Table of Contents
Journal of Discrete Mathematics
Volume 2014 (2014), Article ID 210892, 9 pages
http://dx.doi.org/10.1155/2014/210892
Research Article

Knight’s Tours on Rectangular Chessboards Using External Squares

1University of Wisconsin Oshkosh, Oshkosh, WI 54901, USA
2Saginaw Valley State University, University Center, MI 48710, USA

Received 27 May 2014; Revised 12 November 2014; Accepted 17 November 2014; Published 9 December 2014

Academic Editor: Stavros D. Nikolopoulos

Copyright © 2014 Grady Bullington et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The classic puzzle of finding a closed knight’s tour on a chessboard consists of moving a knight from square to square in such a way that it lands on every square once and returns to its starting point. The 8 × 8 chessboard can easily be extended to rectangular boards, and in 1991, A. Schwenk characterized all rectangular boards that have a closed knight’s tour. More recently, Demaio and Hippchen investigated the impossible boards and determined the fewest number of squares that must be removed from a rectangular board so that the remaining board has a closed knight’s tour. In this paper we define an extended closed knight’s tour for a rectangular chessboard as a closed knight’s tour that includes all squares of the board and possibly additional squares beyond the boundaries of the board and answer the following question: how many squares must be added to a rectangular chessboard so that the new board has a closed knight’s tour?