CPE 07 - Documento

the serious - e.g. Fermat's Last Theorem, the Four Col- THE UNREASONABLE UTILITY OF RECREA- our Theorem or the Mandelbrot Set.) TIONAL MATHEMATICS Secondly, recreational mathematics is mathe- by Prof. David Singmaster matics that is fun and used as either as a diversion from serious mathematics or as a way of making serious For First European Congress of Mathematics, Paris, mathematics understandable or palatable. These are the July, 1992. Amended on 24 Jan 1993 and 7 Sep 1993. pedagogic uses of recreational mathematics. They are already present in the oldest known mathematics and Les hommes ne sont jamais plus ing‚nieux que continue to the present day. dans l'invention des jeux. [Men are never more ingen- Mathematical recreations are as old as mathe- ious than in inventing games.] matics itself. The earliest piece of Egyptian mathemat- Leibniz to De Montmort, 29 Jul 1715. ics, the Rhind Papyrus of c-1800, has a problem (No. 79 - OHPs) where there are 7 houses, each house has 7 cats, Amusement is one of the fields of applied each cat ate 7 mice, each mouse would have eaten 7 ears mathematics. of spelt and each ear of spelt would produce 7 hekat of William F. White; A Scrap-Book of Elemen- spelt. Then 7 + 49 + 343 + 2401 + 16807 is computed. tary Mathematics; 1908. A similar problem of adding powers of 7 occurs in Fibo- nacci (1202), in a few later medieval texts and in the ... it is necessary to begin the Instruction of children's riddle rhyme quot;As I was going to St. Ivesquot;. Youth with the Languages and Mathematicks. These Despite the gaps in the history it is tempting to believe should ... be taught to-gether, the Languages and that quot;St. Ivesquot; is a descendent from the ancient Egyp- Classicks as ... Business and the Mathematicks as ... Di- tians. Though there is some question as to whether this version. Samuel Johnson, first President of Columbia problem is really a fanciful exercise in summing a geo- University, in 1731. metric progression, it has no connection with other problems in the papyrus and seems to be inserted as a diver- My title is a variation on Eugene Wigner's fa- sion or recreation. The earliest mathematical works from Babylo- mous essay 'The unreasonable effectiveness of mathe- nia also date from about -1800 and they include such matics in the physical sciences'. Like Wigner, I origi- problems as the following on AO 8862 (OHP) quot;I know nally did not come up with any explanation, but more the length plus the width of a rectangle is 27, while the recently I have begun to formulate an explanation. But area plus the difference of the length and the width is first let me describe the background and illustrate the 183. Find the length and width.quot; By no stretch of the situation. imagination can this be considered a practical problem - For a decade, I have been working to find rather it is a way of presenting two equations in two un- sources of classical problems in recreational mathemat- knowns which should make the problem more interest- ics. This has led to an annotated bibliography/history of ing for the student. the subject, now covering about 392 topics on about 456 These two aspects of recreational mathematics pages. (404 topics on 500 pp) - the popular and the pedagogic - overlap considerably and there is no clear boundary between them and quot;seri- 1. THE NATURE OF RECREATIONAL ousquot; mathematics. In addition, there are two other inde- MATHEMATICS pendent fields which contain much recreational mathematics: games and mechanical puzzles. To begin with, it is worth considering what is Games of chance and games of strategy also meant by recreational mathematics. An obvious defini- seem to be about as old as human civilization. The tion is that it is mathematics that is fun, but almost any mathematics of games of chance began in the Middle mathematician will say that he enjoys his work, even if Ages and its development by Fermat and Pascal in the he is studying eigenvalues of elliptic differential opera- 1650s rapidly led to probability theory and insurance tors, so this definition would encompass almost all companies based on this theory were founded in the mathematics and hence is too general. There are two, mid-18C. The mathematics of games of strategy only somewhat overlapping, definitions that cover most of started about the beginning of the 20th century, but soon what is meant by recreational mathematics. developed into game theory. First, recreational mathematics is mathematics Mechanical puzzles range widely in mathe- that is fun and popular - that is, the problems should be matical content. Some only require a certain amount of understandable to the interested layman, though the solu- dexterity; others require ingenuity and logical thought; tions may be harder. (However, if the solution is too while others require systematic application of mathe- hard, this may shift the topic from recreational toward

