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Axis-parallel square packing with an exact formula for optimal packing of seven squares

  • Matthew Badger
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Abstract

How large a square is needed to hold axis-parallel squares of prescribed side lengths? We give exact formulas for the optimal side through seven squares, with explicit constructions and geometric lower-bound proofs. For every n≥6, the optimum for n squares is a sum of at most n−3 of the prescribed side lengths. This bound is sharp on an open set of side lengths and yields the optimal Euclidean Lipschitz constant √(n−3) of the optimum as a function of the n side lengths.

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Cite

BibTeXRIS
BibTeX
@misc{badger2026axisparallel,
  title = {{Axis-parallel square packing with an exact formula for optimal packing of seven squares}},
  author = {Matthew Badger},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.22622},
  eprint = {2610.22622},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.22622},
}
RIS
TY  - UNPB
TI  - Axis-parallel square packing with an exact formula for optimal packing of seven squares
AU  - Matthew Badger
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2610.22622
UR  - https://arxiv.org/abs/2610.22622
AB  - How large a square is needed to hold axis-parallel squares of prescribed side lengths? We give exact formulas for the optimal side through seven squares, with explicit constructions and geometric lower-bound proofs. For every n≥6, the optimum for n squares is a sum of at most n−3 of the prescribed side lengths. This bound is sharp on an open set of side lengths and yields the optimal Euclidean Lipschitz constant √(n−3) of the optimum as a function of the n side lengths.
ER  -