Which Chessboards have a Closed Knight's Tour within the Cube?

Which Chessboards have a Closed Knight's Tour within the Rectangular Prism?

A closed knight’s tour of a chessboard uses legal moves of the knight to visit every square exactly once and return to its starting position. In 1991 Schwenk completely classified the m × n rectangular chessboards that admit a closed knight’s tour. In honor of the upcoming twentieth anniversary of the publication of Schwenk’s paper, this article extends his result by classifying the i × j × k r...

Which Rectangular Chessboards Have a Knight's Tour?

The Closed Knight Tour Problem in Higher Dimensions

The problem of existence of closed knight tours for rectangular chessboards was solved by Schwenk in 1991. Last year, in 2011, DeMaio and Mathew provide an extension of this result for 3-dimensional rectangular boards. In this article, we give the solution for n-dimensional rectangular boards, for n > 4.

An Efficient Algorithm for the Knight's Tour Problem

A knight’s tour is a series of moves made by a knight visiting every square of an n x n chessboard exactly once. The knight’s tour problem is the problem of constructing such a tour, given n. A knight’s tour is called closed if the last square visited is also reachable from the first square by a knight’s move, and open otherwise. Define the knight’s graph for an n x n chessboard to be the graph...

Generalised Knight's Tours

The problem of existence of closed knight’s tours in [n]d, where [n] = {0, 1, 2, . . . , n− 1}, was recently solved by Erde, Golénia, and Golénia. They raised the same question for a generalised, (a, b) knight, which is allowed to move along any two axes of [n]d by a and b unit lengths respectively. Given an even number a, we show that the [n]d grid admits an (a, 1) knight’s tour for sufficient...