In combinatorial game theory, star, written as * or * 1, 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 {0|0}. This game is an unconditional first-player win. Mathematicians playing Konane at a Combinatorial game theory workshop (for technical content, see external link) This article is on the theory of combinatorial games. ... In combinatorial game theory, the zero game is the game where neither player has any legal options. ...

Star, as defined by John Conway in Winning Ways for your Mathematical Plays, is a value, but not a number in the traditional sense. Star is not zero, but neither positive nor negative, and is therefore said to be fuzzy and confused with (a fourth alternative that means neither "less than", "equal to", nor "greater than") 0. It is probably less than all positive rational numbers, and greater than all negative rationals. Since the rationals are dense in the reals, this also makes * greater than any negative real, and less than any positive real. John Horton Conway (born December 26, 1937, Liverpool, England) is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. ... Winning Ways for your Mathematical Plays (ISBN 1568811306) by Elwyn R. Berlekamp, John H. Conway, and Richard K. Guy is a compendium of information on mathematical games. ... A number is an abstract idea used in counting and measuring. ... A negative number is a number that is less than zero, such as −3. ... A negative number is a number that is less than zero, such as −3. ... In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ... In topology and related areas of mathematics, a subset A of a topological space X is called dense (in X) if, intuitively, any point in X can be well-approximated by points in A. Formally, A is dense in X if for any point x in X, any neighborhood of... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...

Games other than {0 | 0} may have value *. For example, the game * 2 + * 3, where the values are nimbers, has value * despite each player having more options than simply moving to 0. In mathematics, the proper class of nimbers (occasionally called Grundy numbers) is introduced in combinatorial game theory, where they are defined as the values of nim heaps, but arise in a much larger class of games because of the Sprague–Grundy theorem. ...

Why * 0

A combinatorial game has a positive and negative player; which player moves first is left ambiguous. The combinatorial game 0, or { | }, leaves no options and is a second-player win. Likewise, a combinatorial game is won (assuming optimal play) by the second player if and only if its value is 0. Therefore, a game of value *, which is a first-player win, is neither positive nor negative. However, * is not the only possible value for a first-player win game (see nimbers). This article may be too technical for most readers to understand. ... In combinatorial game theory, the zero game is the game where neither player has any legal options. ... It has been suggested that this article or section be merged with Logical biconditional. ... In mathematics, the proper class of nimbers is introduced in combinatorial game theory, where they arise as the sizes of nim heaps. ...

Star does have the property that * + * = 0, because the sum of two value-* games is the zero game; the first-player's only move is to the game *, which the second-player will win.

Example of a value-* game

Nim, with one pile and one piece, has value *. The first player will remove the piece, and the second player will lose. A single-pile Nim game with one pile of n pieces (also a first-player win) is defined to have value *n. The numbers *z for integers z form an infinite field of characteristic 2, when addition is defined in the context of combinatorial games and multiplication is given a more complex definition. Nim is a two-player mathematical game of strategy in which players take turns removing objects from distinct heaps. ... The integers are commonly denoted by the above symbol. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, the characteristic of a ring R with identity element 1R is defined to be the smallest positive integer n such that n1R = 0, where n1R is defined as 1R + ... + 1R with n summands. ...

See also

• Nimbers