Computer analysis of Sprouts with nimbers

• Published in 2010
• Added on 2012-04-21

Sprouts is a two-player topological game, invented in 1967 in the University of Cambridge by John Conway and Michael Paterson. The game starts with p spots, and ends in at most 3p-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 reached 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.

Links

• http://arxiv.org/abs/1008.2320
• http://arxiv.org/pdf/1008.2320v1

Other information

BibTeX entry

@article{Lemoine2010,
	key = {Lemoine2010},
	type = {article},
	title = {Computer analysis of Sprouts with nimbers},
	author = {Lemoine, Julien and Viennot, Simon},
	abstract = {Sprouts is a two-player topological game, invented in 1967 in the University of Cambridge by John Conway and Michael Paterson. The game starts with p spots, and ends in at most 3p-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 reached 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.},
	comment = {},
	date_added = {2012-04-21},
	date_published = {2010-08-01},
	urls = {http://arxiv.org/abs/1008.2320,http://arxiv.org/pdf/1008.2320v1},
	collections = {Unusual arithmetic,Computational complexity of games},
	archivePrefix = {arXiv},
	arxivId = {1008.2320},
	eprint = {1008.2320},
	journal = {Analysis},
	month = {aug},
	pages = 17,
	url = {http://arxiv.org/abs/1008.2320 http://arxiv.org/pdf/1008.2320v1},
	year = 2010,
	primaryClass = {math.CO},
	urldate = {2012-04-21}
}