Knight's tours on boards with odd dimensions

Abstract

A closed knight’s tour of a board consists of a sequence of knight moves, where each square is visited exactly once and the sequence begins and ends with the same square. For boards of size $m\times n$ where $m$ and $n$ are odd, a tour is impossible because there are unequal numbers of white and black squares. By deleting a square, we can fix this disparity, and we determine which square to remove to allow for a closed knight’s tour.