B. quot;A good problem is worth a thou- matical ideas or patterns, such as Rubik's Cube, the Chi- sand exercises.quot; There is no greater learning experience nese Rings, the Tower of Hanoi, Rubik's Clock. than trying to solve a good problem. Recreational The creation of beauty often leads to questions mathematics provides many such problems and almost of symmetry and geometry which are studied for their every problem can be extended or amended. Hence rec- own sake - e.g. the carved stone balls. reational mathematics is also a treasury of problems for This outlines the conventional scope of recrea- student investigations. tional mathematics, but there is some variation due to C. Because of its long history, recrea- personal taste. tional mathematics is an ideal vehicle for communicat- ing historical and multicultural aspects of mathematics. 2. THE UTILITY OF RECREATIONAL MATHEMATICS Fourthly, recreational mathematics is very useful to the historian of mathematics. Recreational prob- How is recreational mathematics useful? lems often are of great age and usually can be clearly recognised, they serve as useful historical markers, trac- Firstly, recreational problems are often the ba- ing the development and transmission of mathematics sis of serious mathematics. The most obvious fields are (and culture in general) in place and time. The Chinese probability and graph theory where popular problems Remainder Theorem, Magic Squares, the Cistern Prob- have been a major (or the dominant) stimulus to the lem and the Hundred Fowls Problem are excellent ex- creation and evolution of the subject. Further reflection amples of this process. (The original Hundred Fowls shows that number theory, topology, geometry and alge- problem, from 5th century China, has a man buying 100 bra have been strongly stimulated by recreational prob- fowls for 100 cash, roosters cost 5, hens 3 and chicks are lems. (Though geometry has its origins in practical sur- 3 for a cash - how many of each did he buy?) The num- veying, the Greeks treated it as an intellectual game and ber of topics which have their origins in China or India much of their work must be considered as recreational in is surprising and emphasises our increasing realisation nature, although they viewed it more seriously as reflect- that modern algebra and arithmetic derive more from ing the nature of the world. From the time of the Baby- Babylonia, China, India and the Arabs than from Greece. lonians, algebraists tried to solve cubic equations, though they had no practical problems which led to 3. SOME EXAMPLES OF USEFUL REC- cubics.) There are even recreational aspects of calculus - REATIONAL MATHEMATICS e.g. the many curves studied since the 16C. Conse- quently the study of recreational topics is necessary to In this section I will outline a number of ex- understanding the history of many, perhaps most, topics amples to show how recreational mathematics has been in mathematics. useful. (I will stretch recreational a bit to include some Secondly, recreational mathematics has fre- other non-practical topics.) quently turned up ideas of genuine but non-obvious util- A. Perhaps the most obvious example is ity. I will run through examples of these later. the theory of probability and statistics which grew from Such unusual developments, and the more the analysis of gambling bets to the basis of the straightforward developments of the previous paragraph, insurance industry in the 17th and 18th centuries. Much demonstrate the historical principle of quot;The unreason- of combinatorics likewise has its roots in gambling prob- able utility of recreational mathematicsquot;. This and simi- lems. The theory of Latin squares began as a recreation lar ideas are the historical and social justification of but has become an important technique in experimental mathematical research. design. Thirdly, recreational mathematics has great B. Greek geometry, though it had some pedagogic utility. basis in surveying, was largely an intellectual exercise, A. Recreational mathematics is a treas- pursued for its own sake. The conic sections were de- ury of problems which make mathematics fun. These veloped with no purpose in mind, but 2000 years later problems have been tested by generations going back to turned out to be just what Kepler and Newton needed about 1800 BC. In medieval arithmetic texts, recrea- and which now takes men to the moon. tional questions are interspersed with more straightfor- The regular, quasi-regular and Archimedean ward problems to provide breaks in the hard slog of polyhedra were developed long before they became the learning. These problems are often based on reality, basis of molecular structures. Indeed, the regular solids though with enough whimsey so that they have appealed are now known to be prehsitoric. Very recently, chem- to students and mathematicians for years. They illustrate ists have become excited about 'Bucky Balls', carbon the idea that quot;Mathematics is all around you - you only structures in various polyhedral shapes, of which the ar- have to look for it.quot;

