [possibly Toole Stott 300??]; Parlor Theatricals; The Book of 500 Curious Puzzles. However, [Toole Stott 951] is another version of The Book of Riddles and Five Hundred Home Amusements "by the author of "Fireside Games" [Toole Stott 300], "The Parlor Magic" [perhaps Toole Stott 543, 544], "Parlor Tricks with Cards" [Toole Stott 1056 lists this as by Frikell, "abridged from The Secret Out" (see also 547, 1142)], ..."; Dick & Fitzgerald, 1986 [sic, but must mean 1886??]. ÁÁÁÁSee Magician's Own Book for more about the authorship. ÁÁÁÁSee also: Boy's Own Book, Boy's Own Conjuring Book, Illustrated Boy's Own Treasury, Indoor and Outdoor, Landells: Boy's Own Toy©Maker, The Secret Out, Hanky Panky, The Sociable. Book of Merry Riddles. 1629? ÁÁÁÁÁÁThe Book of Merry Riddles. London, 1629. [Santi 235.] ÁÁSeveral reprints. Also known as Prettie Riddles. ÁÁÁÁA Booke of Merry Riddles; Robert Bird, London, 1631. [Mark Bryant; Dictionary of Riddles; Routledge, 1990, p. 100.] ÁÁÁÁBooke of Merry Riddles; John Stafford & W. G., London, 1660. ÁÁÁÁReprint of the 1629 in: J. O. Halliwell; The literature of the sixteenth and seventeenth centuries; London, 1851, pp. 67-102. [Santi 235.] ÁÁÁÁReprint of the 1660 in: J. O. Halliwell; The Booke of Merry Riddles, together with proper questions, and witty proverbs, to make pleasant pastime. Now first reprinted from the unique edition printed at London in the year 1660. For the author, London, 1866. This was a printing of 25 copies. There is a copy at UCL and a MS note at the end says 15 copies were destroyed on 9 Apr 1866, signed: J. O. H., with Number 9 written below. [Santi 307.] I have seen this, but some of the riddles are quoted by other authors and I will date all items as 1629? until I examine other material. ÁÁÁÁReprint of the 1629 in: Alois Brandl; Shakespeares Book of Merry Riddles und die anderen RÀÀthselbÀGÀcher seiner Zeit; Jahrbuch der deutschen Shakespeare©Gesellschaft 42 (1906) 1©64 (with the 1631 ed on pp. 53©63). ??NYR. [Santi 235 & 237.] Borghi. Arithmetica. 1484. ÁÁÁÁÁÁPietro Borghi = Piero Borgo or Borgi (?? © ÀÀ1494). Qui comenza la nobel opera de arithmethica ne la qual se tracta tute cosse amercantia pertinente facta & compilata ÃÃpÄÄ Piero borgi da veniesia. Erhard Ratdolt, Venice, 1484. 2 + 116 numbered ff. This is the second commercial arithmetic printed in Italy and was reprinted many times. See Rara 16©22. This edition was reproduced in facsimile, with notes by Kurt Elfering, as: Piero Borghi; Arithmetica Venedig 1484; Graphos, Munich, 1964; in: VerÀ?Àffentlichungen des Forschungsinstituts des Deutschen Museums fÀGÀr die Geschichte der Naturwissenschaften und der Technik, Reihe C ©© Quellentexte und ÀFÀbersetzunge, Nr. 2, 1965. ÁÁÁÁ The 3rd ed of 1491 had a title: Libro dabacho. From the 4th ed of 1501, the title was Libro de Abacho, so this is sometimes used as the title for the first editions also. Rara indicates that the printing was revised to 100 numbered ff by the 4th ed. of 1491. I have examined a 1509 ed. by Jacomo Pentio, Venice, ??NX. This has 100 numbered ff, but the last three ff contain additional material, though Rara doesn't mention this until the 11th ed of 1540. H&S discusses a problem and the folio in the 1540 ed is the same as in the 1509 ed. The locations of interest in the 1509 ed. are c18ff before the corresponding locations of the 1484. Van Egmond's Catalog 293©297 lists 13 Venetian editions from 1484 to 1567. ÁÁÁÁIt has been conjectured that this was a pseudonym of Luca Pacioli, but there is no evidence for this [R. Emmett Taylor; No Royal Road Luca Pacioli and His Times; Univ. of North Carolina Press, Chapel Hill, 1942, pp. 60 & 349]. ÁÁÁÁSee also: D. E. Smith; The first great commercial arithmetic; Isis 8 (1926) 41©49. Bourdon. AlgÀ/Àbre. 7th ed., 1834. ÁÁÁÁÁÁLouis Pierre Marie Bourdon (1779©1854). À(ÀlÀ)Àmens d'AlgÀ/Àbre. 7th ed., Bachelier, Paris, 1834. (1st ed, 1817; 5th, 1828; 6th, 1831; 8th, 1837; 1840. Undated preface in the 7th ed. describes many changes, so I will cite this as 1834, though much of the material would have occurred earlier.) Boy's Own Book. 1828. ÁÁÁÁÁÁWilliam Clarke, ed. The Boy's