The Optimal Strategy Of Three Kinds Of Last-Nim Game

Last-Nim is a classical game of impartial combinatorial games,which can be described as:There are N piles of counters which are linearly ordered.In turn,each of n players removes any positive integer of counters from the last pile,The player who cannot make a move wins the game.This paper makes deeply study on three kinds of Last-Nim game:(1)Spread the num-ber of participants from two to more people,we can get ’Multi-player Last-Nim’ game;(2)Base on the ’Multi-player Last-Nim’ game,we add pass options to mobile rule,we can get ’Multi-player Last-Nim with Passes’ game;(3)Similarly,Base on the ’Multi-player Last-Nim’ game,we divide the decisions taken by those involved in the Last-Nim game into rational and irrational(random).When the participants contain exactly one random player,we can get ’Multi-player Last-Nim with a Random Player’ game.The main contents of this paper are as follows:Firstly,we study the ’Multi-player Last-Nim’ game with n players and N piles.Under the mis(?)re play convention,we present the corresponding game values and the optimal strategy for n>N+1,n= N + 1,n = N,respectively.At the same time,for some cases of n<N,the corresponding game value and the optimal strategy are also obtained.Secondly,we study the ’Multi-player Last-Nim with Passes’ game with n players and N piles.Under the mis(?)re play convention,we present the corresponding game values and the optimal strategy for n>N + 1,n = N + 1,n = N,respectively.At the same time,for some cases of n<N,the corresponding game value and the winning strategy are also obtained.Finally,we study ’Multi-player Last-Nim with a Random Player’ game with exact-ly one random player.And under the mis(?)re play convention,we present the winning probability and the optimal strategy of each participant.