Ant Colonies Discover Knight’s Tours

  • Philip Hingston
  • Graham Kendall
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3339)


In this paper we introduce an Ant Colony Optimisation (ACO) algorithm to find solutions for the well-known Knight’s Tour problem. The algorithm utilizes the implicit parallelism of ACO’s to simultaneously search for tours starting from all positions on the chessboard. We compare the new algorithm to a recently reported genetic algorithm, and to a depth-first backtracking search using Warnsdorff’s heuristic. The new algorithm is superior in terms of search bias and also in terms of the rate of finding solutions.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Murray, H.J.R.: History of Chess (1913)Google Scholar
  2. 2.
    Euler, L.: Mémoires de l’Academie Royale des Sciences et Belles Lettres. Année 15, 310–337 (1766)Google Scholar
  3. 3.
    Mordecki, E.: On the Number of Knight’s Tours. Pre-publicaciones de Matematica de la Universidad de la Republica, Uruguay, 2001/57 (2001),
  4. 4.
    Löbbing, M., Wegener, I.: The Number of Knight’s Tours Equals 33,439,123,484,294 – Counting with Binary Decision Diagrams. Electronic Journal of Combinatorics 3(1), R5 (1996)Google Scholar
  5. 5.
    McKay, B.D.: Knight’s tours of an 8x8 chessboard, Tech. Rpt. TR-CS-97-03, Dept. Computer Science, Australian National University (1997)Google Scholar
  6. 6.
    Warnsdorff, H.C.: Des Rösselsprungs einfachste und allgemeinste Lösung. Schmalkalden (1823)Google Scholar
  7. 7.
    Gordon, V.S., Slocum, T.J.: The Knight’s Tour – Evolutionary vs. Depth-First Search. In: Proceedings of the Congress of Evolutionary Computation 2004 (CEC 2004), Portland, Oregon, pp. 1435–1440 (2004)Google Scholar
  8. 8.
    Goldberg, D.: Genetic Algorithms in Search, Optimization, and Machine Learning. Addison-Wesley, Reading (1989)MATHGoogle Scholar
  9. 9.
    Dorigo, M.: Optimization, Learning and Natural Algorithms. Ph.D.Thesis, Politecnico di Milano, Italy (in Italian) (1992)Google Scholar
  10. 10.
    Dorigo, M., Maniezzo, V., Colorni, A.: The Ant System: Optimization by a Colony of Cooperating Agents. IEEE Transactions on Systems, Man, and Cybernetics-Part B 26(1), 29–41 (1996)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Philip Hingston
    • 1
  • Graham Kendall
    • 2
  1. 1.Edith Cowan UniversityAustralia
  2. 2.The University of NottinghamUK

Personalised recommendations