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Efficient Packings of Unit Squares in a Large Square

  • Ricky Pollack Memorial Issue
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Abstract

How efficiently can a large square of side length x be packed with non-overlapping unit squares? In this note, we show that the uncovered area W(x) can be made as small as \(O(x^{3/5})\). This improves an earlier estimate which showed that \(W(x) = O\bigl (x^{({3+\sqrt{2}})/{7} }\log x\bigr )\).

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Acknowledgements

We would like to express our appreciation for the energetic enthusiasm we found at the 33rd Bellairs Winter Workshop on Computational Geometry in Barbados where some of this research was carried out.

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Authors and Affiliations

  1. Department of Mathematics, University of California, San Diego, USA

    Fan Chung

  2. Department of Computer Science and Engineering, University of California, San Diego, USA

    Ron Graham

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  1. Fan Chung
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  2. Ron Graham
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Correspondence to Ron Graham.

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Editor in Charge János Pach

Dedicated to the memory of Ricky Pollack.

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Chung, F., Graham, R. Efficient Packings of Unit Squares in a Large Square. Discrete Comput Geom 64, 690–699 (2020). https://doi.org/10.1007/s00454-019-00088-9

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  • DOI: https://doi.org/10.1007/s00454-019-00088-9

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