Editing Combinatorial game theory (section)

==History==
Combinatorial game theory arose in relation to the theory of [[impartial game]]s, in which any play available to one player must be available to the other as well. One such game is [[Nim]], which can be solved completely. Nim is an impartial game for two players, and subject to the ''[[Normal play convention|normal play condition]]'', which means that a player who cannot move loses.  In the 1930s, the [[Sprague–Grundy theorem]] showed that all impartial games are equivalent to heaps in Nim, thus showing that major unifications are possible in games considered at a combinatorial level, in which detailed strategies matter, not just pay-offs.

In the 1960s, [[Elwyn R. Berlekamp]], [[John H. Conway]] and [[Richard K. Guy]] jointly introduced the theory of a [[partisan game]], in which the requirement that a play available to one player be available to both is relaxed. Their results were published in their book ''[[Winning Ways for your Mathematical Plays]]'' in 1982. However, the first work published on the subject was Conway's 1976 book ''[[On Numbers and Games]]'', also known as ONAG, which introduced the concept of [[surreal number]]s and the generalization to games. ''On Numbers and Games'' was also a fruit of the collaboration between Berlekamp, Conway, and Guy.

Combinatorial games are generally, by convention,  put into a form where one player wins when the other has no moves remaining. It is easy to convert any finite game with only two possible results into an equivalent one where this convention applies. One of the most important concepts in the theory of combinatorial games is that of the [[disjunctive sum|sum]] of two games, which is a game where each player may choose to move either in one game or the other at any point in the game, and a player wins when his opponent has no move in either game. This way of combining games leads to a rich and powerful mathematical structure.

Conway stated in ''On Numbers and Games'' that the inspiration for the theory of partisan games was based on his observation of the play in [[Go (board game)|Go]] endgames, which can often be decomposed into sums of simpler endgames isolated from each other in different parts of the board.