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Chess variants (10x8)

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31. Août 2004, 21:53:43

AbigailII

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Sujet: Re: Gothic Chess NIM.......(OT)

It's a win for black. It's a game of parity. Let the game consist of N numbers (i = 10 for the first example), x_0 .. x_N-1. Decompose each number in powers of 2: x_i = a_i,0 * 2^0 + a_i,1 * 2^1 + a_i,2 * 2^2 + ..... Now for 0 = j, look at the sums a_i,j, 0 = i < N. Call this sum A_j. A player wins if at the end of his turn, all A_j's are even. For the game 3464334643, we have A_0 = 4, A_1 = 6, A_2 = 6, and A_k for k > 2 is equal to 0. This means the person going second, black, wins.

4444444444 is also a win for black (this is very easy to see, black can always copy what white does). 8888800000 is a win for white - but not if he plays a8. In that case, black plays a1, resulting in 0888800000, which is a win for black.