Knight’s Tours on Cylindrical and Toroidal Boards with One Square Removed

Abstract

The following two theorems are proved:
A closed knight’s tour exists on all \(m \times n\) boards wrapped onto a cylinder so that the \(m\) rows go around the cylinder, with one square removed, with the exception of the following boards:

(a) \(n\) is even,

(b) \(m \in \{1,2\}\)

(c) \(m = 4\) and the removed square is in row 2 or 3;

(d) \(m \geq 5\), \(n = 1\), and the removed square is in row 2, 3, …, or \(m-1\).

A closed knight’s tour exists on all \(m \times n\) boards wrapped onto a torus with one square removed except boards with \(m\) and \(n\) both even and \(1 \times 1\),\(1 \times 2\) and \(2 \times 1\) boards.