[HOME - BASE Cinque - Appunti di Matematica ricreativa]
by Prof. David Singmaster
Computing, Information Systems and Mathematics 87 Rodenhurst Road South Bank University London, SW4 8AF, England London, SE1 0AA, England Tel/fax: 0181-674 3676 Tel: 0171-815 7411 Fax: 0171-815 7499 E-mail: ZINGMAST@VAX.SBU.AC.UK
For First European Congress of Mathematics, Paris, July,
1992. Amended on 24 Jan 1993 and 7 Sep 1993.
Les hommes ne sont jamais plus ingenieux que dans l'invention des jeux.
[Men are never more ingenious than in inventing games.]
Leibniz to De Montmort, 29 Jul 1715.
Amusement is one of the fields of applied mathematics.
William F. White; A Scrap-Book of Elementary Mathematics; 1908.
... it is necessary to begin the Instruction of Youth with the Languages and Mathematicks.
These should ... be taught to-gether, the Languages and Classicks as ... Business and the Mathematicks as ... Diversion.
Samuel Johnson, first President of Columbia University, in 1731.
My title is a variation on Eugene Wigner's famous essay 'The unreasonable effectiveness of mathematics in the physical sciences'. Like Wigner, I originally did not come up with any explanation, but more recently I have begun to formulate an explanation. But first let me describe the background and illustrate the situation.
For a decade, I have been working to find sources of classical problems in recreational mathematics. This has led to an annotated bibliography/history of the subject, now covering about 392 topics on about 456 pages. (404 topics on 500 pp)
1. THE NATURE OF RECREATIONAL MATHEMATICS
To begin with, it is worth considering what is meant by recreational mathematics. An obvious definition is that it is mathematics that is fun, but almost any mathematician will say that he enjoys his work, even if he is studying eigenvalues of elliptic differential operators, so this definition would encompass almost all mathematics and hence is too general. There are two, somewhat overlapping, definitions that cover most of what is meant by recreational mathematics. First, recreational mathematics is mathematics that is fun and popular - that is, the problems should be understandable to the interested layman, though the solutions may be harder. (However, if the solution is too hard, this may shift the topic from recreational toward the serious - e.g. Fermat's Last Theorem, the Four Colour Theorem or the Mandelbrot Set.)
Secondly, recreational mathematics is mathematics that is fun and used as either as a diversion from serious mathematics or as a way of making serious mathematics understandable or palatable. These are the pedagogic uses of recreational mathematics. They are already present in the oldest known mathematics and continue to the present day.
Mathematical recreations are as old as mathematics itself. The earliest piece of Egyptian mathematics, the Rhind Papyrus of c-1800, has a problem (No. 79 - OHPs) where there are 7 houses, each house has 7 cats, each cat ate 7 mice, each mouse would have eaten 7 ears of spelt and each ear of spelt would produce 7 hekat of spelt. Then 7 + 49 + 343 + 2401 + 16807 is computed. A similar problem of adding powers of 7 occurs in Fibonacci (1202), in a few later medieval texts and in the children's riddle rhyme "As I was going to St. Ives". Despite the gaps in the history it is tempting to believe that "St. Ives" is a descendent from the ancient Egyptians. Though there is some question as to whether this problem is really a fanciful exercise in summing a geometric progression, it has no connection with other problems in the papyrus and seems to be inserted as a diversion or recreation.
The earliest mathematical works from Babylonia also date from about -1800 and they include such problems as the following on AO 8862 (OHP) "I know the length plus the width of a rectangle is 27, while the area plus the difference of the length and the width is 183. Find the length and width." By no stretch of the imagination can this be considered a practical problem - rather it is a way of presenting two equations in two unknowns which should make the problem more interesting for the student.
These two aspects of recreational mathematics - the popular and the pedagogic - overlap considerably and there is no clear boundary between them and "serious" mathematics. In addition, there are two other independent fields which contain much recreational mathematics: games and mechanical puzzles. Games of chance and games of strategy also seem to be about as old as human civilization. The mathematics of games of chance began in the Middle Ages and its development by Fermat and Pascal in the 1650s rapidly led to probability theory and insurance companies based on this theory were founded in the mid-18C. The mathematics of games of strategy only started about the beginning of the 20th century, but soon developed into game theory.
