Sprague-Grundy theorem

The game can be analysed using the Sprague-Grundy theorem.

They arise in a much larger class of games because of the Sprague-Grundy theorem.

The Sprague-Grundy theorem states that every impartial game is equivalent to a nimber.

A position can be expressed as a sum of other positions if it is equivalent (see Sprague-Grundy theorem).

Partisan games are more difficult to analyze than impartial games, as the Sprague-Grundy theorem does not apply.

It seems only a small variation in the rules, but it results in a completely different game, that can be analyzed with the Sprague-Grundy theorem.

By the Sprague-Grundy theorem, is the mex over all possible moves of the nim-sum of the nim-values of the two resulting sections.

The Sprague-Grundy theorem implies that a heap of size n is equivalent to a nim heap of a given size, usually noted G(n).

This is the fundamental operation that is used in the Sprague-Grundy theorem for impartial games and which led to the field of combinatorial game theory for partisan games.

The Sprague-Grundy theorem has been developed into the field of combinatorial game theory, notably by E. R. Berlekamp, John Horton Conway and others.

Since Cram is an impartial game, the Sprague-Grundy theorem indicates that in the normal version any Cram position is equivalent to a nim-heap of a given size, also called the Grundy value.

In the original folklore version of Hackenbush, any player is allowed to cut any edge: as this is an impartial game it is comparatively straightforward to give a complete analysis using the Sprague-Grundy theorem.

For the purposes of the Sprague-Grundy theorem, a game is a two-player game of perfect information satisfying the ending condition (all games come to an end: there are no infinite lines of play) and the normal play condition (a player who cannot move loses).

The Sprague-Grundy theorem applies to impartial games (in which each move may be played by either player) and asserts that every such game has an equivalent Sprague-Grundy value, a "nimber", which indicates the number of pieces in an equivalent position in the game of nim.

Normal play Nim (or more precisely the system of nimbers) is fundamental to the Sprague-Grundy theorem, which essentially says that in normal play every impartial game is equivalent to a Nim heap that yields the same outcome when played in parallel with other normal play impartial games (see disjunctive sum).