Last Nim Game With Shifted Alliance System

Abstract

We introduce a class of impartial combinatorial game which is the multiple-player last Nim game denoted by MLNim(N, n) in which there are N piles of counters which are linearly ordered and the move will be, the n-player will remove any positive integer of counters from the last pile, we will introduce this MLNim(N, n) with Shifted Standard alliance system by r, denoted by SSAMr in which each player will prefer winning for another player over himself. The Aim is to determine the Game value of the positions of MLNim(N, n) where N≥ 1 is the number of piles and n≥ 3 is the number of players and we will present the possible n≤ Nand determine the game value in this case.