Editing Combinatorial game theory (section)

==Game abbreviations==

===Numbers===
Numbers represent the number of free moves, or the move advantage of a particular player. By convention positive numbers represent an advantage for Left, while negative numbers represent an advantage for Right. They are defined recursively with 0 being the base case.
: 0 = {|}
: 1 = {0|}, 2 = {1|}, 3 = {2|}
: <nowiki>−1 = {|0}, −2 = {|−1}, −3 = {|−2}</nowiki>

The [[zero game]] is a loss for the first player.

The sum of number games behaves like the integers, for example 3 + −2 = 1.

Any game number is in the class of the [[surreal numbers]].

===Star===
{{Main article|Star (game theory)}}
''Star'', written as ∗ or {0|0}, is a first-player win since either player must (if first to move in the game) move to a zero game, and therefore win.

: ∗ + ∗ = 0, because the first player must turn one copy of ∗ to a 0, and then the other player will have to turn the other copy of ∗ to a 0 as well; at this point, the first player would lose, since 0 + 0 admits no moves.

The game ∗ is neither positive nor negative; it and all other games in which the first player wins (regardless of which side the player is on) are said to be ''[[fuzzy game|fuzzy]]'' or ''confused with 0''; symbolically, we write ∗ || 0.

The game ∗n is notation for {0, ∗, …, ∗(n−1)| 0, ∗, …, ∗(n−1)}, which is also representative of normal-play [[Nim]] with a single heap of n objects. (Note that ∗0 = 0 and ∗1 = ∗.)

===Up and down===
''Up'', written as ↑, is a position in combinatorial game theory.<ref name=winningways>{{cite book |author1=E. Berlekamp |author2=J. H. Conway |author3=R. Guy | title=[[Winning Ways for your Mathematical Plays]] | volume=I | publisher=Academic Press | year=1982 | isbn=0-12-091101-9}}<br />{{cite book |author1=E. Berlekamp |author2=J. H. Conway |author3=R. Guy | title=Winning Ways for your Mathematical Plays | volume=II | publisher=Academic Press | year=1982 | isbn=0-12-091102-7}}</ref> In standard notation, ↑ = {0|∗}. Its negative is called ''down''.

: −↑ = ↓ (''down'')

Up is strictly positive (↑ > 0), and down is strictly negative (↓ < 0), but both are [[infinitesimal]]. Up and down are defined in ''[[Winning Ways for your Mathematical Plays]]''.

==="Hot" games===
{{Main article|hot game}}
Consider the game {1|−1}. Both moves in this game are an advantage for the player who makes them; so the game is said to be "hot;" it is greater than any number less than −1, less than any number greater than 1, and fuzzy with any number in between. It is written as ±1. Note that a subclass of hot games, referred to as ±n for some numerical game n is a switch game. Switch games can be added to numbers, or multiplied by positive ones, in the expected fashion; for example, 4 ± 1 = {5|3}.