Variants of (s,t)-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.