In combinatorial game theory

Combinatorial game theory

Combinatorial game theory is a branch of applied mathematics and theoretical computer science that studies sequential games with perfect information, that is, two-player games which have a position in which the players take turns changing in defined ways or moves to achieve a defined winning...

, an impartial game is a game

Mathematical game

A mathematical game is a multiplayer game whose rules, strategies, and outcomes can be studied and explained by mathematics. Examples of such games are Tic-tac-toe and Dots and Boxes, to name a couple. On the surface, a game need not seem mathematical or complicated to still be a mathematical game...

 in which the allowable moves depend only on the position and not on which of the two players is currently moving, and where the payoffs are symmetric. In other words, the only difference between player 1 and player 2 is that player 1 goes first.

Impartial games can be analyzed using the Sprague–Grundy theorem

Sprague–Grundy theorem

In combinatorial game theory, the Sprague–Grundy theorem states that every impartial game under the normal play convention is equivalent to a nimber. The Grundy value or nim-value of an impartial game is then defined as the unique nimber that the game is equivalent to...

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Impartial games include nim

Nim

Nim is a mathematical game of strategy in which two players take turns removing objects from distinct heaps. On each turn, a player must remove at least one object, and may remove any number of objects provided they all come from the same heap....

, sprouts

Sprouts (game)

Sprouts is a pencil-and-paper game with interesting mathematical properties. It was invented by mathematicians John Horton Conway and Michael S. Paterson at Cambridge University in 1967.- Rules :...

, kayles

Kayles

In combinatorial game theory, Kayles is a simple impartial game. In the notation of octal games, Kayles is denoted 0.77.- Rules :Kayles is played with a row of tokens, which represent bowling pins. The row may be of any length...

, quarto

Quarto (board game)

Quarto is a board game for two players invented by Swiss mathematician Blaise Müller.It is played on a 4×4 board. There are 16 unique pieces, each of which is either:* tall or short;...

, cram, and chomp

Chomp

Chomp is a 2-player game of strategy played on a rectangular "chocolate bar" made up of smaller square blocks . The players take it in turns to choose one block and "eat it" , together with those that are below it and to its right...

. Go

Go (board game)

Go , is an ancient board game for two players that originated in China more than 2,000 years ago...

 and chess

Chess

Chess is a two-player board game played on a chessboard, a square-checkered board with 64 squares arranged in an eight-by-eight grid. It is one of the world's most popular games, played by millions of people worldwide at home, in clubs, online, by correspondence, and in tournaments.Each player...

 are not impartial, as it is necessary to know whose turn it is in order to categorise the possible moves (for example, in chess only player 1 can move the white pieces). Games like ZÈRTZ

ZÈRTZ

ZÈRTZ is the third game in the GIPF Project of six abstract strategy games. The game features a shrinking board and an object that promotes sacrifice combinations...

 and Chameleon are also not impartial, since although they are played with shared pieces, the payoffs are not necessarily symmetric for any given position.

A game that is not impartial is called a partisan game

Partisan game

In combinatorial game theory, a game is partisan or partizan if it is not impartial. That is, some moves are available to one player and not to the other.Most games are partisan; for example, in chess, only one player can move the white pieces....

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