Ppositions

Ppositions, in the field of combinatorial game theory, are the positions from which the player who just moved (the previous player) can force a win under normal play, assuming perfect play by both players. Equivalently, a P-position is a position in which the player about to move has no winning move.

They are defined recursively: a position is a P-position if all legal moves lead to N-positions (positions from which the next player can win). A position is an N-position if there exists at least one legal move to a P-position. The empty position, where no moves are possible, is by convention a P-position.

In practice, P-positions help structure winning strategies. From a P-position, every possible move goes to an N-position, meaning the current player cannot force a win from that position. From an N-position, there exists a move to a P-position, providing a path to victory with optimal play. Beyond general theory, specific games yield particular classifications. For example, in Nim with several heaps, a position is P if the bitwise XOR (nim-sum) of the heap sizes is zero. In a simple subtraction game where a player may remove 1 or 2 stones, P-positions occur at heap sizes that are multiples of 3.

P-positions underpin the Sprague-Grundy theory that assigns a numeric Grundy value to every position, generalizing the P/N dichotomy. They are defined for normal play; misère variants alter the classification and are typically more complex. The term P stands for previous player, contrasting with N, the next player to move.