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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.
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, however, developed in a playful and unpretentious manner and many chapters are accessible to non-mathematicians. Martin Gardner discussed the book at length, particularly Conway's construction of surreal numbers, in his Mathematical Games column in Scientific American in September 1976.
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.
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}. This game is an unconditional first-player win.
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 proper, and a player who cannot make a legal move loses. The game was described and analysed by John Conway, who attributed it to Colin Vout, in On Numbers and Games.
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.
Miscellaneous Mathematical Symbols-B is a Unicode block containing miscellaneous mathematical symbols, including brackets, angles, and circle symbols.
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 Chinese scholar, estimated in his Dream Pool Essays that the number of possible board positions is around 10172. In more recent years, research of the game by John H. Conway led to the development of the surreal numbers and contributed to development of combinatorial game theory (with Go Infinitesimals being a specific example of its use in Go).
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.
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 players may place their dominoes in either orientation, but it results in a very different game. It has been called by many names, including "plugg" by Geoffrey Mott-Smith, and "dots-and-pairs". Cram was popularized by Martin Gardner in Scientific American.
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.
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.
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 games in which the winner is the last player to have a legal move. Overheating was generalised by Elwyn Berlekamp for the analysis of Blockbusting. Chilling and warming are variants used in the analysis of the endgame of Go.
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
- Albert, Michael H.; Nowakowski, Richard J.; Wolfe, David (2007). Lessons in Play: An Introduction to Combinatorial Game Theory. A K Peters, Ltd. ISBN 1-56881-277-9.
- Berlekamp, Elwyn R.; Conway, John H.; Guy, Richard K. (2003). Winning Ways for Your Mathematical Plays . A K Peters, Ltd.