The Restrictions Of (s,t)-Wythoff’s Game

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 (s, t)-Wythoff’s game:given two integers s≥1, t>1and 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 e>0from the two heaps, say,0<k<e, here k and e constrained byO<k<e<sk+t(the General Wythoff’s rule).This paper makes an in-depth study of the three kinds of restrictions concerning (s,l)-Wythoff’s game. The four new games which belong to the first type of model, obtained by restricting both the moves of the (s,t)-Wythoff’s game, are as follows:ΓOEI game,ΓEOI game, ΓOOI game and ΓEEI game. By restricting "the Nim rule" only, we get four new games belonging to the second type of model, which are ΓOEII game, ΓEOII game,ΓOOII game and ΓEEII game. By restricting "the General Wythoff’s rule" only, we obtain another four new games belonging to the third type of model, which are as follows:ΓOEIII game,ΓEOIII game, ΓOOIII game and ΓEEIII game. This paper is divided into four chapters.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 researches the four new games belonging to the first type of model in depth. Take ΓEEI game as an example, in which the two heaps are marked with1and2 respectively. A player can take even tokens from the heap maked with1or2, or take even tokens of k from1and take even of e from2at the same time, here0<k<e<sk+l. In this chapter, we will show all the P-positions of the first type of model under the normal and the misere conventions respectively, as well as the corresponding winning strategy, thus solving the first type of model thoroughly.The third chapter popularizes the ΓEEI game, ie., to expand the "even" numbers (the integral multiples of2) of allowed take away in the ΓEEI 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 the normal and misere play conventions respectively, as well as the corresponding winning strategy.The fourth chapter mainly studies five new games belonging to the second type and the third type of model, ie., ΓOEII game, ΓOOII game, ΓOOII game, ΓOEIII game and ΓEOIII game. Wc investigate several new games of them. This chapter gives all of the P-positions of the new five mew games under the normal and misere play conventions respectively, as well as the corresponding winning strategy, solving the five new games thoroughly.