【作者】 李海燕；

【导师】 刘文安；

【作者基本信息】 河南师范大学 ， 概率论与数理统计， 2013， 硕士

【摘要】 Wythoff’s游戏是公平组合游戏中的重要组成部分.A.S. Fraenkel(1998)将Wythoff’s游戏进行了扩展,定义了(s, t)-Wythoff’s游戏：给定两个整数s≥1,t≥1和两堆各若干个石头.两个游戏者轮流移动石头,移动方法为：(i)要么从两堆中选定一堆,从中移走任意正整数个石头(称为Nim移法)；(ii)要么同时从两堆中选取,从一堆选取k>0个,从另一堆选取e>0个,这里k和e满足0<k≤e<sk+t(称为一般的Wythoff’s移法).本文深入研究了(s,t)-Wythoff’s游戏的三类限制：通过对两种移动方法都进行限制,得到第Ⅰ类模型的四个新游戏,即ΓOEI游戏,ΓEOI。游戏,ΓOOI游戏和ΓEEI游戏；通过仅仅限制"Nim移法”,得到第Ⅱ类模型的四个新游戏,即ΓOEII游戏,ΓEOII游戏,ΓOOII游戏和ΓEEII游戏：通过仅仅限制“一般的Wythoff’s移法”,得到第Ⅲ类模型的四个新游戏,即ΓOEIII游戏,ΓEOIII游戏,ΓOOIII游戏和ΓEEIII游戏.本文共分四章：第一章绪论,主要介绍公平组合游戏的历史及发展,阐述了基本概念与研究现状.第二章深入研究了第Ⅰ类模型的四个新游戏.以ΓEEI游戏为例：将两堆石头分别标记为1号堆和2号堆,游戏者要么从1号堆取偶数(Even)个或从2号堆取偶数(Even)个,要么从1号堆取偶数k个同时从2号堆取偶数e个,且k和e满足0<k≤e<sk+t.本章给出了第Ⅰ类四个游戏分别在normal规则与misere规则下的所有P位置,并给出了相应的取胜策略,从而彻底解决了第Ⅰ类模型.第三章将ΓEEI游戏进行了推广,即把ΓEEI游戏中允许取走的石头个数从“偶数”(2的正整数倍),放宽为“K的正整数倍”,这里K为任意正整数.本章对于任意正整数K,给出了该新模型分别在normal规则与misere规则下的所有P位置,并给出了相应的取胜策略.第四章主要研究了第Ⅱ类和第Ⅲ类模型中的五个新游戏,即ΓOEII游戏,ΓEOII游戏,ΓOOII游戏,ΓOEIII游戏和ΓEOIII游戏.本章给出了这五个新游戏分别在normal规则与misere规则下的所有P位置,并给出了相应的取胜策略,从而彻底解决了这五个新游戏.更多还原

【Abstract】 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.更多还原