Equal circles in an L
Place n non-overlapping circles of equal radius inside the L-shaped region to maximize their common radius.
Formal definition
- ContainerAn L-shaped region formed by removing the top-right 1 × 1 square (1, 2] × (1, 2] from [0, 2] × [0, 2], with the origin (0, 0) at the lower-left corner
- SubmissionExactly n circles: one common radius and n center coordinates
- ConstraintsEvery circle lies wholly inside the L-shaped region (staying clear of the re-entrant corner at (1, 1) and the notch); no two interiors overlap, and tangency is allowed
- ObjectiveMaximize the common radius
Getting a feel for it
Re-entrant corner and contact graphs
The L-shaped container consists of three 1 × 1 squares meeting at the re-entrant corner (1, 1), which pinches circles transitioning between the two arms. At maximum radius, circles lock into a rigid contact network against the outer walls, the re-entrant corner, and one another, making the allocation of circles across the three unit squares decisive.
Literature benchmarks and proven cases
Erich Friedman's Circles in L's survey records constructions and contributors dating back to 2005. Whereas the source fixes unit-radius circles and minimizes the outer side s of the L, this problem fixes the outer side at 2, relating the common radius by r = 2/s. Optimality for n = 4 has been proven and closed on this site; for n = 5–18, external literature targets and current site records are listed alongside each other.
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 (3) ↓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-an-l.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-an-l.json
BibTeX (click to copy)
@misc{minmaxarena-circles-in-an-l-2026-10,
title = {{Equal circles in an L} (P08)},
author = {{MinMax Arena}},
year = {2026},
note = {Machine-verified records, 2026-10 edition},
url = {https://minmaxarena.com/data/editions/2026-10/circles-in-an-l.json},
license = {CC BY 4.0}
}Discussion
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