Integer Programming Based Algorithms for Peg Solitaire Problems
Purchase on Springer.com
$29.95 / €24.95 / £19.95*
* Final gross prices may vary according to local VAT.
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 Hi-Q.
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, forward-only searching and forward-backward 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. SOCS-96.8, 1996.
- Beasley, J. D.: Some notes on Solitaire, Eureka, 25 (1962), 13–18.
- Berlekamp, E. R., Conway, J. H., and Guy, R. K.: Winning Ways for Mathematical Plays. Academic Press, London, 1982.
- de Bruijn, N. G.:A Solitaire Game and Its Relation to a Finite Field. Journal of Recreational Mathematics, 5 (1972), 133–137.
- Cross, D. C.: Square Solitaire and variations. Journal of Recreational Mathematics, 1 (1968), 121–123.
- Gardner, M.: Scientific American, 206 #6(June 1962), 156-166; 214 #2(Feb. 1966), 112–113; 214 #5(May 1966), 127.
- 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.: Generalized Hi-Q is NP-complete, Trans. IEICE, 73 (1990), 270–273.
- 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
- pp 229-240
- Print ISBN
- Online ISBN
- Series Title
- Lecture Notes in Computer Science
- Series Volume
- Series ISSN
- Springer Berlin Heidelberg
- Copyright Holder
- Springer-Verlag Berlin Heidelberg
- Additional Links
- peg solitaire
- integer programming
- backtrack searching
- Industry Sectors
- eBook Packages
- Editor Affiliations
- 4. Department of Computer Science, University of Alberta
- 5. Future University - Hakodate
- Author Affiliations
- 6. Department of Mathematical Engineering and Information Physics Graduate School of Engineering, University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo, 113-8656, Japan
To view the rest of this content please follow the download PDF link above.