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Analytical Solution for “EinStein würfelt nicht!” with One Stone

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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

“EinStein würfelt nicht!” is a simple board game, played usually on a \(5\times 5\) board with 6 stones per player and a die. In this research, we study the game in the particular case when the players start with only one stone. In that case the random element from the use of a die disappears, so that an analytical analysis is possible. We describe and prove a winning strategy for the first (or second) player for all possible board sizes. In most cases, the first player can force a win, but somewhat surprisingly, depending on a precisely formulated condition on the board size, it is sometimes possible for the second player to win.

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Notes

  1. 1.

    For brevity, we use “he” and “his” whenever “he or she” and “his or her” are meant.

  2. 2.

    For example, let us assume that a player still has stones 1, 5, and 6 on the board. Given a die value of 1, 5, or 6, the player must move the corresponding stone. With a die value of 2, 3, or 4, he can choose to move either stone 1 or stone 5.

  3. 3.

    On some boards, there exist winning strategies that do not follow this pattern.

  4. 4.

    Playing diagonally is a losing move for \(P_1\). Note that playing vertically is also a winning move for \(P_1\).

  5. 5.

    This is the crucial move!

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Acknowledgments

This work is partially supported by JSPS KAKENHI Grant (C)(JP15K00183) and (JP15K00189) and Japan Science and Technology Agency, CREST (JPMJCR1404) and Infrastructure Development for Promoting International S&T Cooperation and Project for Establishing a Nationwide Practical Education Network for IT Human Resources Development, Education Network for Practical Information Technologies. We would also like to thank the anonymous reviewers for their comments that helped us improve the paper.

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Authors and Affiliations

  1. Graduate School of Engineering, Osaka University, Suita, Japan

    François Bonnet

  2. Graduate School of Advanced Science and Technology, JAIST, Nomi, Japan

    Simon Viennot

Authors
  1. François Bonnet
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  2. Simon Viennot
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Corresponding authors

Correspondence to François Bonnet or Simon Viennot .

Editor information

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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Bonnet, F., Viennot, S. (2017). Analytical Solution for “EinStein würfelt nicht!” with One Stone. 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_1

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  • DOI: https://doi.org/10.1007/978-3-319-71649-7_1

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-71648-0

  • Online ISBN: 978-3-319-71649-7

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