Sources in Recreational Mathematics

SOURCES - page 1

SOURCES IN RECREATIONAL MATHEMATICS

AN ANNOTATED BIBLIOGRAPHY

EIGHTH PRELIMINARY EDITION

DAVID SINGMASTER

Copyright ©2003 Professor David Singmaster

contact via http://puzzlemuseum.com

Last updated on 7 August 2013.

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.])

INTRODUCTION

NATURE OF THIS WORK

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 2000BC. 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. 195208. 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.

SIMILAR WORKS

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.

COVERAGE

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 adhoc 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.

STATUS OF THE PROJECT

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.