PreprintCC BY 4.0arXiv
An exact formula for optimal axis-parallel packing of eight squares
Abstract
We determine the smallest square container for eight axis-parallel squares of prescribed side lengths. The optimal side length is the minimum of the costs of 23 explicit packing strategies, and each cost is a maximum of sums of side lengths. No minimum of fewer convex functions represents it. Three strategies suffice for seven squares; the jump to 23 for eight squares witnesses the beginning of a combinatorial explosion. We prove the formula by combining the constructions with geometric lower bounds derived from disjoint coordinate projections, cornering arguments, and a weighted area comparison.
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BibTeX
@misc{badger2026exact,
title = {{An exact formula for optimal axis-parallel packing of eight squares}},
author = {Matthew Badger},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28504},
eprint = {2610.28504},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28504},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - An exact formula for optimal axis-parallel packing of eight squares
AU - Matthew Badger
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28504
UR - https://arxiv.org/abs/2610.28504
AB - We determine the smallest square container for eight axis-parallel squares of prescribed side lengths. The optimal side length is the minimum of the costs of 23 explicit packing strategies, and each cost is a maximum of sums of side lengths. No minimum of fewer convex functions represents it. Three strategies suffice for seven squares; the jump to 23 for eight squares witnesses the beginning of a combinatorial explosion. We prove the formula by combining the constructions with geometric lower bounds derived from disjoint coordinate projections, cornering arguments, and a weighted area comparison.
ER -