Computer analysis of Sprouts with nimbers

Summary

Sprouts is a two-player topological game, invented in 1967 at the University of Cambridge by John Conway and Michael Paterson. The game starts with p spots, and ends in at most 3_p -1 moves. The first player who cannot play loses.

The complexity of the p-spot game is very high, so that the best hand-checked proof only shows who the winner is for the 7-spot game, and the best previous computer analysis reached p = 11.

We have written a computer program, using mainly two new ideas. The nimber (also known as Sprague–Grundy number) allows us to compute separately independent subgames; and when the exploration of a part of the game tree seems to be too difficult, we can manually force the program to search elsewhere. Thanks to these improvements, we have settled every case up to p = 32. The outcome of the 33-spot game is still unknown, but the biggest computed value is the 47-spot game! All the computed values support the Sprouts conjecture: the first player has a winning strategy if and only if p is 3, 4 or 5 modulo 6.

We have also used a check algorithm to reduce the number of positions needed to prove which player is the winner. It is now possible to hand-check all the games until p = 11 in a reasonable amount of time.

Sprouts is a two-player pencil-and-paper game invented in 1967 in the University of Cambridge by John Conway and Michael Paterson [Gardner 1967]. The game starts with p spots and players alternately connect the spots by drawing curves between them, adding a new spot on each curve drawn. A new curve cannot cross or touch any existing one, leading necessarily to a planar graph. The first player who cannot play loses.