Extensions And Restrictions Of Wythoff’s Game

Wythoff’s game is an important part of impartial combinatorial games. Given a2-player game played on two heaps of finitely many tokens. Two types of moves are permitted:(i) Remove any positive number of tokens from one heap.(ii) Remove the same positive number of tokens from both heaps. A.S. Fraenkel (1998) defined a kind of new game by restricting the move of the Wythoff’s game, called α-Wythoff’s game:given an integer α≥1, and two heaps of finitely many tokens. There are two types of moves:(i) take any positive number of tokens from one heap(the Nim rule);(ii) take k>0and l>0from the two heaps, say,0<k<l, here k and l constrained by0<|k-l|<α(the General Wythoff’s rule).This paper makes an in-depth study of the three kinds of restrictions concerning Wythoff’s game and α-Wythoff’s game. The three new games which obtained by restricting the horizontal move and the vertical move of the Wythoff’s game, are as follows:Whq game, Wvq game and Whvq game. By restricting both the horizontal and vertical moves of the α-Wythoff’s game, we get a new game which is α-Whvq game. This paper is divided into three chapters.The first chapter is the introduction which mainly introduce the development of impar-tial combinatorial games, the major achievements of the Wythoff’s game and the history of the Wythoff’s game. In the past decades, many scholars had researched on Wythoff’s game, and they obtained many achievements. There are kinds of deformations by chang-ing the method of the move. These deformations can be divided into two classes:the restriction of the Wythoff’s game, the extension of the Wythoff’s game.The second chapter, on account of the Wythoff’s game, there are three kinds of moving way:the horizontal move, the vertical move and the diagonal move. In this chapter, we got the Wqh game and the Wqv game through to restrict the horizontal and vertical move of the Wythoff’s game respectively. We got the Wqhv game through to restrict the horizontal and vertical move of the Wythoff’s game at the same time. In this chapter, we will show all the P-positions of the three new games under the normal conventions, as well as the corresponding winning strategy.The third chapter, on account of the a-Wythoff’s game, there are three kinds of mov-ing way:the horizontal move, the vertical move and the general slash movement. We got the a-Wqhv game through to restrict the horizontal and vertical move of the a-Wythoff’s game at the same time. In this chapter, we will show all the P-positions of the a-Wqhv game under the normal conventions, as well as the corresponding winning strategy.