Entertaining Mathematical Puzzles

by Martin Gardner

n selecting materials for this collection I have done my best to find puzzles that are unusual and entertaining, that call for only the most elementary knowledge of mathematics, but at the same time provide stimulating glimpses into higher levels of mathematical thinking. 

The puzzles (many of which appeared in my column "On the Light Side" that ran in Science World) have been grouped into sections, each dealing with a different area of mathematics. Brief comments at the beginning of each section suggest something of the nature and importance of the kind of mathematics one must use in tackling the puzzles of that section. In the answers, I have tried to go into as much detail as space permits in explaining how each problem is solved, and pointing out some of the inviting paths that wind away from the problems into lusher areas of the mathematical jungle. 

Perhaps in playing with these puzzles you will discover that mathematics is more delightful than you expected. Perhaps this will make you want to study the subject in earnest, or less hesitant about taking up the study of a science for which a knowledge of advanced mathematics will eventually be required. 

Surely no one today can doubt the enormous practical value of mathematics. Without its use as a tool, the discoveries and achievements of modern science would have been impossible. But many people do not realize that mathematicians actually enjoy mathematics. Take my word for it, there is as much satisfaction in knocking over an interesting problem with a well-aimed thought as there is in knocking over ten wooden pins with a well-aimed bowling ball. 

In one of L. Frank Baum's funniest fantasies, The Emerald City of Oz, Dorothy (together with the Wizard and her uncle and aunt) visit the city of Fuddlecumjig in the Qualding section of Oz. Its remarkable inhabitants, the Fuddles, are made of pieces of painted wood cleverly fitted together like three-dimensional jigsaw puzzles. As soon as an outsider approaches they scatter in a heap of disconnected pieces on the floor so that the visitors will have the pleasure of putting them together again. As Dorothy's party leaves the city, Aunt Em remarks:
     "Those are certainly strange people, but I really can't see what use they are, at all."
     "Why, they amused us for several hours," replies the Wizard. "That is being of use to us, I'm sure."
     "I think they're more fun than playing solitaire or mumbletypeg," Uncle Henry adds. "For my part, I'm glad we visited the Fuddles." 

I hope that you will resist mightily the temptation to look at the answer before you try seriously to work a problem. And I hope that when you finish with these puzzles you will be glad, like Uncle Henry, to have been befuddled by them.

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Last Update
August 28, 2003

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