Computer analysis of Sprouts

Sprouts is a two-player pencil-and-paper game with a topological avor. It was invented in 1967 by Michael Paterson and John Conway, and was popularized by Martin Gardner in the Mathematical Games column of Scienti c American magazine [6]. We have written a computer program to analyze the n-spot game of Sprouts for general n. Our program uses a number of standard techniques to expedite adversary searches such as cutting o the search as soon as the value can be determined, and hashing previously evaluated positions. But the truly innovative feature is our representation of game positions, which provides enough information to generate moves and has the property that many di erent planar graphs collapse into the same representation. This has an enormous impact on the speed of the search. The complexity of n-spot Sprouts grows extremely rapidly with n. According to Gardner [7, page 7], Conway estimated that analysis of the eight-spot game was beyond the reach of present-day computers. Before our program, even the value of the seven-spot game was unknown; we have calculated the value of all games up to and including eleven spots. Our calculation supports the Sprouts Conjecture: The rst player loses if n is 0, 1 or 2 modulo 6 and wins otherwise.