Solving combinatorial problems: The 15-puzzle

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.

[1]  N. Maier.  Reasoning in humans. I. On direction. , 1930 .

[2]  E. Tolman,et al.  Studies in spatial learning: Orientation and the short-cut. , 1946, Journal of experimental psychology.

[3]  F. Bartlett.  Thinking: An Experimental and Social Study , 1958 .

[4]  R. Shepard.  The analysis of proximities: Multidimensional scaling with an unknown distance function. II , 1962 .

[5]  R. Shepard.  The analysis of proximities: Multidimensional scaling with an unknown distance function. I. , 1962 .

[6]  Nils J. Nilsson,et al.  Problem-solving methods in artificial intelligence , 1971, McGraw-Hill computer science series.

[7]  A. Rosenfeld,et al.  Edge and Curve Detection for Visual Scene Analysis , 1971, IEEE Transactions on Computers.

[8]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[9]  M. Garey.  Johnson: computers and intractability: a guide to the theory of np- completeness (freeman , 1979 .

[10]  Keith Price,et al.  Review of "Principles of Artificial Intelligence by Nils J. Nilsson", Tioga Publishing Company, Palo Alto, CA, ISBN 0-935382-01-1. , 1980, SGAR.

[11]  Nils J. Nilsson,et al.  Principles of Artificial Intelligence , 1980, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[12]  Richard E. Korf,et al.  Depth-First Iterative-Deepening: An Optimal Admissible Tree Search , 1985, Artif. Intell..

[13]  Manfred K. Warmuth,et al.  Finding a Shortest Solution for the N × N Extension of the 15-PUZZLE Is Intractable , 1986, AAAI.

[14]  N. Biggs.  THE TRAVELING SALESMAN PROBLEM A Guided Tour of Combinatorial Optimization , 1986 .

[15]  J. Metcalfe,et al.  Intuition in insight and noninsight problem solving , 1987, Memory & cognition.

[16]  A. Ramsay.  Formal Methods in Artificial Intelligence , 1988 .

[17]  Charles A. Bouman,et al.  Multiple Resolution Segmentation of Textured Images , 1991, IEEE Trans. Pattern Anal. Mach. Intell..

[18]  Azriel Rosenfeld,et al.  A Pyramid Framework for Early Vision: Multiresolutional Computer Vision , 1993 .

[19]  Azriel Rosenfeld,et al.  A Pyramid Framework for Early Vision , 1994 .

[20]  Peter Norvig,et al.  Artificial Intelligence: A Modern Approach , 1995 .

[21]  W Richards,et al.  Trajectory Mapping: A New Nonmetric Scaling Technique , 1995, Perception.

[22]  A. Rosenfeld,et al.  An exponential pyramid model of the time-course of size processing , 1995, Vision Research.

[23]  K. Upton,et al.  A modern approach , 1995 .

[24]  Kenji Yamaguchi,et al.  Complete Solution of the Eight-Puzzle , 1995 .

[25]  T. Ormerod,et al.  Human performance on the traveling salesman problem , 1996, Perception & psychophysics.

[26]  A. Rosenfeld,et al.  Curve Detection in a Noisy Image , 1997, Vision Research.

[27]  S. Payne,et al.  The Effects of Operator Implementation Cost on Planfulness of Problem Solving and Learning , 1998, Cognitive Psychology.

[28]  Eric W. Weisstein,et al.  The CRC concise encyclopedia of mathematics , 1999 .

[29]  Horst Bischof,et al.  Hierarchical, Adaptive, and Robust Methods for Image Understanding , 1999 .

[30]  A. Joshi,et al.  The traveling salesman problem: A hierarchical model , 2000, Memory & cognition.

[31]  N. Cowan.  The magical number 4 in short-term memory: A reconsideration of mental storage capacity , 2001, Behavioral and Brain Sciences.

[32]  Zygmunt Pizlo,et al.  Pyramid algorithms as models of human cognition , 2003, IS&T/SPIE Electronic Imaging.

[33]  M. Lee,et al.  The roles of the convex hull and the number of potential intersections in performance on visually presented traveling salesperson problems , 2003, Memory & cognition.

[34]  Zygmunt Pizlo,et al.  Graph pyramids as models of human problem solving , 2004, IS&T/SPIE Electronic Imaging.

[35]  Abraham P. Punnen,et al.  The traveling salesman problem and its variations , 2007 .