Advances in Computer Games

Advances in Computer Games pp 160-176 | Cite as

Draws, Zugzwangs, and PSPACE-Completeness in the Slither Connection Game

  • Édouard Bonnet
  • Florian Jamain
  • Abdallah Saffidine
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9525)


Two features set Slither apart from other connection games. Previously played stones can be relocated and some stone configurations are forbidden. We show that the interplay of these peculiar mechanics with the standard goal of connecting opposite edges of a board results in a game with a few properties unexpected among connection games, for instance, the existence of mutual Zugzwangs. We also establish that, although there are positions where one player has no legal move, there is no position where both players lack a legal move and that the game cannot end in a draw. From the standpoint of computational complexity, we show that the game is pspace-complete, the relocation rule can indeed be tamed so as to simulate a hex game on a Slither board.


  1. 1.
    Arneson, B., Hayward, R.B., Henderson, P.: Monte Carlo tree search in Hex. IEEE Trans. Comput. Intell. AI Games 2(4), 251–258 (2010)CrossRefGoogle Scholar
  2. 2.
    Bonnet, É., Jamain, F., Saffidine, A.: Havannah and TwixT are PSPACE-complete. In: van den Herik, H.J., Iida, H., Plaat, A. (eds.) CG 2013. LNCS, vol. 8427, pp. 175–186. Springer, Heidelberg (2014) Google Scholar
  3. 3.
    Bonnet, É., Jamain, F., Saffidine, A.: On the complexity of trick-taking card games. In: Rossi, F. (ed.) 23rd International Joint Conference on Artificial Intelligence (IJCAI), Beijing, China, August 2013, pp. 482–488. AAAI Press (2013)Google Scholar
  4. 4.
    Browne, C.: Connection Games: Variations on a Theme. A K Peters, Massachusetts (2005)MATHGoogle Scholar
  5. 5.
    Even, S., Tarjan, R.E.: A combinatorial problem which is complete in polynomial space. J. ACM (JACM) 23(4), 710–719 (1976)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Ewalds, T.: Playing and solving Havannah. Master’s thesis, University of Alberta (2012)Google Scholar
  7. 7.
    Furtak, T., Kiyomi, M., Uno, T., Buro, M.: Generalized Amazons is PSPACE-complete. In: Kaelbling, L.P., Saffiotti, A. (eds.) 19th International Joint Conference on Artificial Intelligence (IJCAI), pp. 132–137 (2005)Google Scholar
  8. 8.
    Hearn, R.A., Demaine, E.D.: Games, Puzzles, and Computation. A K Peters, USA (2009)MATHGoogle Scholar
  9. 9.
    Henderson, P.T.: Playing and solving the game of Hex. Ph.D. thesis, University of Alberta, August 2010Google Scholar
  10. 10.
    Hsieh, M.Y., Tsai, S.-C.: On the fairness and complexity of generalized-in-a-row games. Theor. Comput. Sci. 385(1–3), 88–100 (2007)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Reisch, S.: Hex ist PSPACE-vollständig. Acta Informatica 15(2), 167–191 (1981)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Schaefer, T.J.: On the complexity of some two-person perfect-information games. J. Comput. Syst. Sci. 16(2), 185–225 (1978)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Steane, A.M.: Threat, support and dead edges in the Shannon game, October 2012Google Scholar
  14. 14.
    Steane, A.M.: Minimal and irreducible links in the Shannon game, January 2013Google Scholar
  15. 15.
    van Rijswijck, J.: Set colouring games. Ph.D. thesis, University of Alberta, October 2006Google Scholar
  16. 16.
    Yato, T.: On the NP-completeness of the Slither link puzzle. In: Notes of the 74th Meeting of IPSJ SIG ALgorithms, pp. 25–32 (2000)Google Scholar
  17. 17.
    Yoshinaka, R., Saitoh, T., Kawahara, J., Tsuruma, K., Iwashita, H., Minato, S.-I.: Finding all solutions and instances of Numberlink and Slitherlink by ZDDs. Algorithms 5(2), 176–213 (2012)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Édouard Bonnet
    • 1
  • Florian Jamain
    • 2
  • Abdallah Saffidine
    • 3
  1. 1.SZTAKI, Hungarian Academy of SciencesBudapestHungary
  2. 2.LAMSADE, Université Paris-DauphineParisFrance
  3. 3.CSE, The University of New South WalesSydneyAustralia

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