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arXiv · 2610.28504

An exact formula for optimal axis-parallel packing of eight squares

Matthew Badger
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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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Matthew Badger. 2026-10-20. An exact formula for optimal axis-parallel packing of eight squares. https://arxiv.org/abs/2610.28504

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