Abstract
We show that squares of sides of length \( 1\), \(2^{-t}\), \(3^{-t}\), \(4^{-t}\),\(\ldots \) can be packed perfectly into a square, provided \( 1/2 < t \le 2/3\). Moreover, we prove that cubes of edges of length \( 1\), \(2^{-t}\), \(3^{-t}\), \(4^{-t}\), \( \ldots \) can be packed perfectly into a box, provided \( 2/3 < t \le 4/11\).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
K. Ball, On packing unequal squares, J.Combin. Theory Ser.A, 75 (1996), 353–357
A. Chalcraft, Perfect square packings, J.Combin. Theory Ser.A, 92 (2000), 158–172
P. Grzegorek and J. Januszewski, A note on three Moser's problems and two Paulhus' lemmas, J.Combin. Theory Ser.A, 162 (2019), 222–230
A. Joós, On packing of squares in a rectangle, in: Discrete Geometry Fest (May 15–19, 2017), Rényi Institute (Budapest, 2017)
Joós, A.: Perfect packing of cubes. Acta Math. Hungar. 156, 375–384 (2018)
A. Joós, Perfect square packings, Math. Rep., accepted
Meir, A., Moser, L.: On packing of squares and cubes. J. Combin. Theory 5, 126–134 (1968)
M.M. Paulhus, An algorithm for packing squares, J.Combin. Theory Ser.A, 82 (1998), 147–157
Wästlund, J.: Perfect packings of squares using the stack-pack strategy. Discrete Comput. Geom. 29, 625–631 (2003)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Januszewski, J., Zielonka, Ł. A note on perfect packing of squares and cubes. Acta Math. Hungar. 163, 530–537 (2021). https://doi.org/10.1007/s10474-020-01114-6
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10474-020-01114-6