Term in combinatorial game theory

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 the surreal form **{0|0}**. This game is an unconditional first-player win.

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 less than all positive rational numbers, and greater than all negative rationals.

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.

## 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).

Star does have the property that the sum * + *, has value 0, because 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.

## See also[]

## References[]