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Packing Cube Nets into Rectangles with O(1) Holes

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Book cover Discrete and Computational Geometry, Graphs, and Games (JCDCGGG 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 13034))

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Abstract

We show that the 11 hexomino nets of the unit cube (using arbitrarily many copies of each) can pack disjointly into an \(m \times n\) rectangle and cover all but a constant c number of unit squares, where \(4 \le c \le 14\) for all integers \(m, n \ge 2\). On the other hand, the nets of the dicube (two unit cubes) can be exactly packed into some rectangles.

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Notes

  1. 1.

    An n-omino or polyomino is an edge-to-edge joining of n unit squares. The special cases \(n=1,2,3,4,5,6\) are called monominoes, dominoes, trominoes, tetrominoes, pentominoes, and hexominoes, respectively.

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Acknowledgments

This research is partially supported by JSPS KAKENHI Grant Numbers JP17K00017, JP21K11757, JP20H05964, JP17H06287, JP18H04091, and JST CREST Grant Number JPMJCR1402.

Author information

Authors and Affiliations

  1. Massachusetts Institute of Technology, Cambridge, MA, USA

    Erik D. Demaine & Martin L. Demaine

  2. Japan Advanced Institute of Science and Technology, Nomi, Ishikawa, Japan

    Ryuhei Uehara

  3. Osaka Prefecture University, Sakai, Osaka, Japan

    Yushi Uno

  4. University of Texas Rio Grande Valley, Edinburg, TX, USA

    Andrew Winslow

Authors
  1. Erik D. Demaine
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  2. Martin L. Demaine
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  3. Ryuhei Uehara
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  4. Yushi Uno
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  5. Andrew Winslow
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Corresponding author

Correspondence to Yushi Uno .

Editor information

Editors and Affiliations

  1. Tokyo University of Science, Tokyo, Japan

    Jin Akiyama

  2. Ateneo de Manila University, Quezon City, Philippines

    Reginaldo M. Marcelo

  3. Ateneo de Manila University, Quezon City, Philippines

    Mari-Jo P. Ruiz

  4. Osaka Prefecture University, Osaka, Japan

    Yushi Uno

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Demaine, E.D., Demaine, M.L., Uehara, R., Uno, Y., Winslow, A. (2021). Packing Cube Nets into Rectangles with O(1) Holes. In: Akiyama, J., Marcelo, R.M., Ruiz, MJ.P., Uno, Y. (eds) Discrete and Computational Geometry, Graphs, and Games. JCDCGGG 2018. Lecture Notes in Computer Science(), vol 13034. Springer, Cham. https://doi.org/10.1007/978-3-030-90048-9_12

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  • DOI: https://doi.org/10.1007/978-3-030-90048-9_12

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-90047-2

  • Online ISBN: 978-3-030-90048-9

  • eBook Packages: Computer ScienceComputer Science (R0)

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