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) $m\ge 5$, $n=1$, 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.