A Mathematician’s Gambit: “Checking” the Relationship Between Graph Theory and Linear Algebra in Chess

Abstract

Chess has long been used as a tool; from training the human mind for battle strategies to gauging the potential for artificial intelligence and computers, chess is more than a game but a way to learn. While it is unattainable to classify all 10120 possible games (called the Shannon number after mathematician Claude Shannon), special cases of specific board configurations can be studied to provide a broader understanding of the game. This thesis explores the mathematical modeling of chess in terms of dominance and non-attacking scenarios. A configuration of pieces of one specific type (bishops, knights, rooks, queens) is considered to have dominance if all of the squares are controlled on the whole board, with the goal to find the minimum number of pieces that lead to a dominant arrangement. A non-attacking configuration looks at the maximum number of pieces that can be placed on the board without threatening (i.e. attacking) each other. While not part of the game of chess itself, these chess puzzles can be used to learn the mathematical concepts using a format that a student is already familiar with: chess. This thesis is a synthesis of domination and non-attacking modeling strategies using the work of cliques from Foulds and Johnston and graph polynomials from Mertens to study how these opposing methods apply to each other. New board constraints are also an avenue for exploration, as are vector representations of the game