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Tribone Tilings of Triangular Regions that Cover All but Three Holes

Discrete & Computational Geometry volume 53, pages 466–477 (2015)Cite this article

Abstract

This paper extends the literature of tiling \(n\)-row triangular arrays (\(T_n\)) with copies of \(1 \times 3\) trominos discussed by Thurston, Conway, and Lagarias, focusing on tilings which cover all but three holes. We find a set of \(2^{\Omega (n^2)}\) such tilings, disproving the conjecture from 1993 that there are only \(2^{o(n^2)}\) such tilings. Furthermore, we show that if three cells are randomly removed from \(T_n\) when \(n \equiv 0,2 \pmod 3\), then the probability that the remaining region can be tiled by tribones is nonzero.

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References

  1. Conway, J.H., Lagarias, L.C.: Tiling with polyominoes and combinatorial group theory. J. Comb. Theory Ser. A 53(2), 183–208 (1990)

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  2. Lagarias, J.C., Romano, D.S.: A polyomino tiling problem of Thurston and its configurational entropy. J. Comb. Theory Ser. A 63(2), 338–358 (1993)

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Acknowledgments

I would like to sincerely thank Rich Schwartz for introducing me to Conway’s tiling groups, for discussing this and related problems with me, and for all his advice throughout the process. I would also like to thank Brown University for funding this research during Summer 2013 with the Karen T. Romer Undergraduate Teaching and Research Award, and an anonymous referee for helpful comments and suggestions.

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Authors and Affiliations

  1. Harvard Business School, Boston, MA, USA

    Michael Thaler

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  1. Michael Thaler
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Correspondence to Michael Thaler.

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Thaler, M. Tribone Tilings of Triangular Regions that Cover All but Three Holes. Discrete Comput Geom 53, 466–477 (2015). https://doi.org/10.1007/s00454-014-9652-z

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  • DOI: https://doi.org/10.1007/s00454-014-9652-z

Keywords

  • Tiling groups
  • Tiling by tribones
  • Imperfect tilings

Mathematics Subject Classification

  • 05B45
  • 05B50
  • 52C20