Axis-parallel square packing is the problem of placing squares with prescribed side lengths in a square container with
disjoint interiors and sides parallel to those of the container. Unlike square
packing of congruent squares with unrestricted orientations,
the sizes here may differ and rotations are not allowed. Let 
be the least possible container side for nonnegative
side lengths 
(Badger 2026a).
Badger (2026a) gave exact formulas through seven squares. For six,
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For seven,
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More generally, for 
the optimum is a sum of at most 
prescribed side lengths, and this bound is sharp on an open set of side vectors (Badger 2026a). Badger (2026a)
credits Claude with implementing exact searches and assisting the development of
the lower-bound arguments, and Claude and ChatGPT with reviewing the mathematics.
The author states that he checked all proofs.
Badger (2026b) extended the exact optimization to eight prescribed squares. The optimum is the minimum of 23 explicit convex strategy-cost functions. Moreover, it cannot be expressed as the minimum of fewer than 23 convex functions. Thus the minimum complete strategy-family sizes for six, seven, and eight squares are 1, 3, and 23, respectively. Badger (2026b) reports AI assistance with the search for the formula, development of lower bounds, and exact verification of the constructions and certificates.
