Place n circles with disjoint interiors in a rectangle of perimeter 4. Both the rectangle's dimensions and every radius may vary; maximize the sum of the radii.
Formal definition
Containerwidth>0, height>0, and width+height=2; the lower-left corner is fixed at (0,0)
SubmissionExactly n circles, each written [x,y,r], with strictly positive radius
ConstraintsEvery circle lies wholly inside the rectangle; interiors are pairwise disjoint and tangency is allowed
ObjectiveMaximize the exact integer sum of all radii
Getting a feel for it
Optimizing container and cargo together
With a fixed square only the circles move. Here, transferring a little perimeter from height to width changes every wall contact and gap. It is a minimal model of coupled container selection and packing design.
Why uniformity loses
Equal radii usually waste the gaps between curved boundaries. Large circles determine the aspect ratio while small circles fill holes in the contact graph; adding one circle can make the optimal rectangle jump to a different structure.
Published frontier
A 2026 public computational project supplies complete coordinates through n=60. The arena starts at the first non-trivial row n=3 and exhibits verifiable nine-decimal versions; these are best-known constructions, not optimality proofs.
Every sub-problem in this family — authoritative scores, proof status, coordinates and sources — lives at the stable address below, published under CC BY 4.0. Scores move as records fall, so cite the generatedAt timestamp the file carries.
GET https://minmaxarena.com/data/variable-circles-in-fixed-perimeter-rectangle.json
The first frozen edition is published at the start of next month; until then, cite with the date you accessed it.
BibTeX (click to copy)
@misc{minmaxarena-variable-circles-in-fixed-perimeter-rectangle,
title = {{Variable-radius circles in a fixed-perimeter rectangle} (P67)},
author = {{MinMax Arena}},
year = {2026},
note = {Machine-verified records, accessed 2026-08-29},
url = {https://minmaxarena.com/problems/variable-circles-in-fixed-perimeter-rectangle},
license = {CC BY 4.0}
}
DISCUSSION
Discussion
Talk strategy, share methods, ask why you are stuck. Posts carry your public byline, the same name your records use; the #number after it is the account's signup ordinal, so a name cannot be worn by someone else. The floor is earned: break a record once, anywhere, and it is yours for good. New posts appear after an automated review.
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