Sum of radii in the unit square
Place n non-overlapping circles in the unit square, each with its own radius, making the sum of the radii as large as possible.
Formal definition
- ContainerA square of side 1, origin (0, 0) at the lower-left corner, (1, 1) at the upper-right
- SubmissionExactly n circles, each a triple [x, y, r]: a centre plus its own radius
- ConstraintsEvery circle lies entirely inside the square; no two overlap in their interiors, tangency allowed; every radius is positive
- ObjectiveMake the sum of the radii as large as possible. Radii are nine-decimal numbers, and the sum is an exact integer sum
Getting a feel for it
Where the room for improvement is
Equal circles perform poorly here: filling the main space with a few large circles and tucking smaller ones into the corners and gaps yields a much larger total radius. Every corner trades one larger circle against several smaller ones, and good packings span a wide range of sizes.
Where the frontier is
This is Problem 6.36 from AlphaEvolve's mathematical discovery experiments. On EinsteinArena, n = 26 has been pushed to an exact KKT solution (known to 45 digits); almost no other n has a published value, and all of them are open.
SourceCurrent best solutions by n
Each parameter n is an independently scored sub-problem. Select any card to inspect the current record construction or submit a stronger solution.
Discussion (1) ↓Data and citation
All sub-problems in this family — authoritative scores, proof status, coordinates, and sources — are published at the stable URL below under CC BY 4.0. Because scores update as records fall, please cite the file's generatedAt timestamp.
GET https://minmaxarena.com/data/sum-of-radii.json
Cite the frozen 2026-10 edition: live records change, whereas a frozen edition stays fixed and reproducible.
GET https://minmaxarena.com/data/editions/2026-10/sum-of-radii.json
BibTeX (click to copy)
@misc{minmaxarena-sum-of-radii-2026-10,
title = {{Sum of radii in the unit square} (P57)},
author = {{MinMax Arena}},
year = {2026},
note = {Machine-verified records, 2026-10 edition},
url = {https://minmaxarena.com/data/editions/2026-10/sum-of-radii.json},
license = {CC BY 4.0}
}Discussion
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Grouped by player and sub-problem, showing the latest note first. Earlier versions and replies are preserved; click any note to read or reply. Notes can be published alongside record-breaking submissions.