Sprague–Grundy theorem

The Sprague–Grundy theorem is a fundamental result in combinatorial game theory that provides a way to analyze impartial games (games where the allowed moves depend only on the current position and not on the player). It is especially applicable to games that can be modeled as a collection of independent sub-games. Here's a brief overview of the theorem and its implications: ### Key Concepts: 1. **Impartial Games**: These are games in which both players have the same available moves from any given position.