In combinatorial game theory, up, written as , is the value of the game where Left can only move to a zero game, while Right can only move to a star game. It is greater than zero and less than all positive numbers, hence it is an infinitesimal. may be represented by the surreal formBecause , Left always wins a game with value . Its additive inverse, down, written as , may be represented asSimilarly, is less than zero and greater than all negative numbers.
and are equivalent to and , tiny-0 and miny-0.
Value- games
Hackenbush
In Hackenbush, a 1-petaled delphinium, that is, a blue edge on top of a green edge, has value . Hence, a green edge along with a 1-petaled delphinium has value .
Col Because a Col position can have only values or for some number , no Col position can have value .
Toads and Frogs In Toads and Frogs, may be represented by the position
Confusion with nimbers The only nimber confused with is . This is easily illustrated by the Hackenbush example, as a blue delphinium and a green edge along with a position is not fuzzy.
See also
Star Tiny and miny
References