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Dots and boxes is a pencil-and-paper game for two players. It was first published in the 19th century by French mathematician Édouard Lucas, who called it la pipopipette. It has gone by many other names, including dots and dashes, game of dots, dot to dot grid, boxes, and pigs in a pen.
Minimax is a decision rule used in artificial intelligence, decision theory, game theory, statistics, and philosophy for minimizing the possible loss for a worst case scenario. When dealing with gains, it is referred to as "maximin" – to maximize the minimum gain. Originally formulated for several-player zero-sum game theory, covering both the cases where players take alternate moves and those where they make simultaneous moves, it has also been extended to more complex games and to general decision-making in the presence of uncertainty.
Nim is a mathematical game of strategy in which two players take turns removing objects from distinct heaps or piles. 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 or pile. Depending on the version being played, the goal of the game is either to avoid taking the last object or to take the last object.
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 nim. It can therefore be represented as a natural number, the size of the heap in its equivalent game of nim, as an ordinal number in the infinite generalization, or alternatively as a nimber, the value of that one-heap game in an algebraic system whose addition operation combines multiple heaps to form a single equivalent heap in nim.
In mathematics, the surreal number system is a totally ordered proper class containing not only the real numbers but also infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number. Research on the Go endgame by John Horton Conway led to the original definition and construction of surreal numbers. Conway's construction was introduced in Donald Knuth's 1974 book Surreal Numbers: How Two Ex-Students Turned On to Pure Mathematics and Found Total Happiness.
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 symmetric. In other words, the only difference between player 1 and player 2 is that player 1 goes first. The game is played until a terminal position is reached. A terminal position is one from which no moves are possible. Then one of the players is declared the winner and the other the loser. Furthermore, impartial games are played with perfect information and no chance moves, meaning all information about the game and operations for both players are visible to both players.
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 endowed with nimber addition and nimber multiplication, which are distinct from ordinal addition and ordinal multiplication.
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 games that have a position that the players take turns changing in defined ways or moves to achieve a defined winning condition. Combinatorial game theory has not traditionally studied games of chance or those that use imperfect or incomplete information, favoring games that offer perfect information in which the state of the game and the set of available moves is always known by both players. However, as mathematical techniques advance, the types of game that can be mathematically analyzed expands, thus the boundaries of the field are ever changing. Scholars will generally define what they mean by a "game" at the beginning of a paper, and these definitions often vary as they are specific to the game being analyzed and are not meant to represent the entire scope of the field.
In combinatorial game theory, a game is partisan if it is not impartial. That is, some moves are available to one player and not to the other.
Domineering 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 rectangle, an entirely irregular polyomino, or a combination of any number of such components. Two players have a collection of dominoes which they place on the grid in turn, covering up squares. One player places tiles vertically, while the other places them horizontally. As in most games in combinatorial game theory, the first player who cannot move loses.
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 "ground" line.
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, take turns coloring in one uncolored region per turn, subject to various constraints, as in the map-coloring problem. The move constraints and the winning condition are features of the particular game.
SOS is paper and pencil game for two or more players. It is similar to tic-tac-toe and dots and boxes, but has much greater complexity.
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 pins, until all the pins are gone. Using the notation of octal games, Kayles is denoted 0.77.
The octal games are a class of two-player games that involve removing tokens from heaps of tokens. They have been studied in combinatorial game theory as a generalization of Nim, Kayles, and similar games.
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.
In combinatorial game theory, and particularly in the theory of impartial games in misère play, an indistinguishability quotient is a commutative monoid that generalizes and localizes the Sprague–Grundy theorem for a specific game's rule set.
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.
Treblecross is a degenerate tic-tac toe variant. The game is an octal game, played on a one-dimensional board and both players play using the same piece. Each player on their turn plays a piece in an unoccupied space. The game is won if a player on their turn makes a line of three pieces in a row.
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 and in which the allowed moves reduce these numbers. Often, the moves of the game allow any number to be reduced by subtracting a value from a specified subtraction set, and different subtraction games vary in their subtraction sets. These games also vary in whether the last player to move wins or loses. Another winning convention that has also been used is that a player who moves to a position with all numbers zero wins, but that any other position with no moves possible is a draw.
References
- Conway, John Horton (1976), On Numbers and Games , Academic Press.