In combinatorial game theory, up, written as ↑ {\displaystyle \uparrow } , 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. ↑ {\displaystyle \uparrow } may be represented by the surreal form ↑= { 0 ∣ ∗ } . {\displaystyle \uparrow =\{0\mid *\}.} Because ↑> 0 {\displaystyle \uparrow >0} , Left always wins a game with value ↑ {\displaystyle \uparrow } . Its additive inverse, down, written as ↓ {\displaystyle \downarrow } , may be represented as ↓= { ∗ ∣ 0 } . {\displaystyle \downarrow =\{*\mid 0\}.} Similarly, ↓ {\displaystyle \downarrow } is less than zero and greater than all negative numbers.
↑ {\displaystyle \uparrow } and ↓ {\displaystyle \downarrow } are equivalent to + 0 {\displaystyle +_{0}} and − 0 {\displaystyle -_{0}} , tiny-0 and miny-0.
Value- ↑ {\displaystyle \uparrow } games
Hackenbush
In Hackenbush, a 1-petaled delphinium, that is, a blue edge on top of a green edge, has value { 0 , ∗ ∣ 0 } =↑ + ∗ {\displaystyle \{0,*\mid 0\}=\uparrow +*} . Hence, a green edge along with a 1-petaled delphinium has value ↑ + ∗ + ∗ =↑ {\displaystyle \uparrow +*+*=\uparrow } .
Col Because a Col position can have only values z {\displaystyle z} or z + ∗ {\displaystyle z+*} for some number z {\displaystyle z} , no Col position can have value ↑ {\displaystyle \uparrow } .
Toads and Frogs In Toads and Frogs, ↑ {\displaystyle \uparrow } may be represented by the position T ◻ T F F = { ◻ T T F F ∣ T F T ◻ F } = { 0 ∣ { T F ◻ T F ∣ T F T F ◻ } } = { 0 ∣ { 0 ∣ 0 } } =↑ . {\displaystyle T\square TFF=\{\square TTFF\mid TFT\square F\}=\{0\mid \{TF\square TF\mid TFTF\square \}\}=\{0\mid \{0\mid 0\}\}=\uparrow .}
Confusion with nimbers The only nimber confused with ↑ {\displaystyle \uparrow } is ∗ {\displaystyle *} . This is easily illustrated by the Hackenbush example, as a blue delphinium and a green edge along with a ∗ 2 {\displaystyle *2} position is not fuzzy.
See also Star Tiny and miny
References