Computer Hex Algorithm Using a Move Evaluation Method Based on a Convolutional Neural Network

  • Kei TakadaEmail author
  • Hiroyuki Iizuka
  • Masahito Yamamoto
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 818)


In recent years, a move evaluation model using a convolutional neural network (CNN) has been proposed for Go, and it has been shown that CNN can learn professional human moves. Hex is a two-player connection game, which is included in the Computer Olympiad. It is important to consider cell adjacency on the board for a better Hex strategy. To evaluate cell adjacency from various perspectives properly, we propose a CNN model that evaluates all candidate moves by taking as input all sets consisting of 3 mutually adjacent cells. The proposed CNN model is tested against an existing CNN model called “NeuroHex,” and the comparison results show that our CNN model is superior to NeuroHex on a \(13\,\times \,13\) board even though our CNN model is trained on an \(11\,\times \,11\) board. We also use the proposed model as an ordering function and test it against the world-champion Hex program “MoHex 2.0” on an \(11\,\times \,11\) board. The results show that the proposed model can be used as a better ordering function than the ordering function created by minimax tree optimization, and we obtained a win rate of 49.0% against MoHex 2.0 (30 s/move).



The authors would like to thank prof. Ryan Hayward for supporting the program development and the fruitful discussions.


  1. 1.
    Anshelevich, V.V.: A hierarchical approach to computer Hex. Artif. Intell. 134(1–2), 101–120 (2002)CrossRefzbMATHGoogle Scholar
  2. 2.
    Arneson, B., Hayward, R., Henderson, P.: Wolve 2008 wins Hex tournament. ICGA J. 32(1), 49–53 (2009)CrossRefGoogle Scholar
  3. 3.
    Arneson, B., Henderson, P.T., Hayward, R.B.: Benzene (2009–2012).
  4. 4.
    Borgatti, S.P.: Centrality and network flow social networks. Soc. Netw. 27(1), 55–71 (2005)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Browne, C.: Hex Strategy: Making the Right Connections. A. K. Peters, Natick (2000)zbMATHGoogle Scholar
  6. 6.
    Even, S., Tarjan, R.E.: A combinatorial problem which is complete in polynomial space. J. ACM 23(4), 710–719 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Gale, D.: The game of hex and the brouwer fixed-point theorem. Am. Math. Monthly 86(10), 818–827 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hayward, R.B.: Six wins hex tournament. ICGA J. 29(3), 163–165 (2006)MathSciNetGoogle Scholar
  9. 9.
    Hayward, R.B.: MoHex wins hex tournament. ICGA J. 35, 124–127 (2012)CrossRefGoogle Scholar
  10. 10.
    Hayward, R.B., Weninger, N., Young, K., Takada, K., Zhang, T.: MoHex wins Hex 11\(\times \)11 and 13\(\times \)13 tournament. ICGA J. (2017, To appear)Google Scholar
  11. 11.
    He, K., Zhang, X., Ren, S., Sun, J.: Delving deep into rectifiers: Surpassing human-level performance on imagenet classification. In: Proceedings of the 2015 IEEE International Conference on Computer Vision (ICCV 2015), pp. 1026–1034. IEEE Computer Society (2015)Google Scholar
  12. 12.
    Henderson, P., Arneson, B., Hayward, R.: Solving 8\(\times \)8 Hex. In: Proceedings of the 21st International Joint Conference on Artificial Intelligence (IJCAI 2009), pp. 505–510 (2009)Google Scholar
  13. 13.
    Henderson, P.T.: Playing and Solving the Game of Hex. Ph.D. thesis, University of Alberta (2010)Google Scholar
  14. 14.
    Hoki, K., Kaneko, T.: Large-scale optimization for evaluation functions with minimax search. J. Artif. Intell. Res. 49, 527–568 (2014)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Huang, S.-C., Arneson, B., Hayward, R.B., Müller, M., Pawlewicz, J.: MoHex 2.0: A pattern-based MCTS hex player. In: van den Herik, H.J., Iida, H., Plaat, A. (eds.) CG 2013. LNCS, vol. 8427, pp. 60–71. Springer, Cham (2014). Google Scholar
  16. 16.
    Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization. In: Proceedings of the 3rd International Conference on Learning Representations (ICLR) (2014)Google Scholar
  17. 17.
    Nash, J.: Some games and machines for playing them. Technical report, D-1164, RAND (1952)Google Scholar
  18. 18.
    Pawlewicz, J., Hayward, R., Henderson, P., Arneson, B.: Stronger virtual connections in hex. IEEE Trans. Comput. Intell. AI Games 7(2), 156–166 (2015)CrossRefGoogle Scholar
  19. 19.
    Pawlewicz, J., Hayward, R.B.: Scalable parallel DFPN search. In: van den Herik, H.J., Iida, H., Plaat, A. (eds.) CG 2013. LNCS, vol. 8427, pp. 138–150. Springer, Cham (2014). Google Scholar
  20. 20.
    Pawlewicz, J., Hayward, R.B.: Sibling conspiracy number search. In: Proceedings of the 8th International Symposium Combinatorial Search, pp. 105–112. AAAI Press (2015)Google Scholar
  21. 21.
    Silver, D., Huang, A., Maddison, C.J., Guez, A., Sifre, L., van den Driessche, G., Schrittwieser, J., Antonoglou, I., Panneershelvam, V., Lanctot, M., Dieleman, S., Grewe, D., Nham, J., Kalchbrenner, N., Sutskever, I., Lillicrap, T., Leach, M., Kavukcuoglu, K., Graepel, T., Hassabis, D.: Mastering the game of go with deep neural networks and tree search. Nature 529(7587), 484–489 (2016)CrossRefGoogle Scholar
  22. 22.
    Takada, K., Honjo, M., Iizuka, H., Yamamoto, M.: Developing computer hex using global and local evaluation based on board network characteristics. In: Plaat, A., van den Herik, J., Kosters, W. (eds.) ACG 2015. LNCS, vol. 9525, pp. 235–246. Springer, Cham (2015). CrossRefGoogle Scholar
  23. 23.
    Tompson, J., Goroshin, R., Jain, A., LeCun, Y., Bregler, C.: Efficient object localization using convolutional networks. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 648–656 (2015)Google Scholar
  24. 24.
    Young, K., Vasan, G., Hayward, R.: NeuroHex: A deep Q-learning hex agent. In: Cazenave, T., Winands, M.H.M., Edelkamp, S., Schiffel, S., Thielscher, M., Togelius, J. (eds.) CGW/GIGA -2016. CCIS, vol. 705, pp. 3–18. Springer, Cham (2017). CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Kei Takada
    • 1
    Email author
  • Hiroyuki Iizuka
    • 1
  • Masahito Yamamoto
    • 1
  1. 1.Graduate School of Information Science and TechnologyHokkaido UniversitySapporoJapan

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