Optimal algorithms for constructing knight's tours on arbitrary n×m chessboards

Shun Shii Lin*, Chung Liang Wei

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

21 Citations (Scopus)


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(n2) (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.

Original languageEnglish
Pages (from-to)219-232
Number of pages14
JournalDiscrete Applied Mathematics
Issue number3
Publication statusPublished - 2005 Mar 15


  • Divide-and-conquer algorithm
  • Knight's tour problem
  • Optimal algorithm

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics


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