Place n non-overlapping circles inside an equilateral triangle of side 1, each with its own radius, making the sum of the radii as large as possible.
Formal definition
ContainerThe closed equilateral triangle with vertices (0, 0), (1, 0) and (1/2, √3/2); a circle may touch a side
SubmissionExactly n circles, each [x, y, r] with r > 0, every number to at most nine decimals
ConstraintsEvery circle lies inside the triangle: y ≥ r, √3·x − y ≥ 2r and √3·(1 − x) − y ≥ 2r; no two interiors meet: the distance between centres is at least the sum of the radii. The slanted sides are compared as squares of integers; everything is exact, and touching is allowed
ScoreThe sum of the radii, an exact integer sum shown to nine decimals; larger is better
ObjectiveMaximise the sum of the radii over every legal arrangement
Getting a feel for it
Where the room for improvement is
A triangular array of equal circles is among the worst strategies: a large circle in the middle, medium ones tucked into the three corners, and small ones filling the gaps along the sides are what raise the sum. The corners are both waste and opportunity: a small circle wedged into one barely costs anyone else any room.
Where the frontier is
Erich Friedman's maximum-total-perimeter table runs to n = 30. Twenty-six rows come from David W. Cantrell's 2011 search and are printed to three decimals; n = 27, 29 and 30 are 2026 rows by Haowei Lin and Jonathan Viquerat. Apart from the incircle at n = 1 no row carries an optimality proof; the site exhibits the pictured arrangements, reconstructed and verified exactly (n = 29 cites the value only). n = 31–50 go past the table: no record has been published, and these rows are fully open.
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/circles-in-triangle-radii.json
Cite the frozen 2026-09 edition: records move, a frozen edition never does, so the citation is still checkable years later.
GET https://minmaxarena.com/data/editions/2026-09/circles-in-triangle-radii.json
BibTeX (click to copy)
@misc{minmaxarena-circles-in-triangle-radii-2026-09,
title = {{Sum of radii in the equilateral triangle} (P93)},
author = {{MinMax Arena}},
year = {2026},
note = {Machine-verified records, 2026-09 edition},
url = {https://minmaxarena.com/data/editions/2026-09/circles-in-triangle-radii.json},
license = {CC BY 4.0}
}
DISCUSSION
Discussion
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