have discovered that DNA molecules form into closed chetype is the truncated icosahedron, with 60 carbon at- chains which may be knotted, or not knotted. oms at the vertices. Such molecules apparently are the The M”bius strip arose about 1858 in work by basis for the formation of soot particles in the air. The both M”bius and Listing, Listing being apparently a bit idea of making such molecules apparently originated earlier, though a five twist strip may occur in Roman with David Jones, the scientific humorist who writes as mosaics. (OHPs) By 1890, it was already being used as 'Daedalus', in one of his humour columns. Somewhat a magic trick - magic being another application of further in the past, I recall that chemists produced cu- mathematics - indeed some people view all mathematics bane and dodecane - hydrocarbons in the shape of a cube as magic! More recently, such strips have served as the and a dodecahedron. basis of works by M. C. Escher - art being yet another C. Non-Euclidean geometry was devel- application of mathematics. The M”bius strip has also oped long before Einstein considered it as a possible ge- been patented several times! - e.g. as a single-sided ometry for space. D. The problem of the Seven Bridges of conveyor belt which has double the wearing surface. K”nigsberg (OHP), mazes, knight's tours, circuits on the (OHPs) None of the patents that I have seen make any dodecahedron (Icosian Game) (OHP from 2nd reference to any previous occurrence of the concept. lecture) were major sources of graph theory and are the Gardner says it has also been patented as a non-inductive basis of major fields of optimization, leading on to one resistor. Those with dot matrix printers, etc., may (or of the major unsolved problems of the century: NP = P?? may not) know that printer ribbons commonly have a The routes of postmen, streetsweepers and snowplows, twist so they are M”bius strips in order to allow the as well as salesmen are worked out by these methods. printer to use both edges. I first discovered this when I Further, Hamilton's thoughts on the Icosian Game led found one of our technicians trying to put such a ribbon him to the first presentation of a group by generators and back into its cartridge - he had done it several times and relations. (OHP) it kept coming out twisted which he thought was his mis- E. Number theory is another of the take! I. In combinatorics, the pattern of the fields where recreations have been a major source of Chinese Rings puzzle is the binary coding known as the problems and these problems have been a major source Gray Code, patented as an error-minimising code for modern algebra. Fermat's Last Theorem lead to by Frank Gray of Bell Labs in 1953 and already used in Kummer's invention of ideals and most of algebraic the same way by Baudot in the 1870s. number theory. There was a famous application of I would like to present another binary coding primitive roots to the splicing of telephone cables. Pri- which Baudot utilized. Chain codes = memory wheels. mality and factorization were traditionally innocuous recreational pastimes, but since 1978 when Rivest, THE PENROSE PIECES Shamir and Adleman introduced their method of public- key cryptography, my friends in this field get rung up by Penrose's Pieces have led to the discovery of a reporters wanting to know if the national security is new kind of solids - the 'quasicrystals'. threatened. The factorization of a big number or the de- I will only sketch the ideas here, with some termination of the next Mersenne prime are generally references. front page news now. South Bank Polytechnic's coat of arms in- F. A major impetus for algebra has cluded 'the net of half a dodecahedron', i.e. a pentagon been the solving of equations. The Babylonians already surrounded by five other pentagons. (OHP) One of the gave quadratic problems where the area of a rectangle basic results of crystallography is that no crystal struc- was added to the difference between the length and the ture can have five-fold symmetry. In 1973, I wrote to width. This clearly had no practical significance. Simi- Roger Penrose on a Polytechnic letterhead which shows lar impractical problems led to cubic equations and the the half dodecahedron. Penrose had long been interested eventual solution of the cubic. Negative solutions first in tiling the plane with pieces that could not tile the become common in medieval puzzle problems about plane periodically and the letterhead inspired him to try men buying a horse or finding a purse. Galois fields and even polynomials over them to fill the plane with pentagons and other related shapes. are now standard tools for cryptographers. He soon found such a tiling with six kinds of shape G. Even in analysis, the study of curves (OHP) and then managed to reduce it to two shapes (e.g. the cycloid) had some recreational motivation. which could tile the plane in uncountably many ways, H. Topology has much of its origins in but in no periodic way. (OHP) Some of the tilings have a five-fold centre of symmetry, recreational aspects of curves and surfaces. Knots, an- and all have a sort of generalised five-fold symmetry. other field once generally considered of no possible use, They are now called 'quasicrystals'. These tilings fasci- are now of great interest to molecular biologists who

