Two on a Match

Early in the murky ravelings of the current movie Last Year at Marienbad (TIME, March 16) comes a scene in which the cadaverous "X" invites the importunate "M" to play a little game of matchsticks. With insouciant deliberation "X" lays out 16 matches in four rows on a table top--seven in the top row, five in the next, three next and one alone. He explains that they will take turns picking up the matches; each may take as few or as many as he wishes (even a whole row), but all must be taken from the same row. The player who picks up the last remaining match loses. "X" wins--and wins every time the game is played. He will always win, he points out.

Last week the Marienbad game was popping up at cocktail parties (with colored toothpicks), on commuter trains (with paper matches), in offices (paper clips) and in bars (with swizzle sticks). Only two can play, but any number can kibitz--and everyone, it seems, has a system for duplicating "X's" talent for winning.

Actually the Marienbad match game is a variation of one of the most ancient of all two-person mathematical divertissements. Originating in China around 3000 B.C., it was given the name Nim by Harvard Mathematician Charles Leonard Bouton, who found, in 1901, that a strategy using move combinations based on binary numbers would make anyone a winner. All the successful player has to do is memorize them--if he can.

In fact, Nim is more of a trap than a game. The canny con man, with all the possible combinations locked in his head, graciously allows his victim to go first (see diagram). Since Nim's starting setup (7-5-3-1) is a winning combination itself, whoever tampers with it (i.e., the player who makes the first move) is doomed. But even if the wily match-sharp, out of courtesy or cunning, should agree to move first in an occasional game, he can still save the day by resorting to the memorized combinations as soon as the proper situation presents itself.

Many players have developed even simpler, if less foolproof systems of their own, based either on hunches or intuition. One nimble Nim player moves swiftly to reduce the rows of matches into either an odd number of rows each containing an unequal number of matches, or into an even number of rows each containing an equal number of matches. Says Bosley Crowther, Marienbad-applauding motion picture critic of the New York Times: "Once I get the other guy to make the first move, I remove even numbers of matches until he loses--almost always--unless he is playing the same rules." Winning the game is good for a free drink in most bars; in some spots matches are dispensed with, and shot-glasses of bourbon are used as pawns, each being downed as it is removed. This frequently results in the players' being removed also before the game is finished.

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