Optimal algorithms for constructing knight's tours on arbitrary [formula omitted] chessboards

Abstract

The knight's tour problem is an ancient puzzle whose goal is to find out how to construct a series of legal moves made by a knight so that it visits every square of a chessboard exactly once. In previous works, researchers have partially solved this problem by offering algorithms for subsets of chessboards. For example, among prior studies, Parberry proposed a divided-and-conquer algorithm that can build a closed knight's tour on an n × n , an n × ( n + 1 ) or an n × ( n + 2 ) chessboard in O ( n 2 ) (i.e., linear in area) time on a sequential processor. In this paper we completely solve this problem by presenting new methods that can construct a closed knight's tour or an open knight's tour on an arbitrary n × m chessboard if such a solution exists. Our algorithms also run in linear time ( O ( nm ) ) on a sequential processor.