The Winning Strategy Of Nim Game For Multi-player

Nim game is a classical game of many conbinatorial games.The rules of the game are as follows.There are n piles of counters.Two players move alternately.Each removing any positive number of counters from a single pile.Under normal play convention the first player who cannot make a legal move loses.Respectively,under misere play convention,this player wins..This paper makes deeply study on the restrictions and extensions of Nim.By adding the number of players in misere Nim,we get a more interesting and complicated game,the new game we called MiNim game.And on the basis of the MiNim game,the decision made by the people who participate in the Nim game is divided into rational and irrational(random),and a new game model is obtained,which is called OPNim game.This paper is divided into three chapter:The first chapter is the introduction which mainly introduces the development of impartial combinatorial game,it also elaborates the research situation of Nim game.The second chapter mainly studies the MiNim game with N heaps for n-player.Un-der the misere play convention,we present the game values and the optimal strategy of MiNim(N,n)for n>N+1,n = N+1,n = N and n<N,respectively.In the third chapter,we study OPNim game with exactly 1 random player and more than one random players.For many random players,we analyze two cases of non-adjacent random players and adjacent random players.And under the misere play convention,we present the game values and the optimal strategy of OPNim game.