Equal circles in a half-disc
Place n non-overlapping circles of equal radius inside a half-disk of radius 1 to maximize their common radius.
Formal definition
- ContainerAn upper half-disk of radius 1: its diameter lies along y = 0 from (0, 0) to (2, 0), centered at (1, 0), with the circular arc in y ≥ 0
- SubmissionExactly n circles: one common radius and n center coordinates
- ConstraintsEvery circle lies wholly inside the half-disk; no two interiors overlap, and tangency is allowed
- ObjectiveMaximize the common radius
Getting a feel for it
Mixed straight and curved boundary
A half-disk combines a flat diameter, a curved upper arc, and two 90° corners at the ends of the base. Strong packings balance linear rows along the diameter against arched layers along the upper boundary, locking circles into a rigid contact graph while occasionally leaving a rattler in an interior pocket.
Literature benchmarks and open frontier
Best-known values for circle packing in a half-disk are taken from Eckard Specht's csc survey table on Packomania, which marks none of these cases as mathematically proven optimal; all fourteen instances remain 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 (0) ↓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-a-semicircle.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-a-semicircle.json
BibTeX (click to copy)
@misc{minmaxarena-circles-in-a-semicircle-2026-10,
title = {{Equal circles in a half-disc} (P09)},
author = {{MinMax Arena}},
year = {2026},
note = {Machine-verified records, 2026-10 edition},
url = {https://minmaxarena.com/data/editions/2026-10/circles-in-a-semicircle.json},
license = {CC BY 4.0}
}Discussion
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