Map-coloring games
Several map-coloring games are studied in combinatorial game theory. The general idea is that we are given a map with regions drawn in but with not all the regions colored. Two players, Left and Right
Go and mathematics
The game of Go is one of the most popular games in the world. As a result of its elegant and simple rules, the game has long been an inspiration for mathematical research. Shen Kuo, an 11th century Ch
Branching factor
In computing, tree data structures, and game theory, the branching factor is the number of children at each node, the outdegree. If this value is not uniform, an average branching factor can be calcul
Generalized game
In computational complexity theory, a generalized game is a game or puzzle that has been generalized so that it can be played on a board or grid of any size. For example, generalized chess is the game
Poset game
In combinatorial game theory, poset games are mathematical games of strategy, generalizing many well-known games such as Nim and Chomp. In such games, two players start with a poset (a partially order
Nim
Nim is a mathematical game of strategy in which two players take turns removing (or "nimming") objects from distinct heaps or piles. On each turn, a player must remove at least one object, and may rem
Hot game
In combinatorial game theory, a branch of mathematics, a hot game is one in which each player can improve their position by making the next move. By contrast, a cold game is one where each player can
Misère
Misère (French for "destitution"), misere, bettel, betl, beddl or bettler (German for "beggar"; equivalent terms in other languages include contrabola, devole, pobre) is a bid in various card games, a
Genus theory
In the mathematical theory of games, genus theory in impartial games is a theory by which some games played under the misère play convention can be analysed, to predict the outcome class of games. Gen
Disjunctive sum
In the mathematics of combinatorial games, the sum or disjunctive sum of two games is a game in which the two games are played in parallel, with each player being allowed to move in just one of the ga
Grundy's game
Grundy's game is a two-player mathematical game of strategy. The starting configuration is a single heap of objects, and the two players take turn splitting a single heap into two heaps of different s
Cram (game)
Cram is a mathematical game played on a sheet of graph paper. It is the impartial version of Domineering and the only difference in the rules is that each player may place their dominoes in either ori
Kayles
Kayles is a simple impartial game in combinatorial game theory, invented by Henry Dudeney in 1908. Given a row of imagined bowling pins, players take turns to knock out either one pin, or two adjacent
Partisan game
In combinatorial game theory, a game is partisan (sometimes 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
Parsimonious reduction
In computational complexity theory and game complexity, a parsimonious reduction is a transformation from one problem to another (a reduction) that preserves the number of solutions. Informally, it is
Winning Ways for Your Mathematical Plays
Winning Ways for Your Mathematical Plays (Academic Press, 1982) by Elwyn R. Berlekamp, John H. Conway, and Richard K. Guy is a compendium of information on mathematical games. It was first published i
Mex (mathematics)
In mathematics, the mex of a subset of a well-ordered set is the smallest value from the whole set that does not belong to the subset. That is, it is the minimum value of the complement set. The name
Octal game
The octal games are a class of two-player games that involve removing tokens (game pieces or stones) from heaps of tokens.They have been studied in combinatorial game theory as a generalization of Nim
Jenga
Jenga is a game of physical skill created by British board game designer and author Leslie Scott and marketed by Hasbro. Players take turns removing one block at a time from a tower constructed of 54
Toads and Frogs
The combinatorial game Toads and Frogs is a partisan game invented by Richard Guy. This mathematical game was used as an introductory game in the book Winning Ways for your Mathematical Plays. Known f
Chomp
Chomp is a two-player strategy game played on a rectangular grid made up of smaller square cells, which can be thought of as the blocks of a chocolate bar. The players take it in turns to choose one b
Fuzzy game
In combinatorial game theory, a fuzzy game is a game which is incomparable with the zero game: it is not greater than 0, which would be a win for Left; nor less than 0 which would be a win for Right;
Chopsticks (hand game)
Chopsticks is a hand game for two or more players, in which players extend a number of fingers from each hand and transfer those scores by taking turns to tap one hand against another. Chopsticks is a
Game complexity
Combinatorial game theory has several ways of measuring game complexity. This article describes five of them: state-space complexity, game tree size, decision complexity, game-tree complexity, and com
Bulgarian solitaire
In mathematics and game theory, Bulgarian solitaire is a card game that was introduced by Martin Gardner. In the game, a pack of cards is divided into several piles. Then for each pile, remove one car
Game tree
In the context of Combinatorial game theory, which typically studies sequential games with perfect information, a game tree is a graph representing all possible game states within such a game. Such ga
Pebble game
In mathematics and computer science, a pebble game is a type of mathematical game played by placing "pebbles" or "markers" on a directed acyclic graph according to certain rules:
* A given step of th
Angel problem
The angel problem is a question in combinatorial game theory proposed by John Horton Conway. The game is commonly referred to as the Angels and Devils game. The game is played by two players called th
Subtraction game
In combinatorial game theory, a subtraction game is an abstract strategy game whose state can be represented by a natural number or vector of numbers (for instance, the numbers of game tokens in piles