Mechanical puzzles range widely in mathematical content. Some only require a certain amount of dexterity; others require ingenuity and logical thought; while others require systematic application of mathematical ideas or patterns, such as Rubik's Cube, the Chinese Rings, the Tower of Hanoi, Rubik's Clock. The creation of beauty often leads to questions of symmetry and geometry which are studied for their own sake - e.g. the carved stone balls.
This outlines the conventional scope of recreational mathematics, but there is some variation due to personal taste.
2. THE UTILITY OF RECREATIONAL MATHEMATICS
How is recreational mathematics useful?
Firstly, recreational problems are often the basis of serious mathematics. The most obvious fields are probability and graph theory where popular problems have been a major (or the dominant) stimulus to the creation and evolution of the subject. Further reflection shows that number theory, topology, geometry and algebra have been strongly stimulated by recreational problems. (Though geometry has its origins in practical surveying, the Greeks treated it as an intellectual game and much of their work must be considered as recreational in nature, although they viewed it more seriously as reflecting the nature of the world. From the time of the Babylonians, algebraists tried to solve cubic equations, though they had no practical problems which led to cubics.) There are even recreational aspects of calculus - e.g. the many curves studied since the 16C. Consequently the study of recreational topics is necessary to understanding the history of many, perhaps most, topics in mathematics.
Secondly, recreational mathematics has frequently turned up ideas of genuine but non-obvious utility. I will run through examples of these later.
Such unusual developments, and the more straightforward developments of the previous paragraph, demonstrate the historical principle of "The unreasonable utility of recreational mathematics". This and similar ideas are the historical and social justification of mathematical research.
Thirdly, recreational mathematics has great pedagogic utility.
A. Recreational mathematics is a treasury of problems which make mathematics fun. These problems have been tested by generations going back to about 1800 BC. In medieval arithmetic texts, recreational questions are interspersed with more straightforward problems to provide breaks in the hard slog of learning. These problems are often based on reality, though with enough whimsey so that they have appealed to students and mathematicians for years. They illustrate the idea that "Mathematics is all around you - you only have to look for it." B. "A good problem is worth a thousand exercises." There is no greater learning experience than trying to solve a good problem. Recreational mathematics provides many such problems and almost every problem can be extended or amended. Hence recreational mathematics is also a treasury of problems for student investigations.
C. Because of its long history, recreational mathematics is an ideal vehicle for communicating historical and multicultural aspects of mathematics.
Fourthly, recreational mathematics is very useful to the historian of mathematics. Recreational problems often are of great age and usually can be clearly recognised, they serve as useful historical markers, tracing the development and transmission of mathematics (and culture in general) in place and time. The Chinese Remainder Theorem, Magic Squares, the Cistern Problem and the Hundred Fowls Problem are excellent examples of this process. (The original Hundred Fowls problem, from 5th century China, has a man buying 100 fowls for 100 cash, roosters cost 5, hens 3 and chicks are 3 for a cash - how many of each did he buy?) The number of topics which have their origins in China or India is surprising and emphasises our increasing realisation that modern algebra and arithmetic derive more from Babylonia, China, India and the Arabs than from Greece.
3. SOME EXAMPLES OF USEFUL RECREATIONAL MATHEMATICS
In this section I will outline a number of examples to show how recreational mathematics has been useful. (I will stretch recreational a bit to include some other non-practical topics.)
A. Perhaps the most obvious example is the theory of probability and statistics which grew from the analysis of gambling bets to the basis of the insurance industry in the 17th and 18th centuries. Much of combinatorics likewise has its roots in gambling problems. The theory of Latin squares began as a recreation but has become an important technique in experimental design.
B. Greek geometry, though it had some basis in surveying, was largely an intellectual exercise, pursued for its own sake. The conic sections were developed with no purpose in mind, but 2000 years later turned out to be just what Kepler and Newton needed and which now takes men to the moon.
The regular, quasi-regular and Archimedean polyhedra were developed long before they became the basis of molecular structures. Indeed, the regular solids are now known to be prehsitoric. Very recently, chemists have become excited about 'Bucky Balls', carbon structures in various polyhedral shapes, of which the archetype is the truncated icosahedron, with 60 carbon atoms at the vertices. Such molecules apparently are the basis for the formation of soot particles in the air. The idea of making such molecules apparently originated with David Jones, the scientific humorist who writes as 'Daedalus', in one of his humour columns. Somewhat further in the past, I recall that chemists produced cubane and dodecane - hydrocarbons in the shape of a cube and a dodecahedron.
