Master of Science (MS)
Chess, Knight's Tours, Zeta Function
Given an m × n chessboard, we get an associated graph by letting each square represent a vertex and by joining two vertices if there is a valid move by a knight between the corresponding squares. A knight’s tour is a sequence of moves in which the knight lands on every square exactly once, i.e., a Hamiltonian path on the associated graph. Knight’s tours have an interesting history. One interesting mistake regarding Knight’s Tours was made by the famous mathematician Euler. His mistake led to the further study of knight’s tours on 3 × n chessboards. We will explore and explain a method found by Donald Knuth for enumerating the number k(n) of all closed knight’s tours on a 3 × (2n + 8) chessboard for an integer n ≥ 1. Interestingly, there is a 21-term recurrence relation for k(n) discovered independently by Knuth and Elkies. This relation comes from studying generating functions which can be interpreted in the context of the Ihara zeta function of a certain graph.
Brown, Alfred James, "Knight's Tours and Zeta Functions" (2017). Master's Theses. 4836.