Domineering
Domineering (also called Stop-Gate or Crosscram) is a mathematical game that can be played on any collection of squares on a sheet of graph paper. For example, it can be played on a 6×6 square, a rect
Combinatorial game theory
Combinatorial game theory is a branch of mathematics and theoretical computer science that typically studies sequential games with perfect information. Study has been largely confined to two-player ga
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 one-heap game of nim, or to an infinite generalization of
Zero game
In combinatorial game theory, the zero game is the game where neither player has any legal options. Therefore, under the normal play convention, the first player automatically loses, and it is a secon
Cooling and heating (combinatorial game theory)
In combinatorial game theory, cooling, heating, and overheating are operations on hot games to make them more amenable to the traditional methods of the theory,which was originally devised for cold ga
Variation (game tree)
A variation can refer to a specific sequence of successive moves in a turn-based game, often used to specify a hypothetical future state of a game that is being played. Although the term is most commo
Monte Carlo tree search
In computer science, Monte Carlo tree search (MCTS) is a heuristic search algorithm for some kinds of decision processes, most notably those employed in software that plays board games. In that contex
Blockbusting (game)
Blockbusting is a solved combinatorial game introduced in 1987 by Elwyn Berlekamp illustrating a generalisation of overheating. The analysis of Blockbusting may be used as the basis of a strategy for
Sylver coinage
Sylver coinage is a mathematical game for two players, invented by John H. Conway. It is discussed in chapter 18 ofWinning Ways for Your Mathematical Plays. This article summarizes that chapter. The t
Star (game theory)
In combinatorial game theory, star, written as or , is the value given to the game where both players have only the option of moving to the zero game. Star may also be denoted as the surreal form {0|0
Col (game)
Col is a pencil and paper game, specifically a map-coloring game, involving the shading of areas in a line drawing according to the rules of graph coloring. With each move, the graph must remain prope
Combinatorial explosion
In mathematics, a combinatorial explosion is the rapid growth of the complexity of a problem due to how the combinatorics of the problem is affected by the input, constraints, and bounds of the proble
Infinite chess
Infinite chess is any variation of the game of chess played on an unbounded chessboard. Versions of infinite chess have been introduced independently by multiple players, chess theorists, and mathemat
Fibonacci nim
Fibonacci nim is a mathematical subtraction game, a variant of the game of nim. Players alternate removing coins from a pile, on each move taking at most twice as many coins as the previous move, and
Hackenbush
Hackenbush is a two-player game invented by mathematician John Horton Conway. It may be played on any configuration of colored line segments connected to one another by their endpoints and to a "groun
On Numbers and Games
On Numbers and Games is a mathematics book by John Horton Conway first published in 1976. The book is written by a pre-eminent mathematician, and is directed at other mathematicians. The material is,
Impartial game
In combinatorial game theory, an impartial game is a 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 sy
Indistinguishability quotient
In combinatorial game theory, and particularly in the theory of impartial games in misère play, an indistinguishability quotient is a commutative monoidthat generalizes and localizes the Sprague–Grund
Futile game
In game theory, a futile game is a game that permits a draw or a tie when optimal moves are made by both players. An example of this type of game is the classical form of Tic-tac-toe, though not all v
Games, Puzzles, and Computation
Games, Puzzles, and Computation is a book on game complexity, written by Robert Hearn and Erik Demaine, and published in 2009 by A K Peters. It is revised from Hearn's doctoral dissertation, which was
Clobber
Clobber is an abstract strategy game invented in 2001 by combinatorial game theorists Michael H. Albert, J.P. Grossman and Richard Nowakowski. It has subsequently been studied by Elwyn Berlekamp and E
Notakto
Notakto is a tic-tac-toe variant, also known as neutral or impartial tic-tac-toe. The game is a combination of the games tic-tac-toe and Nim, played across one or several boards with both of the playe
Meshulam's game
In graph theory, Meshulam's game is a game used to explain a theorem of Roy Meshulam related to the homological connectivity of the independence complex of a graph, which is the smallest index k such
Solved game
A solved game is a game whose outcome (win, lose or draw) can be correctly predicted from any position, assuming that both players play perfectly.This concept is usually applied to abstract strategy g
Shannon number
The Shannon number, named after the American mathematician Claude Shannon, is a conservative lower bound of the game-tree complexity of chess of 10120, based on an average of about 103 possibilities f
Surreal number
In mathematics, the surreal number system is a totally ordered proper class containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value
Tiny and miny
In mathematics, tiny and miny are operators that yield infinitesimal values when applied to numbers in combinatorial game theory. Given a positive number G, tiny G (denoted by ⧾G in many texts) is equ
Nimber
In mathematics, the nimbers, also called Grundy numbers, are introduced in combinatorial game theory, where they are defined as the values of heaps in the game Nim. The nimbers are the ordinal numbers
Subtract a square
Subtract-a-square (also referred to as take-a-square) is a two-player mathematical subtraction game. It is played by two people with a pile of coins (or other tokens) between them. The players take tu