Mathematical Puzzles and Diversions

Martin Gardner

4 ideas

1. Arthur Stone's Trimmed Paper Becomes Hexaflexagon

   In 1939 Arthur Stone, a British graduate student at Princeton, trimmed his American notebook paper to fit English binders and idly folded the leftover strips. One strip folded into a hexagon that showed different faces when flexed, and this trihexaflexagon led Stone, Richard Feynman, John Tukey and Bryant Tuckerman to form a 'flexagon committee' that worked out its theory. Gardner's account of this episode was his first Scientific American column in December 1956.

2. Three-Card Swindle Exploits Miscounted Equiprobability

   Three cards are shown: one red on both sides, one white on both sides, and one red on one side and white on the other. A card is drawn and shows red, and the operator offers even money that the hidden side is also red. Gamblers reason there are two cards it could be, but there are three equally likely red faces, and two of them sit on the red-red card. The hidden side is therefore red with probability 2/3, and the operator profits from the victim's wrong choice of equally likely cases.

3. Strategy-Stealing Proves Hex Favors First Player

   Hex, invented independently by Piet Hein and John Nash, can never end in a draw. Suppose the second player had a winning strategy. The first player could make an arbitrary opening move and then follow that strategy, and an extra stone never hurts, which is a contradiction. So a first-player win must exist, yet on full-size boards the proof gives no clue what the winning strategy actually is.

4. Binary Parity Solves the Game of Nim

   Write each Nim pile size in binary and add each column without carrying (XOR). A position is a loss for the player about to move exactly when every column sum is even (zero). The winning strategy is always to move to such a zero position, which is always possible from a nonzero one, while any move from a zero position creates a nonzero one.

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