An impartial game is a game in which the possible moves are the same for each player in any position. For finite impartial games in which the last player wins, the nim-value of the sum of two games is the nim-sum of their nim-values. In this sum, each move is made in exactly one component. The nim-sum is obtained by applying exclusive disjunction to corresponding binary digits. The equivalence classes obtained by identifying positions with the same nim-value form an additive Abelian group (Fenner and Rogers 2015). If the last player loses, the game is said to be in misère form and the analysis is much more difficult.
Impartial Game
See also
Chomp, Fair Game, Game, Misère-Form Game, Nim-Value, Partisan Game, SproutsExplore with Wolfram|Alpha
References
Fenner, S. A. and Rogers, J. "Combinatorial Game Complexity: An Introduction with Poset Games." Bull. EATCS 116, 2015. Extended version: https://arxiv.org/abs/1505.07416.Referenced on Wolfram|Alpha
Impartial GameCite this as:
Weisstein, Eric W. "Impartial Game." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ImpartialGame.html