C. Non-Euclidean geometry was developed long before Einstein considered it as a possible geometry for space.
D. The problem of the Seven Bridges of K"nigsberg (OHP), mazes, knight's tours, circuits on the dodecahedron (Icosian Game) (OHP from 2nd lecture) were major sources of graph theory and are the basis of major fields of optimization, leading on to one of the major unsolved problems of the century: NP = P?? The routes of postmen, streetsweepers and snowplows, as well as salesmen are worked out by these methods. Further, Hamilton's thoughts on the Icosian Game led him to the first presentation of a group by generators and relations. (OHP)
E. Number theory is another of the fields where recreations have been a major source of problems and these problems have been a major source for modern algebra. Fermat's Last Theorem lead to Kummer's invention of ideals and most of algebraic number theory. There was a famous application of primitive roots to the splicing of telephone cables. Primality and factorization were traditionally innocuous recreational pastimes, but since 1978 when Rivest, Shamir and Adleman introduced their method of public-key cryptography, my friends in this field get rung up by reporters wanting to know if the national security is threatened. The factorization of a big number or the determination of the next Mersenne prime are generally front page news now.
F. A major impetus for algebra has been the solving of equations. The Babylonians already gave quadratic problems where the area of a rectangle was added to the difference between the length and the width. This clearly had no practical significance. Similar impractical problems led to cubic equations and the eventual solution of the cubic. Negative solutions first become common in medieval puzzle problems about men buying a horse or finding a purse.
Galois fields and even polynomials over them are now standard tools for cryptographers.
G. Even in analysis, the study of curves (e.g. the cycloid) had some recreational motivation.
H. Topology has much of its origins in recreational aspects of curves and surfaces. Knots, another field once generally considered of no possible use, are now of great interest to molecular biologists who have discovered that DNA molecules form into closed chains which may be knotted, or not knotted. The M"bius strip arose about 1858 in work by both M"bius and Listing, Listing being apparently a bit earlier, though a five twist strip may occur in Roman mosaics. (OHPs) By 1890, it was already being used as a magic trick - magic being another application of mathematics - indeed some people view all mathematics as magic! More recently, such strips have served as the basis of works by M. C. Escher - art being yet another application of mathematics. The M"bius strip has also been patented several times! - e.g. as a single-sided conveyor belt which has double the wearing surface. (OHPs) None of the patents that I have seen make any reference to any previous occurrence of the concept. Gardner says it has also been patented as a non-inductive resistor. Those with dot matrix printers, etc., may (or may not) know that printer ribbons commonly have a twist so they are M"bius strips in order to allow the printer to use both edges. I first discovered this when I found one of our technicians trying to put such a ribbon back into its cartridge - he had done it several times and it kept coming out twisted which he thought was his mistake!
I. In combinatorics, the pattern of the Chinese Rings puzzle is the binary coding known as the Gray Code, patented as an error-minimising code by Frank Gray of Bell Labs in 1953 and already used in the same way by Baudot in the 1870s.
I would like to present another binary coding which Baudot utilized. Chain codes = memory wheels.
THE PENROSE PIECES
Penrose's Pieces have led to the discovery of a new kind of solids - the 'quasicrystals'.
I will only sketch the ideas here, with some references.
