Variants of [formula omitted]-Wythoff’s game.

In this paper, we study four games, they are all restrictions of ( s , t ) -Wythoff’s game which was introduced by A.S. Fraenkel. The first one is a modular type restriction of ( s , t ) -Wythoff’s game, where a player is restricted to remove a multiple of K tokens in each move ( K is a fixed positive integer). The others we called rook type restrictions of ( s , t ) -Wythoff’s game, including Odd-Arbitrary-Nim ( s , t ) -Wythoff’s Game, Odd–Odd-Nim ( s , t ) -Wythoff’s Game and Odd–Even-Nim ( s , t ) -Wythoff’s Game. In these three games, the restrictions are only made on horizontal and vertical moves, but not on the extended diagonal moves. For any K , s , t ≥ 1 , the sets of P -positions of our games are given in both normal and misère play. [ABSTRACT FROM AUTHOR]