Integer Programming Based Algorithms for Peg Solitaire Problems
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.
Keywordspeg solitaire integer programming backtrack searching
Unable to display preview. Download preview PDF.
- 1.Avis, D., and Deza, A.: Solitaire Cones, Technical Report No. SOCS-96.8, 1996.Google Scholar
- 2.Beasley, J. D.: Some notes on Solitaire, Eureka, 25 (1962), 13–18.Google Scholar
- 4.de Bruijn, N. G.:A Solitaire Game and Its Relation to a Finite Field. Journal of Recreational Mathematics, 5 (1972), 133–137.Google Scholar
- 5.Cross, D. C.: Square Solitaire and variations. Journal of Recreational Mathematics, 1 (1968), 121–123.Google Scholar
- 6.Gardner, M.: Scientific American, 206 #6(June 1962), 156-166; 214 #2(Feb. 1966), 112–113; 214 #5(May 1966), 127.Google Scholar
- 7.Kanno, E.: Linear Programming Algorithm for Peg Solitaire Problems, Bachelor thesis, Department of Mathematical Engineering, Faculty of Engineering, University of Tokyo, 1997 (in Japanese).Google Scholar
- 8.Uehara, R., Iwata, S.: Generalized Hi-Q is NP-complete, Trans. IEICE, 73 (1990), 270–273.Google Scholar