Racing a rook to the corner

Question

A single rook sits on the top-right square of an ordinary 8 by 8 chessboard. Two players alternate moving it. A move slides the rook any number of squares to the left or any number of squares down (never right, never up). The player who moves the rook onto the bottom-left corner square wins. Who wins with best play, and what is the strategy? What if the board were 8 squares wide but only 6 squares tall?

Accepted Answer

On the square board the second player wins by always returning the rook to the diagonal. On an 8 by 6 board the first player wins. Set up coordinates. Let the rook be at (a, b), where a is the number of columns to the left it still needs to travel and b the number of rows down. The target is (0, 0). Moving left reduces a; moving down reduces b; a move must reduce one of them by at least 1 and leaves the other unchanged. The losing positions are those with a = b. From (k, k) with k at least 1, any move produces unequal coordinates. From unequal coordinates (a, b) with a above b, the mover can …

Racing a rook to the corner

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