Combinatorics of Go

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4630)


We present several results concerning the number of positions and games of Go. We derive recurrences for L(m,n), the number of legal positions on an m ×n board, and develop a dynamic programming algorithm which computes L(m,n) in time O(m 3 n 2 λ m ) and space O(m λ m ), for some constant λ< 5.4. An implementation of this algorithm enables us to list L(n,n) for n ≤ 17. For larger boards, we prove the existence of a base of liberties \(\lim \sqrt[mn]{L(m,n)}\) of ~2.9757341920433572493. Based on a conjecture about vanishing error terms, we derive an asymptotic formula for L(m,n), which is shown to be highly accurate.

We also study the Game Tree complexity of Go, proving an upper bound on the number of possible games of (mn)L(m,n) and a lower bound of \(2^{2^{n^2/2\,-O(n)}}\) on n×nboards and \(2^{2^{n-1}}\) on 1 ×n boards, in addition to exact counts for mn ≤ 4 and estimates up to mn = 9. Most proofs and some additional results had to be left out to observe the page limit. They may be found in the full version available at [8].


State Count Edge State Simple Path Border State Legal Position 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Blahut, R.E.: Theory and Practice of Error Control Codes. Addison-Wesley, London (1983)zbMATHGoogle Scholar
  2. 2.
    Chen, P.: Heuristic Sampling: A Method for Predicting the Performance of Tree Searching Programs. SIAM J. Comput. 21(2), 295–315 (1992)zbMATHCrossRefGoogle Scholar
  3. 3.
    Crâşmaru, M., Tromp, J.: Ladders are PSPACE-complete. In: Marsland, T., Frank, I. (eds.) CG 2001. LNCS, vol. 2063, pp. 241–249. Springer, Heidelberg (2002)Google Scholar
  4. 4.
    Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete Mathematics, 2nd edn. Addison-Wesley, London (1994)zbMATHGoogle Scholar
  5. 5.
    Lichtenstein, D., Sipser, M.: GO is Polynomial-Space Hard. Journal of the ACM 27(2), 393–401 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Robson, J.M.: The Complexity of Go. In: Proc. IFIP (International Federation of Information Processing), pp. 413–417 (1983)Google Scholar
  7. 7.
    Tromp, J.: The Game of Go (2006),
  8. 8.
    Tromp, J.: Number of Legal Go Positions (2006),
  9. 9.
    Woeginger, G.: Personal communicationGoogle Scholar
  10. 10.
    Wolfe, D.: Go endgames are PSPACE-hard. In: Nowakowski, R.J. (ed.) More Games of No Chance, vol. 42, pp. 125–136. MSRI Publications (2002)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  1. 1.CWI, Amsterdam 
  2. 2.Laboratory of Mathematics in Imaging, Harvard Medical School, Boston, Massachusetts 

Personalised recommendations