Preservation Of P-positions Of A Class Of Impartial Combinatorial Games

Wythoff’s game is an important part of impartial combinatorial games. A.S. Fraenkel (1998) defined a kind of new game by restricting the move of the Wythoff’s game, called a-Wythoff’s game and Extended Wythoff’s game.This paper mainly studied the new game models which get through Some sort of extension or restriction about the α-Wythoff’s game and Extended Wythoff’s game, and how to determine its all P positions, and studied the preservation of P position between the models.The first chapter is the introduction which mainly introduces the development of impartial combinatorial games, and the basic conceptions and the research status.The second chapter is based on the Extended Wythoff’s game, we studied the new model ΓΚ which get through extending P position of K of the Extended Wythoff’s game as legal movement. This chapter first get the set (?)1of all P position of Γ1, Secondly proved:for any integer K≥1, we have (?)K=(?)1, i.e., the set of all P position of ΓK stay the same.The third chapter is based on a-Wythoff’s game, we studied the new model βαK which get through extending P position of K of a-Wythoff’s game as legal movement. In this chapter, distinguish between α=2and α>2and α=1, we respectively obtained all P position of βαK for arbitrary integer K≥1, and studied the preservation of P position between the models.The fourth chapter studies the question of the preservation of P position, if we restrict the game. E. Duchene and others studied such following questions:whether there is such a game, which is limited by Wythoff’s game and its P position is the same to Wythoff’s game. Their results show that this game does not exist.In this chapter, we find a couple of games:a game is another limitation, and they have the same P position. We also find the other pair of game:a game is another limitation, but they are not the same P position. This means that a game and it restricted the Pposition between properties depends on the game itself. This means that the preservation of P position of a game and its restricted game depends on the game itself.