Own Book. The bibliography of this book is extremely complex ©© by 1880, it was described as having gone through scores of editions. My The Bibliography of Some Recreational Mathematics Books has 11 pages listing 76 English (40 UK, 37 US, 1 Paris) versions and a Danish version, implying 88 English (50 UK, 37 US, 1 Paris) versions, and 10 (or 11) related versions, and giving a detailed comparison of the versions that I have seen. Because of the multiplicity of versions, I have cited it by title rather than by the original editor's name, which is not in any of the books (except the modern facsimile) though this attribution seems to be generally accepted. I have examined the following versions, sometimes in partial photocopies or imperfect copies. ÁÁÁÁVizetelly, Branston and Co., London, 1828, 448pp.; 2nd ed., 1828, 462pp.; 3rd ed., 1829, 464pp (has an inserted advertisement sheet); 6th ed??, c1830, 462pp?? (my copy lacks TP, pp. 417©418, 431©436, 461©462); 9th ed., 1834, 462pp. Longman, Brown & Co., London, 24th ed., 1846, 462pp. [The latter five are identical, except for a bit in the Prelude (and the extra sheet in 3rd ed), so I will just cite the first of these as 1828-2. It seems that all editions from the 2nd of 1828 through the 29th of 1848, 462pp. are actually identical except for a bit of the Prelude (and the advertisement sheet in the 3rd ed.)] ÁÁÁÁFirst American Edition. Munroe & Francis, Boston & Charles S. Francis, NY, 1829, 316pp. Facsimile by Applewood Books, Bedford, Massachusetts, nd [1998?]. This is essentially an abridgement of the 2nd ed of 1828, copying the Prelude and adding "So far the London Preface. The American publishers have omitted a few articles, entirely useless on this side of the Atlantic, ...." The type is reset, giving some reduction in pages. A number of the woodcuts have been omitted. The section title pages are omitted. Singing Birds, Silkworms, White Mice, Bantams, Magnetism, Aerostatics, Chess and Artificial Fireworks are omitted. Angling, Rabbits, Pigeons, Optics are reduced. Rosamond's Bower is omitted from Paradoxes and Puzzles. Surprisingly, The Riddler is increased in size. The 2pp Contents is omitted and an 8pp Index is added. ÁÁÁÁBaudry's European Library & Stassin & Xavier, Paris, 1843, 448pp. [The existence of a Paris edition was previously unknown to the vendor and myself, but it is Heyl 354 and he cites Library of Congress. It is very different than the English and US editions, listing J. L. Williams as author. Even when the topic is the same, the text, and often the topic's name, are completely rewritten. See my The Bibliography of Some Recreational Mathematics Books for details ©© in it I have found it generally necessary to treat this book separately from all other editions. I will cite it as 1843 (Paris). Much of this, including almost all of the material of interest is copied exactly in Anon: Boy's Treasury, 1844, qv, and in translated form in de Savigny, Livre des À(Àcoliers, 1846, qv. The problem of finding the number of permutations of the letters of the alphabet assumes 24 letters, which makes me wonder if these books are based on some earlier French work. Heyl 355 is probably the same book, with slight variations in the title, by Dean and Munday, London, c1845.] ÁÁÁÁDavid Bogue, London, 1855, 611pp. [It seems that this version first appears in 1849 and continues through about 1859, when two sections were appended.] ÁÁÁÁ[W. Kent (late D. Bogue), London, 1859, 624pp??. For almost all material of interest, this is identical to the 1855 ed, so I will rarely (if ever?) cite it.] ÁÁÁÁ[Lockwood & Co., London, 1861, 624pp. Identical to the 1859 ed., so I will not cite it.] ÁÁÁÁLockwood & Co., London, 1868, 696pp. ÁÁÁÁ[Lockwood & Co., London, 1870, 716pp. Identical to 1868 with 20pp of Appendices, so page numbers for material of interest are the same as in 1868, so I will not cite it.] ÁÁÁÁ[Crosby Lockwood & Co., London, 1880, 726pp. Identical to 1870, but having the Appendices and 20 more pages incorporated into a new section. For almost all material of interest, the page numbers are 30 ahead of the 1868 & 1870 page numbers, so I