The Restriction And Extension Of The Wythoff's Game

Wythoff's game is an important part of impartial combinatorial games. It is played on two heaps of finitely many tokens, two types of moves are allowed:(ⅰ) Remove any positive number of tokens from one heap(the Nim rule); Remove the same positive number of tokens from both heaps(the Wythoff's rule).This paper gets two kinds of new games by restricting the move of the Wythoff's game and expanding the Wythoff's game, called:OE-Wythoff's game and Matrix-Nim game. This paper is divided into three chapters:In the first chapter, we mainly introduce the development of impartial combinatorial games and the situation 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 changing 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.In the second chapter, we mainly research on the restriction of the Wythoff's game. Defining OE-Wythoff's game based on the limit of the move in the Wythoff's game. OE-Wythoff's game is different from Wythoff's game in the number of tokens you can take from a single pile:A player chooses a heap, if it is marked 1, he can take odd tokens from this heap; if it is marked 2, he can take even tokens from this heap. Other moves are unchanged. we show all of P-positions of OE-Wythoff's game under the normal and misere form in this paper.In the third chapter, we research on the extension of the Wythoff's game. T.S. Fer-guson proposed a new form of 2-dimensional Nim:The positions are m×n matrices of nonnegative integers, where m and n are fixed positive integers. A move consists of choos-ing a row or a column and subtracting same positive integer k from each integer of the chosen row or column. The case m=1 and n=2 is just Wythoff's game, i.e., Matrix-Nim game is the extension of the Wythoff's game. We investigate the case m=n=2 of Matrix-Nim under the misere form. All P-positions of 2×2 Matrix-Nim are determined.