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Two-Player Domino Games

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Twenty Years of Theoretical and Practical Synergies (CiE 2024)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14773))

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Abstract

We introduce a 2-player game played on an infinite grid, initially empty, where each player in turn chooses a vertex and colours it. The first player aims to create some pattern from a target set, while the second player aims to prevent it.

We study the problem of deciding which player wins, and prove that it is undecidable. We also consider a variant where the turn order is not alternating but given by a balanced word, and we characterise the decidable and undecidable cases.

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Notes

  1. 1.

    The name “Domino game” has sometimes been used for the one-player version.

  2. 2.

    A strategy does not need to be defined on unreachable positions, e.g. infinite patterns.

  3. 3.

    Choosing \(\llbracket -n-1, n+1\rrbracket \times \llbracket -n, n \rrbracket ^{d-1}\) as the support for forbidden patterns would be enough, but we made the support slightly larger for clarity.

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Acknowledgements

The authors received financial support from IZES, an ANR project.

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Authors and Affiliations

  1. Université Paris-Saclay, CNRS, Laboratoire Interdisciplinaire des Sciences du Numérique, 91400, Orsay, France

    Benjamin Hellouin de Menibus

  2. Université Paris-Saclay, ENS Paris-Saclay, 91190, Gif-sur-Yvette, France

    Rémi Pallen

Authors
  1. Benjamin Hellouin de Menibus
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  2. Rémi Pallen
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Correspondence to Benjamin Hellouin de Menibus .

Editor information

Editors and Affiliations

  1. The Mathematics Institute of Jussieu, Paris, France

    Ludovic Levy Patey

  2. University College London, London, UK

    Elaine Pimentel

  3. University of Amsterdam, Amsterdam, The Netherlands

    Lorenzo Galeotti

  4. University of Göttingen, Göttingen, Germany

    Florin Manea

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Hellouin de Menibus, B., Pallen, R. (2024). Two-Player Domino Games. In: Levy Patey, L., Pimentel, E., Galeotti, L., Manea, F. (eds) Twenty Years of Theoretical and Practical Synergies. CiE 2024. Lecture Notes in Computer Science, vol 14773. Springer, Cham. https://doi.org/10.1007/978-3-031-64309-5_12

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  • DOI: https://doi.org/10.1007/978-3-031-64309-5_12

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-64308-8

  • Online ISBN: 978-3-031-64309-5

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