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Cohen, who proved the opposite, both believe that Cantor's Continuum Hypothesis will ultimately be proven false. The questions remain, and they rest at the very foundations of mathematics. But what does all this have to do with mathematical games, puzzles, and amuse-ments? First of all, Zeno's paradoxes are among the most commonly included subject matter in recreational mathematics, and Cantor's theories serve as the basis for numerous problems included in recreational mathematics.
Dictionary Of Mathematical , Puzzles, And Amusements (Karan 007)
Furthermore, since the first third of the twentieth century, there has been a shift in recreational mathematics from less sophisticated amusements in number curiosities, mazes, simple geometric puzzles, arithmatic story problems, card tricks, magic squares, trisection of the angle, squaring the circle, duplication of the cube, Pythagorean triples, board games, and so on, to at times highly sophisticated explorations of number theory, game theory, graph theory, topological problems, flexagons, logical paradoxes, fallacies of logic, paradoxes of the infinite, and so on.
Mathematical games, puzzles, and amusements go back to the beginning of recorded history, many of them, such as Zeno' s paradoxes, reappearing time and time again throughout history.
- Dictionary of Mathematical Games, Puzzles, and Amusements.
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Little is known about the mathematical recrea-tions in existence during the so-called Dark Ages, but by the Middle Ages, notable names, such as Fibonacci Leonardo of Pisa; , who is famous for the Fibonacci series, and Rabbiben Ezra , appear. Gerolamo Cardan and Nicholas Tartaglia Niccolo Fontana; took part in a mathematical duel as the result of Cardan stealing Tartaglia's solution to a cubic equation. In the seventeenth century, books began appearing devoted solely to recrea-tional mathematics. Claude Gaspard Bachet de Meziriac wrote Problemes plaisans et delectables qui se font par les nombres , which went through five editions, the most recent in , and served as a source for similar collections.
Dictionary of Mathematical Games, Puzzles, and Amusements
It included card tricks, watch-dial puzzles, weight problems, and difficult crossings. It went through thirty editions before Claude Mydorge wrote Examen du livre des recreations mathematiques , and Denis Henrion wrote Les Recreations mathematiques avec l' examen de ses problemes en arithmetique, geometrie, mechanique, cosmographie. In Germany, Daniel Schwenter put together a collection of recreational problems based on Leurechon' s book. These were published after his death as Deliciae Physico-mathematicae oder Mathematische und Philosoph-ische Erquickstunden and were extremely popular.
Subsequently, an enlarged edition was published. In the eighteenth century, England began publishing mathematics recreations in earnest. The most important work of the time, however, was that of Jacques Ozanam, Recreations mathe-matiques et physiques , a four-volume set that went through numerous editions, eventually being translated into English by Charles Hutton , and again by Edward Riddle , The second half of the nineteenth century produced a number of important writers in recreational mathematics, the most famous being Lewis Carroll C.
But others, notably Edouard Lucas, author of Recreations mathemati-ques , a four-volume set, were also producing classic works in the field. These creators and collectors were the forerunners of contemporary recreational mathe-matics. Others, however, were also important. Rouse Ball published Mathematical Recreations and Essays in , using a more scholarly approach; it became a classic and is still a standard reference.
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Maurice Kraitchik, editor of Sphilu, published Mathematical Recreations , which has since been re-vised and remains a standard reference. Fred Schuh published Wonderlijke Prob-lemen; Leerzaam Tijdverdrijf Door Puzzle en Spel , another important text on mathematical recreations, translated and republished in English by Dover Publications in , once again, a more esoteric approach. These books, and others, reveal the shift to more sophisticated recreations. Steinhaus' Mathematical Snapshots , Joseph S.
Madachy's Mathematics. This brief history would not be complete without mentioning the excellent, continually updated bibliography of recreational mathematics put out by W. Schaaf, an indication of the popularity of the field today, as are the numerous magazines containing articles on recreational mathematics, for example, Ameri-can Mathematical Monthly, Arithmetic Teacher, Fibonacci Quarterly, Journal of Recreational Mathematics, Mathematics Magazine, Recreational Mathemat-ics Magazine, Scientific American, and Scripta Mathematica.
Game theory has recently become an important branch of mathematics. It attempts to analyze problems of conflict by abstracting common quantifiable strategic features. These features can then be reformulated in mathematical lan-guage as games, since they are patterned on such real games as bridge and poker.
The French mathematician Emile Borel first put forth a theory of games of strategy in However, it was John von Neumann who independently de-rived the cornerstone for the theory and presented it in This cornerstone is what is referred to as the Min-Max theorem, i. It is not necessary to go into the complex-ities of the theory here, but rather simply to point out that game theory opens up the door to a mathematical analysis of such wide-ranging concerns as sociology, psychology, politics, and war-certainly matters of serious concern, and all tied together with recreational mathematics-which leads me to the following pas-sage written by Henri Poincare directed to science, but also applicable to mathe-matics and even to recreational mathematics :.
Science keeps us in constant relation with something which is greater than ourselves; it offers us a spectacle which is constantly renewing itself and growing always more vast.
Behind the great vision it affords us, it leads us to guess at something greater still; this spectacle is a joy to us, but it is a joy in which we forget ourselves and thus it is morally sound. He who has tasted of this, who has seen, if only from afar, the splendid harmony of the natural laws will be better disposed than another to pay little attention to his petty, egoistic interests. He will have an ideal which he will value more than himself, and that is the only ground on which we can build an ethics.
He will work for this ideal without sparing himself and without expecting any of those vulgar rewards which are everything to some persons; and when he has assumed the habit of disinterestedness, this habit will follow him everywhere; his entire life will remain as if flavored with it. It is the love of truth even more than passion which inspires him. And is not such a love an entire code of morality? Is there anything which is more important than to combat lies because they are one of the most common vices in primitive man and one of the most degrading?
When we have acquired the habit of scientific methods, of their scru-pulous exactitude, of the horror of all attempts to deflect the course of experiment; when. And is this not the best means of acquiring the rarest, the most difficult of all sincerities, the one which consists in not deceiving oneself? These lofty realms truly are a part of recreational mathematics, not just in the paradoxes of Zeno, but in the joy to be found in Pascal's triangle or the discovery of a new knight's tour or the successful squaring of the square or the recent proof of the four-color map problem or the development of the Franklin squares or the discovery of an additional prime number or, perhaps, even a resolution to Can-tor's Theory of the Continuum, leading to a better understanding of the problems Zeno contemplated as he gazed over the beautiful country of ancient Greece.
This book does not offer a solution to Zeno's paradoxes or, for that matter, even a new knight's tour. Rather, it contains a collection of the various mathe-matical games, puzzles, and amusements that have led people to such new solutions.
The goal has been to bring together the creations of others. Certainly, not all of the mathematical recreations of all time are included, nor are all of the contemporary mathematical recreations included. That would be an impossible task. The attempt, rather, has been to collect a good number of recreations of various types.