There are 8 white pawns on the squares at one edge of an chessboard and 8 black pawns on the squares at the opposite edge. On each move, a player shifts one of his pawns by one or more squares forward (toward the opponent's piece) or backward, but moving a pawn to a square containing the opponent's pawn or over such a square is prohibited. Moves are performed alternately, with white starting. The player who cannot make a move loses. Which player has a winning strategy?
Solution
Black can use the following strategy. If white moves his th pawn counting from his left, by squares forward, black moves his th pawn counting from his left, by squares forward. If white moves his pawn by squares backward, black moves his pawn on the same file by squares forward. After each move made by black, the distance between two pawns on each file is the same as that on the file symmetric w.r.t. the midpoint of the board. Thus whenever a white pawn has moved forward, the black can make the move determined by the strategy described. As the black pawns move forward only, white will be paralyzed sooner or later.
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