Richardson extrapolation

2013-11-30T15:10:15Z

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{{About|the concept from combinatorial game theory|the board game Star|Star (board game)|the board game *Star|*Star}}
In [[combinatorial game theory]], '''star''', written as '''<math>*</math>''' or '''<math>*1</math>''', 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 Horton Conway|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 number|positive]] nor [[negative number|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 number]]s, and greater than all negative rationals. Since the rationals are [[Dense set|dense]] in the [[real number|reals]], this also makes * greater than any negative real, and less than any positive real.

Games other than {0 | 0} may have value *. For example, the game <math>*2 + *3</math>, 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 [[zero (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 [[nimber]]s).

Star does have the property that * + * = 0, because the [[sum of combinatorial games|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 [[integer]]s ''z'' form an infinite [[field (mathematics)|field]] of [[Characteristic (algebra)|characteristic]] 2, when addition is defined in the context of combinatorial games and multiplication is given a more complex definition.

==See also==
* [[Nimber]]s
* [[Surreal number]]s
==References==
*[[John Horton Conway|Conway, J. H.]], ''[[On Numbers and Games]],'' [[Academic Press]] Inc. (London) Ltd., 1976

{{DEFAULTSORT:Star (Game)}}
[[Category:Combinatorial game theory]]

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Richardson extrapolation