Variant of Wythoff's Game-Corner Two Rooks

Abstract

This article presents an impartial game, “Corner Two Rooks.” This game is a variant of “Corner the Queen” that is mathematically equivalent to Wythoff's game. In “Corner the Queen, ” a single chess queen is placed on a large grid of squares. Each player can move the queen any number of steps toward the upper-left corner of the grid, vertically, horizontally, or diagonally. The player who moves the queen into the upper-left corner is the winner. In this work, the authors use two rooks of chess instead of the queen, and a rook can jump over another rook but not onto another. There is a restriction on the distance that a rook can travel in each turn. This game can be considered as a misère game of the traditional Nim game with four piles and a restriction of the number of stones to be removed in each turn. The authors present the set of P-positions of the game using a theorem for misère games. When there is no restriction on the distance that a rook can travel in each turn, we obtain a similar result in which the set of P-positions is simpler.