Set Matching: An Enhancement of the Hales-Jewett Pairing Strategy

  • Jos W. H. M. UiterwijkEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10664)


When solving k-in-a-Row games, the Hales-Jewett pairing strategy is a well-known strategy to prove that specific positions are (at most) a draw. It requires two empty squares per possible winning line (group) to be marked, i.e., with a coverage ratio of 2.0.

In this paper we present a new strategy, called Set Matching. A matching set consists of a set of nodes (the markers), a set of possible winning lines (the groups), and a coverage set indicating how all groups are covered after every first initial move. This strategy needs less than two markers per group. As such it is able to prove positions in k-in-a-Row games to be draws, which cannot be proven using the Hales-Jewett pairing strategy.

We show several efficient configurations with their matching sets. These include Cycle Configurations, BiCycle Configurations, and PolyCycle Configurations involving more than two cycles. Depending on configuration, the coverage ratio can be reduced to 1.14.

Many examples in the domain of solving k-in-a-Row games are given, including the direct proof (not based on search) that the empty \(4 \times 4\) board is a draw for 4-in-a-Row.


Matching Set Polycyclic Configurations Cycle Configuration Coverage Ratio Empty Squares 
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.


  1. 1.
    Beck, J.: Positional games. Comb. Probab. Comput. 14, 649–696 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Beck, J.: Combinatorial Games: Tic-Tac-Toe Theory. Cambridge University Press, Cambridge (2008)CrossRefzbMATHGoogle Scholar
  3. 3.
    Gardner, M.: Ticktacktoe games. In: Wheels, Life, and Other Mathematical Amusements, pp. 94–105. WH Freeman, New York (1983). (Chap. 9)Google Scholar
  4. 4.
    Hales, A.W., Jewett, R.I.: Regularity and positional games. Trans. Am. Math. Soc. 106, 222–229 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Hefetz, D., Krivelevich, M., Stojaković, M., Szabó, T.: Positional Games. Birkhäuser/Springer, Basel (2014). CrossRefzbMATHGoogle Scholar
  6. 6.
    Uiterwijk, J.W.H.M.: Perfectly solving domineering boards. In: Cazenave, T., Winands, M.H.M., Iida, H. (eds.) CGW 2013. CCIS, vol. 408, pp. 97–121. Springer, Cham (2014). CrossRefGoogle Scholar
  7. 7.
    Uiterwijk, J.W.H.M.: Solving Four-in-a-row (2017, submitted)Google Scholar
  8. 8.
    Uiterwijk, J.W.H.M., van den Herik, H.J.: The advantage of the initiative. Inf. Sci. 122, 43–58 (2000)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Data Science and Knowledge EngineeringMaastricht UniversityMaastrichtThe Netherlands

Personalised recommendations