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We have a regular chessboard $ 8 * 8 $ and a coin. There are two players: A, B.
Firstly, A places the coin somewhere on the chessboard
Then, B can move the coin in a cell in the same line or the same column with the cell where A placed initially the coin. Afther the first two moves, A and B both follow the same rule, namely: each player must move the coin in a cell o the same row or the same column with the cell where the previous player moved, and the coin shouldn't be placed twice on the same cell during the game. The last rule means that if A moves to (1, 3) during the game, neither B, nor again A can move to (1, 3) during the game. Prove that B can win each game, if both players are flowless.

I was thinking to prove that B can move always in the same color as the previous move made by A. Then, I should prove a lemma that B can always move somewhere, using the parity of the number of closed cells on a line/column as an invariant, maybe monovariant.

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2 Answers 2

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$B$ can always move $(x,y)\mapsto (9-x,y)$.

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B can always move horizontally to an available square.

After every one of B's turns, the number of blocked off spaces in every row will be even.

After every one of A's turns, the number of blocked off spaces in every row except the one with the coin will be even, and the number of blocked off spaces in the row with the coin will be odd. This means that there will always be at least one such space for B to move to

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