Combinatorial Games

This paper examines two classical combinatorial games, Nim and the Domino Game on Linear Strips, through the shared principles of parity, recursion, and binary structure. It begins by formalizing Nim, where each position’s outcome is determined by the Nim-sum (bitwise XOR) of heap sizes: positions with zero Nim-sum are losing, and optimal play involves forcing this condition. The analysis then extends to the Domino Game, where players alternately place 2-cell dominoes on a linear board. Using recursive classification, the paper identifies winning and losing positions and introduces the mirror strategy, a symmetry-based method ensuring victory on even-length boards. Finally, the study unifies both games within the framework of impartial game theory, showing how binary parity in Nim parallels recursive decomposition in Domino tiling. The discussion concludes by suggesting extensions to multidimensional and misère variants using Grundy numbers and recursive combinatorial methods.