Integer Programming Based Algorithms for Peg Solitaire Problems
 Masashi Kiyomi,
 Tomomi Matsui
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Abstract
Peg solitaire is a one player game using pegs and a board with some holes. The game is classical, and nowadays sold in many parts of the world under the trade name of HiQ.
In this paper, we dealt with the peg solitaire problem as an integer programming problem. We proposed algorithms based on the backtrack search method and relaxation methods for integer programming problem.
The algorithms first solve relaxed problems and get an upper bound of the number of jumps for each jump position. This upper bound saves much time at the next stage of backtrack searching. While solving the relaxed problems, we can prove many peg solitaire problems are infeasible. We proposed two types of backtrack searching, forwardonly searching and forwardbackward searching. The performance of these two methods highly depends on the symmetricity and the length of the sequence of required jumps. Our algorithm can solve all the peg solitaire problem instances we tried and the total computational time is less than 20 minutes on an ordinary notebook personal computer.
 Avis, D., and Deza, A.: Solitaire Cones, Technical Report No. SOCS96.8, 1996.
 Beasley, J. D. (1962) Some notes on Solitaire. Eureka 25: pp. 1318
 Berlekamp, E. R., Conway, J. H., Guy, R. K. (1982) Winning Ways for Mathematical Plays. Academic Press, London
 Bruijn, N. G. (1972) A Solitaire Game and Its Relation to a Finite Field. Journal of Recreational Mathematics 5: pp. 133137
 Cross, D. C. (1968) Square Solitaire and variations. Journal of Recreational Mathematics 1: pp. 121123
 Gardner, M. (1966) Scientific American 214: pp. 112113
 Kanno, E.: Linear Programming Algorithm for Peg Solitaire Problems, Bachelor thesis, Department of Mathematical Engineering, Faculty of Engineering, University of Tokyo, 1997 (in Japanese).
 Uehara, R., Iwata, S. (1990) Generalized HiQ is NPcomplete. Trans. IEICE 73: pp. 270273
 Title
 Integer Programming Based Algorithms for Peg Solitaire Problems
 Book Title
 Computers and Games
 Book Subtitle
 Second International Conference, CG 2000 Hamamatsu, Japan, October 26–28, 2000 Revised Papers
 Book Part
 Part 3
 Pages
 pp 229240
 Copyright
 2001
 DOI
 10.1007/3540455795_15
 Print ISBN
 9783540430803
 Online ISBN
 9783540455790
 Series Title
 Lecture Notes in Computer Science
 Series Volume
 2063
 Series ISSN
 03029743
 Publisher
 Springer Berlin Heidelberg
 Copyright Holder
 SpringerVerlag Berlin Heidelberg
 Additional Links
 Topics
 Keywords

 peg solitaire
 integer programming
 backtrack searching
 Industry Sectors
 eBook Packages
 Editors

 Tony Marsland ^{(4)}
 Ian Frank ^{(5)}
 Editor Affiliations

 4. Department of Computer Science, University of Alberta
 5. Future University  Hakodate
 Authors

 Masashi Kiyomi ^{(6)}
 Tomomi Matsui ^{(6)}
 Author Affiliations

 6. Department of Mathematical Engineering and Information Physics Graduate School of Engineering, University of Tokyo, 731, Hongo, Bunkyoku, Tokyo, 1138656, Japan
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