Multi-player End-Nim games

Abstract W.O. Krawec (2012) [17] introduced a method of analyzing multi-player impartial games, and derived a recursive function capable of determining which of the n players has a winning strategy. The present paper is devoted to “Multi-player End-Nim games”, abbreviated by ENim( N , n ), assuming that the standard alliance matrix is adopted. The game values of ENim( N , n ) are completely determined for three cases n > N + 1 , n = N + 1 and n = N . For the case n N , by letting d = N − n ≥ 1 , the game values of ENim( n + d , n ) are completely determined if n ≥ d + 4 . The case 3 ≤ n ≤ d + 3 is more complicated, we present the game values for d = 1 .