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

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Advances in Computer Games (ACG 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10664))

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Abstract

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.

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References

  1. Beck, J.: Positional games. Comb. Probab. Comput. 14, 649–696 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Beck, J.: Combinatorial Games: Tic-Tac-Toe Theory. Cambridge University Press, Cambridge (2008)

    Book  MATH  Google Scholar 

  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. Hales, A.W., Jewett, R.I.: Regularity and positional games. Trans. Am. Math. Soc. 106, 222–229 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  5. Hefetz, D., Krivelevich, M., Stojaković, M., Szabó, T.: Positional Games. Birkhäuser/Springer, Basel (2014). https://doi.org/10.1007/978-3-0348-0825-5

    Book  MATH  Google Scholar 

  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). https://doi.org/10.1007/978-3-319-05428-5_8

    Chapter  Google Scholar 

  7. Uiterwijk, J.W.H.M.: Solving Four-in-a-row (2017, submitted)

    Google Scholar 

  8. Uiterwijk, J.W.H.M., van den Herik, H.J.: The advantage of the initiative. Inf. Sci. 122, 43–58 (2000)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Data Science and Knowledge Engineering, Maastricht University, Maastricht, The Netherlands

    Jos W. H. M. Uiterwijk

Authors
  1. Jos W. H. M. Uiterwijk
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Correspondence to Jos W. H. M. Uiterwijk .

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Editors and Affiliations

  1. Department of Data Science and Knowledge Engineering, Maastricht University, Maastricht, Limburg, The Netherlands

    Mark H.M. Winands

  2. Leiden Centre of Data Science, Leiden University, Leiden, Zuid-Holland, The Netherlands

    H. Jaap van den Herik

  3. Leiden Institute of Advanced Computer Science, Leiden University, Leiden, Zuid-Holland, The Netherlands

    Walter A. Kosters

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Uiterwijk, J.W.H.M. (2017). Set Matching: An Enhancement of the Hales-Jewett Pairing Strategy. In: Winands, M., van den Herik, H., Kosters, W. (eds) Advances in Computer Games. ACG 2017. Lecture Notes in Computer Science(), vol 10664. Springer, Cham. https://doi.org/10.1007/978-3-319-71649-7_4

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  • DOI: https://doi.org/10.1007/978-3-319-71649-7_4

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-71648-0

  • Online ISBN: 978-3-319-71649-7

  • eBook Packages: Computer ScienceComputer Science (R0)

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