Memory & Cognition

, Volume 33, Issue 6, pp 1069–1084 | Cite as

Solving combinatorial problems: The 15-puzzle

  • Zygmunt Pizlo
  • Zheng Li


We present a series of experiments in which human subjects were tested with a well-known combinatorial problem called the15-puzzle and in different-sized variants of this puzzle. Subjects can solve these puzzles reliably by systematically building a solution path, without performing much search and without using distances among the states of the problem. The computational complexity of the underlying mental mechanisms is very low. We formulated a computational model of the underlying cognitive processes on the basis of our results. This model applied a pyramid algorithm to individual stages of each problem. The model’s performance proved to be quite similar to the subjects’ performance. Partial support for this research was provided by the Air Force Office of Scientific Research.


Reference State Travel Salesman Problem Goal State Combinatorial Problem Solution Path 
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  1. Bartlett, F. C. (1958).Thinking: An experimental and social study. London: Allen & Unwin.Google Scholar
  2. Bouman, C., &Liu, B. (1991). Multiple resolution segmentation of textured images.IEEE Transactions on Pattern Analysis & Machine Intelligence,13, 99–113.CrossRefGoogle Scholar
  3. Cowan, N. (2001). The magical number 4 in short-term memory: A consideration of mental storage capacity.Behavioral & Brain Sciences,24, 87–185.CrossRefGoogle Scholar
  4. Garey, M. R., &Johnson, D. S. (1979).Computers and intractability: A guide to the theory of NP-completeness. New York: Freeman.Google Scholar
  5. Graham, S. M., Joshi, A., &Pizlo, Z. (2000). The traveling salesman problem: A hierarchical model.Memory & Cognition,28, 1191–1204.Google Scholar
  6. Gutin, G., &Punnen, A. P. (2002).The traveling salesman problem and its variations. Boston: Kluwer.Google Scholar
  7. Jolion, J.-M., &Rosenfeld, A. (1994).A pyramid framework for early vision: Multiresolutional computer vision. Dordrecht: Kluwer.Google Scholar
  8. Korf, R. E. (1985). Depth-first iterative-deepening: An optimal admissible tree search.Artificial Intelligence,27, 97–109.CrossRefGoogle Scholar
  9. Kropatsch, W. G., Leonardis, A., &Bischof, H. (1999). Hierarchical, adaptive, and robust methods for image understanding.Surveys on Mathematics for Industry,9, 1–47.Google Scholar
  10. Lawler, E. L., Lenstra, J. K., Rinnooy Kan, A. H. G., &Shmoys, D. B. (1985).The traveling salesman problem: A guided tour of combinatorial optimization. New York: Wiley.Google Scholar
  11. MacGregor, J. N., &Ormerod, T. (1996). Human performance on the traveling salesman problem.Perception & Psychophysics,58, 527–539.Google Scholar
  12. Maier, N. R. F. (1930). Reasoning in humans: On direction.Journal of Comparative Psychology,10, 115–143.CrossRefGoogle Scholar
  13. Metcalfe, J., &Wiebe, D. (1987). Intuition in insight and noninsight problem solving.Memory & Cognition,15, 238–246.Google Scholar
  14. Newell, A., &Ernst, G. (1965). The search for generality. In W. A. Kalenich (Ed.),Information processing: Proceedings of IFIP Congress (Vol. 1, pp. 17–24). Washington, DC: Spartan.Google Scholar
  15. Nilsson, N. J. (1971).Problem-solving methods in artificial intelligence. New York: McGraw-Hill.Google Scholar
  16. Nilsson, N. J. (1980).Principles of artificial intelligence. Palo Alto, CA: Morgan Kaufmann.Google Scholar
  17. O’Hara, K. P., &Payne, S. J. (1998). The effects of operator implementation cost on planfulness of problem solving and learning.Cognitive Psychology,35, 34–70.CrossRefPubMedGoogle Scholar
  18. Pizlo, Z., &Li, Z. (2003). Pyramid algorithms as models of human cognition. In C. A. Bouman & R. L. Stevenson (Eds.),Computational imaging (Proceedings of SPIE-IS&T Conference on Electronic Imaging, Vol. 5016, pp. 1–12). Bellingham, WA: SPIE.Google Scholar
  19. Pizlo, Z., &Li, Z. (2004). Graph pyramids as models of human problem solving. In C. A. Bouman & E. L. Miller (Eds.),Computational imaging II (Proceedings of the SPIE-IS&T Conference on Electronic Imaging, Computational Imaging, Vol. 5299, pp. 205–215). Bellingham, WA: SPIE.Google Scholar
  20. Pizlo, Z., Rosenfeld, A., &Epelboim, J. (1995). An exponential pyramid model of the time-course of size processing.Vision Research,35, 1089–1107.CrossRefPubMedGoogle Scholar
  21. Pizlo, Z., Salach-Golyska, M., &Rosenfeld, A. (1997). Curve detection in a noisy image.Vision Research,37, 1217–1241.CrossRefPubMedGoogle Scholar
  22. Ratner, D., &Warmuth, M. K. (1986).Finding a shortest solution for then × n extension of the 15-puzzle is intractable. Proceedings of the Fifth National Conference on Artificial Intelligence (Vol. 1, pp. 168–172). San Mateo, CA: Morgan Kaufmann.Google Scholar
  23. Richards, W., &Koenderink, J. J. (1995). Trajectory mapping: A new nonmetric scaling technique.Perception,24, 1315–1331.CrossRefPubMedGoogle Scholar
  24. Rosenfeld, A., &Thurston, M. (1971). Edge and curve detection for visual scene analysis.IEEE Transactions on Computers,C-20, 562–569.CrossRefGoogle Scholar
  25. Russell, S. J., &Norvig, P. (1995).Artificial intelligence. A modern approach. Upper Saddle River, NJ: Prentice-Hall.Google Scholar
  26. Schofield, P. D. A. (1967). Complete solution of the eight puzzle. In E. Dale & D. Michie (Eds.),Machine intelligence (Vol. 2, pp. 125–133). New York: Elsevier.Google Scholar
  27. Shepard, R. N. (1962). The analysis of proximities: Multidimensional scaling with an unknown distance function. Part I.Psychometrika,27, 125–140.CrossRefGoogle Scholar
  28. Tolman, E. C., Ritchie, B. F., &Kalish, D. (1946). Studies in spatial learning: I. Orientation and the short-cut.Journal of Experimental Psychology,36, 13–24.CrossRefPubMedGoogle Scholar
  29. Vickers, D., Lee, M. D., Dry, M., &Hughes, P. (2003). The roles of the convex hull and the number of potential intersections in performance on visually presented traveling salesperson problems.Memory & Cognition,31, 1094–1104.Google Scholar
  30. Weisstein, E. W. (2003).CRC concise encyclopedia of mathematics. New York: Chapman & Hall.Google Scholar

Copyright information

© Psychonomic Society, Inc. 2005

Authors and Affiliations

  1. 1.Department of Psychological SciencesPurdue UniversityWest Lafayette

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