Equal-circle packing in a right triangle
Also known ascircles in a trianglepacking equal circles in a right triangle
Place n non-overlapping equal circles inside the right triangle with legs 1 and 0.75 to maximize their common radius.
Formal definition
- ContainerA 3-4-5 right triangle with its right angle at the origin (0, 0) and legs along the axes to (1, 0) and (0, 0.75)
- SubmissionExactly n circles: one common radius and n center coordinates
- ConstraintsEvery circle lies wholly inside the right triangle; no two interiors overlap, and tangency is allowed
- ObjectiveMaximize the common radius
Getting a feel for it
Unequal acute angles and asymmetric contact graphs
Because the three angles of a 3-4-5 right triangle are all different, the narrower acute corners force circle centers farther from the vertices and leave unequal dead space. At maximum radius, circles form asymmetric contact networks braced against the three sides, occasionally leaving a rattler in an interior pocket.
Literature context and open status
No dedicated benchmark table has been published for equal-circle packing in the 1 : 0.75 : 1.25 right triangle (closely related containers such as the isosceles right triangle are surveyed on Erich Friedman's Packing Center). NUE_13 #31's proof, completed by the site, settles the continuous optimal radii 5/28, 25/164 and 5/36 for n = 2, 3 and 4; those three cases retain their history and close without claiming nine-decimal grid optimality. From n = 5 on every instance is 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/circles-in-right-triangle.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/circles-in-right-triangle.json
BibTeX (click to copy)
@misc{minmaxarena-circles-in-right-triangle-2026-10,
title = {{Equal-circle packing in a right triangle} (P11)},
author = {{MinMax Arena}},
year = {2026},
note = {Machine-verified records, 2026-10 edition},
url = {https://minmaxarena.com/data/editions/2026-10/circles-in-right-triangle.json},
license = {CC BY 4.0}
}Discussion
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