A Study of Recursion Relations and Supermex Operators in Simple Combinatorial Games for Application in Renormalization.

This study investigates recursion relations and supermex operators within the frameworks of simple combinatorial games: Three-Stack Nim, Three-Row Chomp, and a variant of Three-Stack Nim. Game positions were mapped to 3D coordinates [x, y, z] and represented as separately indexed, semi-infinite 2D sheets. Loser sheets (L-sheets) and Instant Winner sheets (IN-sheets or W-sheets) were defined to identify P-positions (previous player wins) and N-positions (next player to move wins). These sheets were related by the recursion operator (Wx+1 = RWx), and supermex operator, (Lx = MWx), to generate all the possible P-positions. For the variant Nim game, auxiliary sheets were required to explain the recursion relation. Deriving these operators is the base for applying renormalization, a method which allows one to calculate detailed probabilities of the geometries of a game’s P-positions.