Multi-player End-Nim games

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. (C) 2018 Elsevier B.V. All rights reserved.