columns for about 15 years and then monthly columns nated both geometers and crystallographers and were for about 20 years. Martin Gardner's columns were a extensively studied from the mid-1970s. Penrose's 'kites major factor in the popularity of Scientific American and and darts' shapes were simplified further to 'fat and thin probably inspired more students to study mathematics rhombuses' (OHP) and extended to three dimensions than any other influence. I have heard that circulation where they are related to the rhombic triacontahedron dropped significantly when he retired. Other major (OHPs). Though the tilings are not periodic, they have names in the field are the following. In English: Lewis quasi-axes and quasi-planes, which can cause diffrac- Carroll, Sam Loyd, Professor Hoffmann, Hubert Phil- tion. (OHPs) Using these, crystallographers determined lips, Tom O'Beirne, Douglas Barnard. In German: the diffraction pattern which a hypothetical quasicrystal Wilhelm Ahrens, Hermann Schubert, Walther would produce - it has a ten-fold centre of symmetry. In 1984, such diffraction patterns were discovered by Lietzmann. In French: douard Lucas, Pierre Berloquin. Shechtman in a sample of rapidly cooled alloy now [I am now trying to carry on this tradition by contribut- known as Shechtmanite and some 20 substances are now ing to the Daily Telegraph and the new magazine Fo- known to have quasi-crystalline forms. Indeed, exam- cus.] ples were found about 30 years earlier but the diffraction There really is considerable interest in mathe- patterns were discarded as being erroneous! It is not yet matics out there and if we enjoy our subject, it should be known whether such materials will be useful but they our duty and our pleasure to try to encourage and feed may be harder or stronger than other forms of the alloys this interest. Indeed, it may be necessary for our self- and hence may find use on aeroplanes, rockets, etc. So a preservation. mathematical flight of fancy has led to the discovery of a new kind of matter on which we may be flying in the WHY IS RECREATIONAL MATHEMAT- future! ICS SO USEFUL? [See Scientific American for January 1979 and August 1986 for expositions of this topic.] As I said earlier, I have only a tentative answer If there is time, I will cover the following as a to this, but it also partly answers Wigner's question. further utility. Mathematics has been described as a search for pattern - An additional utility of recreational mathemat- and that certainly describes much of what we do and also ics is that it provides us a way to communicate mathe- much of what most scientists do. But how do we find matical ideas to the public at large. Mathematicians tend patterns? The real world is messy and patterns are diffi- to underestimate the public interest in mathematics. cult to see. As we begin to see a pattern, we tend to re- [Lee Dembart of the Los Angeles Times wrote that when move all the inessential details and get to an ideal or he told people he was going to a conference on recrea- model situation. These models may be so removed from tional mathematics, they replied that it was a contradic- reality that they become fanciful or even recreational. tion in terms! And we all know the social situation E.g. physicists deal with frictionless perfectly elastic when you confess that you are a mathematician and the particles, weightless strings, ideal gases, etc. Then such response is quot;Oh. I was never any good at maths.quot;] Yet models get modified and adapted into a large variety of somewhere approaching 200 million Rubik Cubes were models. Now one of the ways in which a science pro- sold in three years! Indeed there have been more Rubik gresses is by seeing analogies between reality and sim- Cubes sold in Hungary than there are people. The best pler situations. E.g. the idea of the circulation of the known example of a best-selling game is Monopoly blood could not be developed until the idea of a pump which has taken 50 years to sell about 90 million exam- was known and somewhat understood. The behaviour of ples. a real system cannot be developed until one can see sim- Another measure of the popularity of recrea- pler models within it. But what are these simpler mod- tional mathematics is the number of books that appear in els? They are generally among the large variety of mod- the field each year - perhaps 50 in English alone. The els which have been created in the past, often recrea- long term best-seller in English must be Ball's Mathe- tional or fanciful. Perhaps the clearest example is graph matical Recreations and Essays now in its 101st year and theory, where Euler made a simple model of the reality its 13th edition. It has rarely been out of print in that that he was studying, then later workers found that time. And there are many older books, such as Bachet's model useful in other situations. Thus recreational book of 1612 which had three editions in the late 19C, mathematics helps as a major source of mathematical the last of which has been reprinted several times in this models, which are the raw material for mathematical re- century. search Many newspapers and professional magazines run regular mathematical puzzles, though this was more common in the past. Henry Dudeney published weekly

CPE 07 - Documento

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