Geometrical extensions of Wythoff’s game

In 1905 Bouton gave the complete theory of a two-player combinatorial game: the game of Nim. Two years later, Wythoff defined his game as “a modification” of the game of Nim. In this paper, we give the sets of the losing positions of geometrical extensions of Wythoff’s game, where allowed moves are considered according to a set of vectors (v1,…,vn)(v1,…,vn). When n=3n=3, we present algorithms and algebraic characterizations to determine the losing positions of such games. In the last part, we investigate a bounded version of Wythoff’s game, and give a polynomial way to decide whether a game position is losing or not.