Dynamic programming and board games: A survey

European Journal of Operational Research

Volumn 176, Issue 3, 2007, Pages 1299-1318

  Smith, David K ^a  

^a UNIVERSITY OF EXETER   (United Kingdom)

Author keywords

Dynamic programming; Games; Leisure

Indexed keywords

GAMES; LEISURE; OPTIMAL STRATEGY;

GAME THEORY; OPERATIONS RESEARCH; OPTIMAL CONTROL SYSTEMS; PROBLEM SOLVING;

DYNAMIC PROGRAMMING;

EID: 33750004140     PISSN: 03772217     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.ejor.2005.10.026     Document Type: Article

References (48)

1. Aldous D., and Diaconis P. Longest increasing subsequences: From patience sorting to the Baik-Deift-Johansson theorem. Bulletin of the American Mathematical Society 36 4 (1999) 413-432
2. Allen J.D. A note on the computer solution of Connect-Four. In: Levy D.N.L., and Beal D.F. (Eds). Heuristic Programming in Artificial Intelligence, vol. 1 (1989), Ellis Horwood, Chichester, UK 134-135
3. L. Victor Allis, Searching for solutions in games and artificial intelligence, Ph.D. Thesis, University of Limburg, Maastricht, Netherlands, 1994.
4. Victor Allis L., van der Meulen M., and van den Herik H.J. Proof-number search. Artificial Intelligence 66 1 (1994) 91-124
5. Anon, The Freecell Project, This was documented on the Web, but all links to the proof now appear to be missing. There is a useful web-site about Freecell and its history at: .
6. Ascher M. Mu Torere: An analysis of a Maori game. Mathematics Magazine 60 2 (1987) 90-100
7. Beasley J.D. The Ins and Outs of Peg Solitaire (1992), Oxford Paperbacks
8. J. Bell, D.K. Smith, Iterative dynamic programming and the end of some board games, in: Presented at the EURO Conference, Istanbul July 2003. Available from: , July 2003.
9. Bell R., and Cornelius M. Board Games Around the world: A Resource Book for Mathematical Investigations (1988), Cambridge University Press, Cambridge, UK
10. Bell R.C. Board and Table Games from Many Civilizations (1980), Dover Publications
11. Bellman R. On the application of dynamic programming to the determination of optimal play in chess and checkers. Proceedings of the National Academy of Sciences 53 (1965) 244-247
12. Berlekamp E.R., Conway J.H., and Guy R.K. Winning Ways for your Mathematical Plays vols. 1&2 (1982), Academic Press, London
13. Bouton C.L. Nim, a game with a complete mathematical theory. Annals of Mathematics 3 (1902) 35-39
14. Campbell P.J. Farmer klaus and the mouse. The UMAP Journal 23 2 (2002) 121-134
15. Chlond M.J., and Akyol O. A Nimatron. INFORMS Transactions on Education 3 3 (2003). Available from:
16. S. Fish, Freecell solver. A program in C and discussion. Available from: .
17. Gasser R. Solving nine men's Morris. Games of No Chance 29 (1999), Cambridge University Press for MSRI 101-113. Available from:
18. Guy R.K. What is a game?. In: Nowakowski R.J. (Ed). Games of No Chance (1996), Cambridge University Press, Cambridge, UK 43-60
19. Irving G., Donkers J., and Uiterwijk J.W.H.M. Solving kalah. International Computer Games Association Journal 23 4 (2000) 139-148
20. Kohler D. Optimal strategies for the game of darts. Journal of the Operational Research Society 33 10 (1982) 871-884
21. Koole G. An optimal dice rolling policy for Risk. Nieuw Archief voor Wiskunde 12 1-2 (1994) 49-52
22. Koole G. Stochastisch dynamisch programmeren. Nieuwe Wiskrant. Tijdschrift voor Nederlands Wiskundeonderwijs. 21 3 (2002) 23-26
23. Kostreva M.M., and Hartman R. Multiple objective solutions for Tetris. Journal of Recreational Mathematics 32 3 (2003-2004) 214-225
24. Kuykendall C. Analyzing solitaire. Science 283 5403 (1999) 794-795
25. Little J.D.C., Murty K.G., Sweeney D.W., and Karel C. An algorithm for the traveling salesman problem. Operations Research 11 (1963) 972-989
26. Maliphant S.A., and Smith D.K. Mini-Risk: Strategies for a simplified board game. Journal of the Operational Research Society 41 1 (1990) 9-16
27. Moore E.H. A generalization of the game called Nim. Annals of Mathematics 11 (1910) 93-94
28. Neller T.W., and Presser C.G.M. Optimal play of the dice game pig. The UMAP Journal 25 1 (2004) 25-47
29. Nievergelt J., Gasser R., Maser F., and Wirth C. All the needles in a haystack: Can exhaustive search overcome combinatorial chaos?. Lecture Notes in Computer Science vol. 1000 (1995), Springer-Verlag 254-274. Available from:
30. Parlett D. Oxford History of Board Games (1999), Oxford University Press
31. Pudaile P.R. Problem 10801. American Mathematical Monthly 107 4 (2000) 368
32. Puterman M.L. Markov Decision Processes (1994), Wiley
33. A. Rabb, A probabilistic analysis of the game of Solitaire, Undergraduate honors thesis, Harvard University, 1988.
34. Rapoport A. Fights, Games and Debates (1960), University of Michigan Press, Ann Arbor, Michigan
35. Romein J.W., and Bal H.E. Notes: Awari is solved. Journal of the International Computer Games Association 25 September (2002) 162-165
36. Rump C.M. Who wants to see a $ million error. INFORMS Transactions on Education 1 3 (2001) 102-111
37. Schaeffer J. One Jump Ahead (1997), Springer-Verlag
38. Schuh F. The Master Book of Mathematical Recreations (1968), Dover
39. Smith D.K. Dynamic Programming: A Practical Introduction (1991), Ellis Horwood, Chichester, Sussex, UK
40. Smith D.K. Dynamic programming and two problems of Dudeney. Mathematical Spectrum May (1996)
41. Sneidovich M. OR/MS Games: 4. The joy of egg-dropping in Braunschweig and Hong Kong. INFORMS Transactions on Education 4 1 (2003) 48-64
42. Tan B. Markov chains and the RISK board game. Mathematics Magazine 70 5 (1997) 349-357
43. Thomas L.C. The best banking strategy when playing The Weakest Link. Journal of the Operational Research Society 54 7 (2003) 747-750
44. Trick M. Building a better game through dynamic programming: A flip analysis. INFORMS Transactions on Education 2 1 (2001). http://ite.informs.org/vol2no1/trick/ Available from:
45. Uehara R., and Iwata S. Generalized hi-q is np-complete. Transactions of the IEICE 73 (1990) 270-273
46. van Hentenryck P. Constraint Satisfaction in Logic Programming (1989), The MIT Press, Cambridge, Mass, USA
47. Various. Rubik's official online site. Available from: , 2003.
48. Zwick U., and Paterson M.S. The Memory game. Theoretical Computer Science 110 1 (1993) 169-196