Advertisement

International Journal of Game Theory

, Volume 44, Issue 3, pp 761–767 | Cite as

On the number of positions in chess without promotion

  • Stefan Steinerberger
Article

Abstract

The number of different legal positions in chess is usually estimated to be between \(10^{40}\) and \(10^{50}\). Within this range, the best upper bound \(10^{46}\) is some orders of magnitude bigger than the estimate \(5 \times 10^{42}\) made by Claude Shannon in his seminal 1950 paper, which is usually considered by computer scientists to be a better approximation. We improve Shannon’s estimate and show that the number of positions where any number of chessmen may have been captured but no promotion has occured is bounded from above by \(2 \times 10^{40}\). The actual number should be quite a bit smaller than that and outline possible ways towards improving our result.

Keywords

Shannon’s number Chess State space 

Notes

Acknowledgments

The author is grateful for the very valuable remarks by two anonymous referees, which greatly improved the paper.

References

  1. Allis V (1994) Searching for solutions in games and artificial intelligence. Ph.D. Thesis, University of LimburgGoogle Scholar
  2. Chinchalkar S (1996) An upper bound for the number of reachable positions. ICCA J 19(3):181–183Google Scholar
  3. Schaeffer J, Bjornsson Y, Burch N, Kishimoto A, Muller M, Lake R, Lu P, Sutphen S (2005) Solving checkers. In: International joint conference on artificial intelligence (IJCAI), pp. 292–297Google Scholar
  4. Schaeffer J, Burch N, Bjrnsson Y, Kishimoto A, Müller M, Lake R, Lu P, Sutphen S (2007) Checkers is solved. Nature 317(5844):1518–1522Google Scholar
  5. Schaeffer J, Culberson J, Treloa N, Knight B, Lu P, Szafron D (1991) Reviving the game of checkers. In: Levy DNL, Beal DF (eds) Heuristic programming in artificial intelligence 2: the second computer olympiad. Ellis Horwood Ltd., Chichester, p 119Google Scholar
  6. Shannon C (1950) Programming a computer for playing chess, Philosophical Magazine 41Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of MathematicsYale UniversityNew HavenUSA

Personalised recommendations