"There remains one more game." "What is it?"

"Ennui," I said. "The easiest of all. No rules, no boards, no equipment." "What is Ennui?" Amanda asked. "Ennui is the absence of games."

—Donald Barthelme, Guilty Pleasures.

Unfortunately, as recent studies of education in this country have made clear, one of the chief characteristics of mathematical classes, especially on the lower levels of public education, is ennui. Some teachers may be poorly trained in mathematics and others not) trained at all. If mathematics bores them, can you blame their students for being bored?

Like science, mathematics is a kind of game that we play with the universe. The best mathematicians and the best teachers of mathematics obviously are those who both understand the rules of the game, and who relish the excitement of playing it. Raymond Smullyan, who has enormous zest for the games of philosophy and mathematics, once taught an elementary course in geometry. In his delightful book 5000 B.C. and Other Philosophical Fantasies (1983) he tells how he once introduced his students to the Pythagorean theorem:

I drew a right triangle on the board with squares on the hypotenuse and legs and said, "Obviously, the square on the hypotenuse has a larger area than either of the other two squares. Now suppose these three squares were made of beaten gold, and you were offered either the one large square or the two small squares. Which would you choose?"

Interestingly enough, about half the class opted for the one large square and half for the two small ones. A lively argument began. Both groups were equally amazed when told that it would make no difference.

It is this sense of surprise that all great mathematicians feel, and all great teac tie-rs of mathematics are able to communicate. I know of no better way to do this, especially for beginning students, than by way of games, puzzles, paradoxes, magic tricks, and all the other curious paraphernalia of "recreational mathematics."

"Puzzles and games provide a rich source of example problems useful for illustrating and testing problem-solving methods," wrote Nils Nilsson in his widely used textbook Problem-Solving Methods in Artificial Intelligence. He quotes Marvin Minsky: "It is not that the games and mathematical problems are chosen because they are clear and simple; rather it is that they give us, for- the smallest initial structures, the greatest complexity, so that one can engage some really formidable situations after a relatively minimal diversion into programming."

Nilsson and Minsky had in mind the value of recreational mathematics in learning how to solve problems by computers, but its value in learning how to solve problems by hand is just as great. In this book, the tenth collection of the Mathematical Games columns that I wrote for Scientific American, you will find an assortment of mathematical recreations of every variety. The last three chapters (the third was written especially for this volume) deal with John H. Conway's fantastic game of Life, the full wonders of which are still being explored.

The two previously published articles on Life, in which I had the privilege of introducing this game for the first time, aroused more interest among computer buffs around the world than any other columns I have written. Now that Life software is becoming available for home-computer screens, there has been a renewed interest in this remarkable recreation. Although Life rules are incredibly simple, the complexity of its structure is so awesome that no one can experiment with its "life forms" without being overwhelmed by a sense of the infinite range and depth and mystery of mathematical structure. Few have expressed this emotion more colorfully than the British-American mathematician James J. Sylvester:

Mathematics is not a book confined within a cover and bound between brazen clasps, whose contents it needs only patience to ransack; it is not a mine, whose treasures may take long to reduce into possession, but which fill only a limited number of veins and lodes; it is not a soil, whose fertility can be exhausted by the yield of successive harvests; it is not a continent or an ocean, whose area can be mapped out and its contour defined: it is limitless as that space which it finds too narrow for its aspirations; its possibilities are as infinite as the worlds which are forever crowding in and multiplying upon the astronomer's gaze; it is as incapable of being restricted within assigned boundaries or being reduced to definitions of permanent validity, as the consciousness, the life, which seems to slumber in each monad, in every atom of matter, in each leaf and bud and cell, and is forever ready to burst forth into new forms of vegetable and animal existence.

Martin Gardner

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