The Extension Of The Wythoff's Game Under The Misere Play Convention

Wythof's game is a significant part impartial combinatorial games. Given a2-playergame played on two heaps of finitely many tokens. Two types of moves are permitted:(i) Remove any positive number of tokens from one heap.(ii)Remove the same positivenumber of tokens from both heaps.This paper gets three kinds of new games model by expanding the move of theWythof's game, called:(s, t)-Wythof's game, g-Wythof's game and n-heap game. Thispaper is divided into four chapters:The first chapter is the introduction which mainly introduce the development of im-partial combinatorial games,the major achievements of the Wythof's game and the historyof the Wythof's game. In the past decades, many scholars had researched on Wythof'sgame, and they obtained many achievements. There are kinds of deformations by chang-ing the method of the move. These deformations can be divided into two classes: therestriction of the Wythof's game, the extension of the Wythof's game.In the second chapter, we mainly research on the extension of the Wythof's game.A.S. Fraenkel[20]introduce a new (s, t)-Wythof's game., and determines all its P-positionsof (s, t)-Wythof's game under normal play convention. In this chapter, the (s, t)-Wythof'sgame under the mis`ere play convention is investigated. Thereby the optimal strategy of(s, t)-Wythof's model is given under the Mis`ere play convention. Moreover, we supplied another method of expression all its P-positions of (s, t)-Wythof's game under the Mis`ereplay convention.In the third chapter, mainly researches on g-Wythof's game model under the Mis`ereplay convention. We intruduce yet Boole game and Galois game briefly. A.S. Fraenkel[31]considered a new impartial combinatorial game in2004, and supplied all the P-positionsunder the Normal play convention. In this chapter, all the P-positions of g-Wythof'smodel are determined under the Mis`ere play convention, thereby the optimal strategy ofg-Wythof's model is given under the Mis`ere play convention.The fourth chapter mainly researches on n-heap game model under the Mis`ere playconvention. A.S. Fraenkel[32]considered a new impartial combinatorial game in2001, andsupplied all the P-positions under the Normal play convention. In this chapter, all theP-positions of n-heap model are determined under the Mis`ere play convention, thereby theoptimal strategy of n-heap model is given under the Mis`ere play convention.