Combinatorial game theory

Combinatorial game theory studies finite games with perfect information, no chance, and defined legal moves. It analyzes positions by the moves available and the outcomes optimal play permits.

Impartial game: A game in which both players have the same legal moves from every position. Impartial games provide the setting for the theory’s classic nim-value analysis.
Nim: An impartial game played by removing objects from heaps according to specified rules. Nim is the canonical model for impartial-game analysis.
Hackenbush: A family of games played by deleting colored edges from a graph connected to the ground. Its colored edges produce concrete examples of partizan game values.
Misère play: A play convention in which the player making the last move loses. It reverses normal play’s ending condition and disrupts some standard results.
Partizan game: A game in which the players can have different legal moves from a position. Partizan games require distinctions that impartial-game methods cannot capture.
Nim-sum: The bitwise exclusive-or of two or more nonnegative integers. Its zero or nonzero value identifies losing or winning positions in sums of Nim heaps.
Green Hackenbush: A Hackenbush game in which every edge is green and either player may remove one. Its impartial positions illustrate nim-values within a graphical game.
Chess endgame: The final phase of a chess game, with few pieces remaining on the board. Chess has perfect information but is not generally finite under unrestricted repetition rules.
Normal play: A convention in which the player with no legal move loses. Many standard results assume this ending rule.
Mex: The least nonnegative integer absent from a given set of nonnegative integers. The mex rule recursively determines a position’s Sprague–Grundy value.

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