The Restrictions Of Wythoff-like Game

(s, t)-Wythoff's game is an important part of impartial combinatorial games. A.S. Fraenkel (2012) defined a kind of new game by extending the moves of the (s,t)-Wythoff's game, called Wythoff-like game:Given a function f:Z?0?-Z?0 and two heaps of finitely many tokens. There are two types of moves:(i) take any positive number of tokens from one hea.p(Nim-rule); (ii) take k(> 0) from the first heap and l(> 0) from the second heap, restricted to the condition 0< k?l< f(k)(Wyt(f)-rule).This paper makes an in-depth study of the four kinds of restrictions concerning Wythoff-like game. Four new games which belong to the first type of model, obtained by restricting both moves of the Wythoff-like game, are as follows:OOW(f) game, OEW(f) game, EOW(f)game and EEW(f) game. Take OEW(f) game as an example, in which the two heaps are marked with 1 and 2 respectively. A player can take odd tokens from the heap maked with 1 or take even tokens from the heap maked with2, or take odd tokens of k from 1 and take even of l from 2 at the same time, and 0< k< l< f(k). This paper is divided into five chapters:The first chapter is the introduction which mainly introduces the development of impartial combinatorial games, the basic conceptions and the research status.The second chapter researches the four new games belonging to the first type of model in depth. Take OEW(f) game as an example, in which the two heaps are marked with 1 and 2 respectively. A player can take odd tokens from the heap maked with 1 or take even tokens from the heap maked with 2, or take odd tokens of k from 1 and take even tokens of l from 2 at the same time, and 0< k< l< f(k). In this chapter, we will show all P-positions of the first type of model under normal convention, as well as the corresponding winning strategy.The third chapter popularizes the OOW(f) game and EEW(f) game. Take ?IEE game as an example, expanding the "even" numbers (the integral multiples of 2) of allowed take away in the EEW(f) game to "the integral multiples of K". Here K is any positive integer. For any K, the chapter gives all of the P-positions of the new model under normal convention, as well as the corresponding winning strategy.The fourth chapter mainly studies seven new games belonging to the third type of model, ie., TOOEE game, TOOOE game, TOOEO game?TOEOO game?TEOOO game, TOEEO game and TEOOE game. This chapter gives all of the P-positions of the seven new games under normal conventions, as well as the corresponding winning strategy.The fifth chapter mainly studies three new games belonging to the forth type of model, ie., TOO(?) game, TOE(?) game and TEO(?) game. This chapter gives all of the P-positions of these new games under normal convention, as well as the corresponding winning strategy.