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# CPE 07 - Documento

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The unreasonable utility of recreational mathematics - Documento del Prof. Renato Galicia Brito en el marco del 3º Congreso Provincial de Educación desarrollado los días 18, 19 y 20 de Julio de 2007 en la ciudad de Trelew, Chubut bajo la temática "Calidad Educativa: Un Proceso de Construcción Conjunta."

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### CPE 07 - Documento

1. 1. 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 prob- lems 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 mathe- matics: 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
2. 2. 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 use- ful 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;