Editing Combinatorial game theory (section)

==Nimbers==
An [[impartial game]] is one where, at every position of the game, the same moves are available to both players. For instance, [[Nim]] is impartial, as any set of objects that can be removed by one player can be removed by the other. However, [[domineering]] is not impartial, because one player places horizontal dominoes and the other places vertical ones. Likewise Checkers is not impartial, since the players own different colored pieces. For any [[ordinal number]], one can define an impartial game generalizing Nim in which, on each move, either player may replace the number with any smaller ordinal number; the games defined in this way are known as [[nimber]]s. The [[Sprague–Grundy theorem]] states that every impartial game under the [[normal play convention]] is equivalent to a nimber.

The "smallest" nimbers – the simplest and least under the usual ordering of the ordinals – are 0 and ∗.