Exploring Chess Through Linear Algebra

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; it is a way to learn. While it is unattainable to classify all possible 10120games (called the Shannon number1 after mathematician Claude Shannon), certain 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 dominance configuration consists of pieces of one specific type (bishops, knights, rooks, queens) which control the whole board; the player’s goal is to achieve that control using the minimum number of pieces. 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 puzzles can be used to learn mathematical concepts using a familiar format. This thesis is a synthesis of domination and non-attacking modeling strategies using the work of cliques from Foulds and Johnston to study domination. New board constraints are also an avenue for exploration, as are vector representations of the game