Imagine a 1000 x 1000 chessboard on which a white king and 499 black rooks are placed at random such that no rook threatens the king. And suppose the king goes bonkers and wants to kill himself. Can he reach a threatened square in a finite number of moves if Black is trying actively to avoid this?
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Yes. Imagine that the king is on the lower left square and that on each move he takes a step along the main diagonal toward the upper right. After his first move and Black’s response, there must be no rooks on the lowest three ranks or the leftmost three files, or the king could immediately reach a threatened square. After his 998th move, the king arrives at a similar situation in the upper right: At this point there must be no rooks on the uppermost three ranks or the rightmost three files, for the same reason. This means that when the king begins his journey, all the rooks must be northeast of him, and when he finishes, all must be southwest. During the trip, then, each of Black’s 499 rooks must change both its rank and its file, in order to keep out of his way. This requires 499 x 2 = 998 moves. So, at the latest, Black will run out of time just a half-move too soon: White will squeeze Black’s maneuvering room down to zero on his own 998th move, just before Black evacuates the last rook.
From Miodrag Petkovic, Mathematics and Chess, 1997.
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