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OP-12136Status unassessedGeometry and topology

Circle packing in a square / rectangle (AlphaEvolve problem 36)

Circle packing in a square / rectangle (AlphaEvolve problem 36). Current research status is unassessed; the repository records AlphaEvolve’s result on this question.

MSC 52C15 · Packing and covering in 22 dimensions (aspects of discrete geometry)

Editorial MSC2020 classification. Original article codes and source evidence are recorded in the literature directory.

Licensed source statementAs of Repository at commit 8f44745 (2026-07-11); the repository records AlphaEvolve results, not current research statusSource & evidence ↓
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Problem statement

For any n≥1n \geq 1, let C(n)C(n) denote the largest sum ∑i=1nri\sum_{i=1}^n r_i of radii such that one can place nn disjoint open disks of radius r1,…,rnr_1,\dots,r_n inside the unit square, and let C′(n)C'(n) denote the largest sum ∑i=1nri\sum_{i=1}^n r_i of radii such that one can place nn disjoint open disks of radius r1,…,rnr_1,\dots,r_n inside a rectangle of perimeter 44. Establish upper and lower bounds for C(n)C(n) and C′(n)C'(n) that are as strong as possible.

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Background

Repository result category: former record — AlphaEvolve held the record, which has since been improved. See Section 6.19 in "Mathematical Exploration and Discovery at Scale " paper: arXiv:2511.02864. Experiment notebook.

Progress & source assessment

Slight improvements to the packings generated by AlphaEvolve have since been constructed by Timo Berthold and by Arnaud Deza.

References
  1. [1]Georgiev, Bogdan; Gómez-Serrano, Javier; Tao, Terence; Wagner, Adam Zsolt. Repository of problems accompanying “Mathematical exploration and discovery at scale” (arXiv:2511.02864), github.com/google-deepmind/alphaevolve_repository_of_problems (commit 8f44745, 11 July 2026). — Circle packing in a square / rectangle (AlphaEvolve problem 36) linkApache-2.0 (repository license); problem pages reproduced with attribution.

Recorded progress

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