Eighth preliminary edition

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Copyright ©2003 Professor David Singmaster

contact via http://puzzlemuseum.com

Last updated on 23 December 2016.
This 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.
A 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 .

If I have perchance omitted anything more or less proper or necessary, I beg indulgence, since there is no one who is blameless and utterly provident in all things. [Fibonacci, translated by Grimm.])


Recreational 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).

In 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.

Consequently, 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.

When 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 History of the Theory of Numbers. Like that work, the present work divides the subject into a number of topics and treats them chronologically.

I 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.

This 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.

A fuller description of this project in 1984-1985 is given in my article Some early sources in recreational mathematics, in: C. Hay et al., eds.; Mathematics from Manuscript to Print; Oxford Univ. Press, 1988, pp. 195 208. A more recent description is in my article: Recreational mathematics; in: Encyclopedia of the History and Philosophy of the Mathematical Sciences; ed. by I. Grattan-Guinness; Routledge & Kegan Paul, 1993; pp. 1568-1575.

Below I compare this work with Dickson and similar works and discuss the coverage of this work.

As already mentioned, the work which the present most resembles is Dickson's History of the Theory of Numbers.
The 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.
E. T. Bell. The Search for Truth. George Allen & Unwin, London, 1935, p. 131.
Dickson 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.

The best previous book covering the history of recreational mathematics is the second edition of Wilhelm Ahrens's Mathematische Unterhaltungen und Spiele 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.

Another similar book is the 4th edition of J. Tropfke's Geschichte der Elementarmathematik, 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.

Another book in the field is W. L. Schaaf's Bibliography of Recreational Mathematics, 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.

Collecting 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.

Another related bibliography is Santi's Bibliografia della Enigmistica, 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.

Other relevant bibliographies are listed in Section 3.B.

In selecting topics, I tend to avoid classical number theory and classical geometry. These are both pretty well known. Dickson's History of the Theory of Numbers and Leveque's and Guy's Reviews in Number Theory 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 Winning Ways and Frankel's Bibliography.

The 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.

I 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.

Some 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.

Recreational 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.

When 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.

Recently 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.

Some 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.

I 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.

In 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 Scripta Mathematica (but I have previously read about half of it), Schwenter-Harsdörffer's Deliciæ Physico Mathematicae, Schott's Joco-Seriorum and Murray's History of Board Games Other Than Chess. I have recently acquired the early issues of Eureka, but there are later issues that I have not yet read and they persist in not sending the current copies I have paid for!

I 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.

In 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: The Magician's Own Book (1857); Parlour Pastime, by "Uncle George" (1857); The Sociable (1858); The Boy's Own Toymaker, by Landells (1858); The Book of 500 Curious Puzzles (1859); The Secret Out (1859); Indoor and Outdoor Games for Boys and Girls (c1859); The Boy's Own Conjuring Book (1860); The Illustrated Boy's Own Treasury (1860, but see below); The Parlor Magician (1863); The Art of Amusing, by Bellew (1866); Parlour Pastimes (1868); Hanky Panky (1872); Within Doors, by Elliott (1872); Magic No Mystery (1876), just to name those that I know. Most of these are of uncertain authorship and went through several editions and versions. The Magician's Own Book, The Book of 500 Curious Puzzles, The Secret Out, The Sociable, The Parlor Magician, Hanky Panky, and Magic No Mystery 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 The Secret Out says it is based on Le Magicien des Salons. This is probably Le Magicien des Salons ou le Diable Couleur de Rose, 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: Le Magicien de Société, Delarue, Paris, c1860? and Le Manuel des Sorciers (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 Magician's Own Book (1871?) says it is translated from Le Magicien des Salons 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 Le Magicien des Salons 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.

Christopher 242 cites the following article on this series.

Charles L. Rulfs. Origins of some conjuring works. Magicol 24 (May 1971) 3-5. He discusses the various books, saying that Cremer essentially pirated the Dick & Fitzgerald productions. He says The Magician's Own Book draws on Wyman's Handbook (1850, ??NYS), Endless Amusement, Parlour Magic (by W. Clarke?, 1830s, ??NYS), Brewster's Natural Magic (??NYS). He says The Secret Out is largely taken, illustrations and all, from Blismon de Douai's Manuel du Magicien (1849, ??NYS) and Richard & Delion's Magicien des salons ou le diable couleur de rose (1857 and earlier, ??NYS).

Christopher 622 says Harold Adrian Smith [Dick and Fitzgerald Publishers; Books at Brown 34 (1987) 108-114] has studied this series and concludes that Williams was the author of Magician's Own Book, 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 The Secret Out book is taken from DeLion. He states that The Boy's Own Conjuring Book is a London pirate edition.

Several 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 The Family Friend, 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 The Illustrated Boy's Own Treasury 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 Boy's Own Book and Magician's Own Book. 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.

The 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 Weekly Dispatch 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.

The 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.

I 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.

I have prepared a CD containing this and much else of my material. I divided Sources 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.)

This 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.

Script 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.

Script 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!)

Reading 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.

However, this still leaves some problems. I use em dashes a bit, i.e. –, 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 Mathematical Reviews 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. ĭ, which is the transliteration of й 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.

It is very tedious to convert the underlines back to em dashes, so I will convert every em dash to a double hyphen --.

Finally, 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.

Later 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.

The 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!

The conversion to (Loco)Script provided many improved features compared to my earlier DEC versions. I am using an A4 page (8¼ by 11⅔ inches) rather than an 8½  by 11 inch page, which gives 60 lines of text per page, four more or 7% more than when using the DEC or VAX.

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