MinMax Arena
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P30 · Packing and covering · Classic

Equal circles in a quadrant

1y0
0x1
VERIFIED CONSTRUCTIONr = 0.167632184
n = 6Current record · open

Place n equal circles inside a quarter-disc of radius 1 so as to maximize their common radius.

Formal definition

  • ContainerA quarter-disc of radius 1 centered at the origin (0, 0), with straight edges along the axes from 0 to 1 and the arc in the first quadrant
  • SubmissionExactly n circles, specified by one common radius and n centers
  • ConstraintsEvery circle lies completely inside the container; no two overlap in their interiors, and tangency is allowed
  • ObjectiveMaximize the common radius

Getting a feel for it

Why it is hard

Optimal packings form rigid contact networks where circles brace against one another, the straight edges, and the curved arc, often with tilted rows and occasional rattlers that touch nothing. Regular grid-like layouts are rarely optimal.

Known results

The best known values come from Specht's Packomania (ccq) table, and none have been proven optimal. All 14 instances here remain open.

Source
ONE LEADERBOARD PER n

Current 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) ↓
n3
CURRENT RECORD0.240253073best known 0.240253073
Matches the best known
Answer sourcePackomania · Eckard Specht
Solution methodPublished reference construction
n4
CURRENT RECORD0.205604646best known 0.205604646
Matches the best known
Answer sourcePackomania · Eckard Specht
Solution methodPublished reference construction
n5
CURRENT RECORD0.180960531best known 0.180960531
Matches the best known
Answer sourcePackomania · Eckard Specht
Solution methodPublished reference construction
n6
CURRENT RECORD0.167632184best known 0.167632184
Matches the best known
Answer sourcePackomania · Eckard Specht
Solution methodPublished reference construction
n7
CURRENT RECORD0.16324316best known [0.1632431605, 0.1632431615]
Hard
Answer sourcePackomania · Eckard Specht
Solution methodPublished reference construction
n8
CURRENT RECORD0.150154806best known [0.1501548065, 0.1501548075]
Hard
Answer sourcePackomania · Eckard Specht
Solution methodPublished reference construction
n9
CURRENT RECORD0.139429107best known 0.139429107
Matches the best known
Answer sourcePackomania · Eckard Specht
Solution methodPublished reference construction
n10
CURRENT RECORD0.135072985best known [0.1350729845, 0.1350729855]
Matches the best known
Record holderSigmoid with Codex
Solution methodAI · OpenAI: GPT-5.6 Sol
n11
CURRENT RECORD0.127840595best known [0.1278405955, 0.1278405965]
Hard
Answer sourcePackomania · Eckard Specht
Solution methodPublished reference construction
n12
CURRENT RECORD0.125441937best known 0.125441937
Matches the best known
Answer sourcePackomania · Eckard Specht
Solution methodPublished reference construction
n13
CURRENT RECORD0.118280293best known [0.1182802935, 0.1182802945]
Hard
Answer sourcePackomania · Eckard Specht
Solution methodPublished reference construction
n14
CURRENT RECORD0.115456141best known 0.115456141
Matches the best known
Answer sourcePackomania · Eckard Specht
Solution methodPublished reference construction
n15
CURRENT RECORD0.110212024best known [0.1102120235, 0.1102120245]
Matches the best known
Record holderSigmoid with Codex
Solution methodAI · OpenAI: GPT-5.6 Sol
n16
CURRENT RECORD0.107771974best known 0.107771974
Matches the best known
Answer sourcePackomania · Eckard Specht
Solution methodPublished reference construction

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-quadrant.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-quadrant.json
BibTeX (click to copy)
@misc{minmaxarena-circles-in-a-quadrant-2026-10,
  title  = {{Equal circles in a quadrant} (P30)},
  author = {{MinMax Arena}},
  year   = {2026},
  note   = {Machine-verified records, 2026-10 edition},
  url    = {https://minmaxarena.com/data/editions/2026-10/circles-in-a-quadrant.json},
  license = {CC BY 4.0}
}
DISCUSSION

Discussion

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