South Bank Polytechnic's coat of arms included 'the net of half a dodecahedron', i.e. a pentagon surrounded by five other pentagons. (OHP) One of the basic results of crystallography is that no crystal structure can have five-fold symmetry. In 1973, I wrote to Roger Penrose on a Polytechnic letterhead which shows the half dodecahedron. Penrose had long been interested in tiling the plane with pieces that could not tile the plane periodically and the letterhead inspired him to try to fill the plane with pentagons and other related shapes. He soon found such a tiling with six kinds of shape (OHP) and then managed to reduce it to two shapes which could tile the plane in uncountably many ways, but in no periodic way. (OHP) Some of the tilings have a five-fold centre of symmetry, and all have a sort of generalised five-fold symmetry. They are now called 'quasicrystals'. These tilings fascinated both geometers and crystallographers and were extensively studied from the mid-1970s. Penrose's 'kites and darts' shapes were simplified further to 'fat and thin rhombuses' (OHP) and extended to three dimensions where they are related to the rhombic triacontahedron (OHPs). Though the tilings are not periodic, they have quasi-axes and quasi-planes, which can cause diffraction. (OHPs) Using these, crystallographers determined the diffraction pattern which a hypothetical quasicrystal would produce - it has a ten-fold centre of symmetry. In 1984, such diffraction patterns were discovered by Shechtman in a sample of rapidly cooled alloy now known as Shechtmanite and some 20 substances are now known to have quasi-crystalline forms. Indeed, examples were found about 30 years earlier but the diffraction patterns were discarded as being erroneous! It is not yet known whether such materials will be useful but they may be harder or stronger than other forms of the alloys and hence may find use on aeroplanes, rockets, etc. So a mathematical flight of fancy has led to the discovery of a new kind of matter on which we may be flying in the future!
[See Scientific American for January 1979 and August 1986 for expositions of this topic.]
If there is time, I will cover the following as a further utility.
An additional utility of recreational mathematics is that it provides us a way to communicate mathematical ideas to the public at large. Mathematicians tend to underestimate the public interest in mathematics. [Lee Dembart of the Los Angeles Times wrote that when he told people he was going to a conference on recreational mathematics, they replied that it was a contradiction in terms! And we all know the social situation when you confess that you are a mathematician and the response is "Oh. I was never any good at maths."] Yet somewhere approaching 200 million Rubik Cubes were sold in three years! Indeed there have been more Rubik Cubes sold in Hungary than there are people. The best known example of a best-selling game is Monopoly which has taken 50 years to sell about 90 million examples. Another measure of the popularity of recreational mathematics is the number of books that appear in the field each year - perhaps 50 in English alone. The long term best-seller in English must be Ball's Mathematical Recreations and Essays now in its 101st year and its 13th edition. It has rarely been out of print in that time. And there are many older books, such as Bachet's book of 1612 which had three editions in the late 19C, the last of which has been reprinted several times in this century.
Many newspapers and professional magazines run regular mathematical puzzles, though this was more common in the past. Henry Dudeney published weekly columns for about 15 years and then monthly columns for about 20 years. Martin Gardner's columns were a major factor in the popularity of Scientific American and probably inspired more students to study mathematics than any other influence. I have heard that circulation dropped significantly when he retired. Other major names in the field are the following. In English: Lewis Carroll, Sam Loyd, Professor Hoffmann, Hubert Phillips, Tom O'Beirne, Douglas Barnard. In German: Wilhelm Ahrens, Hermann Schubert, Walther Lietzmann. In French: Edouard Lucas, Pierre Berloquin. [I am now trying to carry on this tradition by contributing to the Daily Telegraph and the new magazine Focus.] There really is considerable interest in mathematics out there and if we enjoy our subject, it should be our duty and our pleasure to try to encourage and feed this interest. Indeed, it may be necessary for our self-preservation.
WHY IS RECREATIONAL MATHEMATICS SO USEFUL?
As I said earlier, I have only a tentative answer to this, but it also partly answers Wigner's question. Mathematics has been described as a search for pattern - and that certainly describes much of what we do and also much of what most scientists do. But how do we find patterns? The real world is messy and patterns are difficult to see. As we begin to see a pattern, we tend to remove all the inessential details and get to an ideal or model situation. These models may be so removed from reality that they become fanciful or even recreational. E.g. physicists deal with frictionless perfectly elastic particles, weightless strings, ideal gases, etc. Then such models get modified and adapted into a large variety of models. Now one of the ways in which a science progresses is by seeing analogies between reality and simpler situations. E.g. the idea of the circulation of the blood could not be developed until the idea of a pump was known and somewhat understood. The behaviour of a real system cannot be developed until one can see simpler models within it. But what are these simpler models? They are generally among the large variety of models which have been created in the past, often recreational or fanciful. Perhaps the clearest example is graph theory, where Euler made a simple model of the reality that he was studying, then later workers found that model useful in other situations. Thus recreational mathematics helps as a major source of mathematical models, which are the raw material for mathematical research.
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