will not cite it except when the page numbers are not as expected.] ÁÁÁÁ[5th (US?) ed., Worthington, NY, 1881, 362pp. For almost all material of interest, this is identical to the 1829 (US) ed., so I will rarely (if ever?) cite it.] ÁÁI will cite pages with edition dates and edition numbers or locations if needed (e.g. 1828©2: 410 or 1829 (US): 216). See also: Book of 500 Puzzles, Boy's Own Conjuring Book, Illustrated Boy's Own Treasury. ÁÁÁÁAnonymous. The Riddler; A Collection of Puzzles, Charades, Rebusses, Conundrums, Enigmas, Anagrams, &c. for the Amusement of Little Folks. S. Babcock, New Haven, Connecticut, 1835. 22pp. My copy has leaf 11/12 half missing and leaf 17/18 missing; NUC & Toole Stott 1392 say it should be 24pp, so presumably leaf 23/24 is also missing here. [Toole Stott 1392 has The Riddler: or, Fire©Side Recreations; a collection ..., 1838, also listed in NUC.] Paradoxes and Puzzles section consists of the introduction and 11 puzzles copied almost exactly from the Paradoxes and Puzzles section of Boy's Own Book, 2nd ed. of 1828 and this material is all in the first American edition of 1829. Other material is charades, etc. and is all in both these versions of Boy's Own Book. Shortz states that this is the first American book with puzzles ©© but there were at least five American versions of Boy's Own Book before this and all the material in The Riddler, except some woodcuts, is taken from Boy's Own Book, so this pamphlet seems to be a pirate version. NUC also lists a 1838 version. Boy's Own Conjuring Book. 1860. ÁÁÁÁÁÁThe Boy's Own Conjuring Book: Being a Complete Hand©book of Parlour Magic; and Containing over One Thousand Optical, Chemical, Mechanical, Magnetical, and Magical Experiments, Amusing Transmutations, Astonishing Sleights and Subtleties, Celebrated Card Deceptions, Ingenious Tricks with Numbers, Curious and Entertaining Puzzles, Charades, Enigmas, Rebuses, etc., etc., etc. Illustrated with nearly two hundred engravings. Intended as a source of amusement for one thousand and one evenings. Dick and Fitzgerald, NY, 1860. 384pp. [Toole Stott 115, corrected, lists this as (1859), and under 114, describes it as an extended edition of The Magician's Own Book ©© indeed the running head of the book is The Magician's Own Book! ©© but see below. Toole Stott 481 cites a 1910 letter from Harris B. Dick, of the publishers Dick & Fitzgerald. He describes The Boy's Own Conjuring Book as a reprint of Magician's Own Book "evidently gotten up and printed in London, but singularly enough it had printed in the book on the title©page ©© New York, Dick & Fitzgerald." Indeed, all the monetary terms are converted into British. Harold Adrian Smith [Dick and Fitzgerald Publishers; Books at Brown 34 (1987) 108©114] states that this is a London pirate edition. BMC has 384pp, c1860. NUC has a 384pp version, nd. Christopher 145©149 are five versions from 1859 and 1860, though none has the blue cover of my copy. Christopher 145 says the 1859 versions were printed by Milner & Sowerby, Halifax, and describes it as an extraction from Magician's Own Book, but see below. Christopher 148 cites Smith's article.] I also have a slightly different version with identical contents except omitting the date and frontispiece, but with a quite different binding, probably Christopher 149. [NUC lists 334pp, nd; 416pp, nd and 416pp, 1860. Toole Stott 114 is a 416pp version, 1861. Toole Stott 959 is a 534pp version, 1861. C&B cite a New York, 1859 with 416pp, a New York, nd, 334pp and London, c1850 (surely too early?).] ÁÁÁÁI have now compared this with The Magician's Own Book of 1857 and it is essentially a minor reworking of that book. The Magician's Own Book has 17 chapters and an answers chapter and a miscellaneous chapter of items which are almost all also listed in the Contents under earlier sections. All together, there are some 635 items. The Boy's Own Conjuring Book copies about 455 of these items essentially directly, completely omitting the chapters on Electricity, Galvanism, Magnetism, Geometry, Art, Secret Writing and Strength, and almost completely omitting the chapter on Acoustics. Of the 488 items in the other chapters, 453 are copied into the Boy's Own Conjuring Book, and this has in addition two of the acoustic problems, 125 new miscellaneous problems and 38pp of charades, riddles, etc. (The later UK edition of Magician's Own Book is very different from the US edition.) Many of the problems are identical to the Boy's Own Book or the Illustrated Boy's Own Treasury. See also: Book of 500 Puzzles, Boy's Own Book, Illustrated Boy's Own Treasury, Landells: Boy's Own Toy-Maker. Boy's Treasury. 