# Zermelo's stones

• Aug 2nd 2011, 11:10 PM
godelproof
Zermelo's stones
It is said the following problem was posed (or solved?) by Zermelo. I'm not sure about that. I read it in an economic forum. But no one gives an answer there!

m*n stones are arranged in an m*n matrix A. Two players play a game: they choose a stone in the matrix, say the stone in position A(i, j), then this stone together with all the stones to the northeast of it are removed (i.e., stones in the submatrix [A1j:Ain] are removed). Two players take turns to choose and remove the stones. Whoever removes the last stone (or stones) loses the game.

Prove: There's a strategy for the first player to win.

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I can show that when m=n, the first player has a winning strategy. But I don't know how show the same result for m<n.
• Aug 3rd 2011, 03:01 AM
godelproof
Re: Zermelo's stones
Interestingly, one can play the game in 3 dimensional space with the stone matrix size being m*n*k, in which one removes stones to the northeast and "behind" the chosen stone. I don't know whether anyone has thought about this, or the first player here can still win the game.
• Aug 3rd 2011, 05:36 AM
godelproof
Re: Zermelo's stones
Sorry, I just find that the game is call Chomp. Chomp - Wikipedia, the free encyclopedia
And my questions are answered there.