Researches On Extensitions And Restrictions Of Wythoff's Game

Wythof's game is an important part of impartial combinatorial games. It is playedon two heaps of fnitely many tokens,two persons move alternately, two types of movesare allowed: Remove any positive number of tokens from one heap; Or remove the samepositive number of tokens from both heaps.This paper mainly discusses two kinds of Games:(1)[α, α,1] Game,(2)R-radius Game.The paper is divided into four chapters:The frst chapter mainly introduces the history and the development of impartialcombinatorial games. It also explicits the research situation of impartial combinatorialgames. The background and the main results are given generally in this chapter.The second chapter mainly researches polynomial algorithmm of P-positions of [α, α,1]game under the normal play convention. Eric Duch ene,Sylvain Gravier researched a newimpartial combinatorial game which is [α, α,1] game, and made use of polynomial algo-rithmm to supply P-positions of [2,2,1] game. In this chapter, we make use of polynomialalgorithmm to supply P-positions of [4,4,1] game.The third chapter mainly researches the [α, α,1] game under the mis`ere play con-vention. This chapter is divided into two parts, The frst part gives concrete algorithm ofP-positions of the game, and all the P-positions when α is any positive integer. The secondpart makes use of polynomial algorithmm to give P-positions of [α, α,1] game when α=2. The fourth chapter mainly researches R-radius game under the mis`ere play convention,R-radius game is restricted move of Wythof's game. each player can either remove at mostR tokens from a single heap, or remove the same number of tokens from both heaps (It isalso at most R tokens). The paper uses its concrete algorithm to get all the P-positionsof this game under the mis`ere play convention. thus it has thoroughly solved the R-radiusgame under this move convention.