1844. ÁÁÁÁÁÁAnonymous. The Boy's Treasury of Sports, Pastimes, and Recreations. With four hundred engravings. By Samuel Williams. [The phrasing on the TP could be read as saying Williams is the author, but the NUC entry shows he was clearly listed as the designer in later editions and his name appears on the Frontispiece.] D. Bogue, London, 1844. Despite the similarity of title, this is quite different from Illustrated Boy's Own Treasury and the similar books of c1860. [Toole Stott 116. Toole Stott 117 is another ed., 1847, 'considerably extended'. Toole Stott gives US editions: 959; 960; 118; 199 & 961©965 are 1st, 1847; 2nd, 1847; 3rd, 1848; 6 versions of the 4th, 1850, 1848, 1849, 1852, 1854, 1848. Hall, BCB 37 is a US ed. of 1850 = Toole Stott 119. Christopher 151 is a US version of 1850? NUC lists 9 versions, all included in Toole Stott. Toole Stott cites some BM copies, but I haven't found this in the BMC. A section of this, with some additional material, was reissued as Games of Skill and Conjuring: ..., in 1860, 1861, 1862, 1865, 1870 ©© see Toole Stott 312©317.] ÁÁÁÁI have now found that much of this, including all the material of interest, is taken directly from the 1843 Paris edition of Boy's Own Book, qv, by J. L. Williams, including many of the illustrations © indeed they have the same Frontispiece, with S. Williams' name on it. BR. c1305.ÁÁGreek MS, c1305, Codex Par. Suppl. Gr. 387, fol. 118v-140v. Transcribed, translated and annotated by Kurt Vogel as: Ein Byzantinisches Rechenbuch des frÀGÀhen 14.Jahrhunderts; Wiener Byzantinistische Studien, Band VI; Hermann BÀ?Àhlaus Nachf., Wien, 1968. I will cite problem numbers and pages from this ©© Vogel gives analysis of the methods on pp. 149-153 and historical comments on pp. 154-160, but I will not cite these. Brahmagupta, c628. See: Brahma-sphuta-siddhanta; Colebrooke. Brahma-sphuta-siddhanta. ÁÁÁÁÁÁBrÀÀhma-sphuta-siddhÀÀnta of Brahmagupta, 628 (see Colebrooke). He only states rules, which are sometimes obscure. It appears from Colebrooke, p. v, and Datta (op. cit. under Bakhshali, p. 10), that almost all the illustrative examples and all the solutions are due to Chaturveda PrthudakasvÀÀmÀ3À in 860. Brahmagupta's rules are sometimes so general that one would not recognise their relevance to these examples and I have often not cited Brahmagupta. E.g. cistern problems are given as examples to Brahmagupta's verse on how to add and subtract fractions. (See also Datta & Singh, I, p. 248.) Some of these comments are taken from Bhaskara I in 629. Brush.ÁÁÁÁHubert Phillips. Brush Up Your Wits. Dent, London, 1936. BSHM.ÁÁÁÁBritish Society for the History of Mathematics. The produce a useful Newsletter. Buteo. Logistica. 1559. ÁÁÁÁÁÁJohannes Buteo (= Jean Borrel, c1485©c1560 or c1492©1572). Ioan. Buteonis Logistica, quÀ%À & Arithmetica vulgÀAÀ dicitur in libros quinque digesta: quorum index summatim habetur in tergo. Gulielmus Rovillius, Lyons, 1559. Most of the material is in books IV and V. H&S cites some problems in the 1560 ed with the same pages as in the 1559 ed, so these editions are presumably identical. See Rara 292©294. c.ÁÁÁÁÁÁcirca, e.g. c1300. Also c= means "approximately equal", though ÀsÀ will be used in mathematical contexts. C.ÁÁÁÁÁÁCentury, e.g. 13C, ©5C. Calandri. Arimethrica. 1491. ÁÁÁÁÁÁPhilippo Calandri. Untitled. Frontispiece is labelled "Pictagoras arithmetrice introductor". Text begins: "Philippi Calandri ad nobilem et studiosus Julianum Laurentii MedicÀoÀ de arimethrica opusculÀÁÀ." Lorenzo de Morgiani & Giovanni Thedesco da Maganza, Florence, 1491. Van Egmond's Catalog 298©299. The Graves collect