MinMax Arena
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P87 · Packing and covering · Classic

Equal discs covering a disc

VERIFIED CONSTRUCTIONVERIFIED CONSTRUCTION
n = 11Current record · open

Cover the largest possible disc with n unit-radius discs. Maximise the covered disc radius.

Formal definition

  • ContainerThe editor and certificate normalize to the closed unit disc centred at (1,1). This is a coordinate convention only; scores use the literature's unit-piece scale. Boundary contact is allowed.
  • SubmissionSubmit {radius, placements:[{x,y,turn},…]}, with decimal strings of at most nine places. x,y are centres; circles require turn=0, polygons use turn=tan(θ/2). Coordinates and turn lie in ±4; 0 < radius ≤ 4.
  • ConstraintsThe n distinct centres lie in the target disc. Covering discs may overlap and protrude, but must cover every target point.
  • PrecisionInput decimals are exact rationals; regular-polygon vertices are algebraic, not rounded templates. Feasibility has no floating-point tolerance. A finite-decimal certificate is not a continuous optimality proof.
  • ObjectiveCovered disc radius R = 1/r, larger is better.

Getting a feel for it

Source and certificate

External targets are converted with their printed precision; decimal coordinate certificates are separate. Reconstruction loss never lowers the literature target, and a source construction is not automatically an optimality proof.

Source
ONE LEADERBOARD PER n

Current best solutions by n

Each n is an independent record with a page of its own. Open any of them to inspect the current construction, then challenge it.

Discussion (0) ↓
n11
CURRENT RECORD2.631690766best known 2.631
Matches the best known
Answer sourceHans Melissen
Solution methodPublished reference construction
n12
CURRENT RECORD2.76929324best known 2.769
Matches the best known
Answer sourceHans Melissen
Solution methodPublished reference construction
n13
CURRENT RECORD2.884791872best known 2.884
Matches the best known
Answer sourceJeremy Tan
Solution methodPublished reference construction
n14
CURRENT RECORD3.014481251best known 3.014
Matches the best known
Answer sourceJeremy Tan
Solution methodPublished reference construction
n15
CURRENT RECORD3.14324129best known 3.143
Matches the best known
Answer sourceJeremy Tan
Solution methodPublished reference construction
n16
CURRENT RECORD3.244437866best known 3.244
Matches the best known
Answer sourceJeremy Tan
Solution methodPublished reference construction
n17
CURRENT RECORD3.349729253best known 3.349
Matches the best known
Answer sourceJeremy Tan
Solution methodPublished reference construction
n18
CURRENT RECORD3.446282746best known 3.446
Matches the best known
Answer sourceJeremy Tan
Solution methodPublished reference construction
n19
CURRENT RECORD3.605551263best known 3.605
Matches the best known
Answer sourceJeremy Tan
Solution methodPublished reference construction
n20
CURRENT RECORD3.69210531best known 3.692
Matches the best known
Answer sourceJeremy Tan
Solution methodPublished reference construction
n21
CURRENT RECORD3.804522521best known 3.804
Matches the best known
Answer sourceJeremy Tan
Solution methodPublished reference construction
n22
CURRENT RECORD3.948644545best known 3.948
Matches the best known
Answer sourceJeremy Tan
Solution methodPublished reference construction
n23
CURRENT RECORD4.000739688best known 4.000
Matches the best known
Answer sourceJeremy Tan
Solution methodPublished reference construction
n24
CURRENT RECORD4.07661341best known 4.076
Matches the best known
Answer sourceJeremy Tan
Solution methodPublished reference construction
n25
CURRENT RECORD4.181540528best known 4.181
Matches the best known
Answer sourceJeremy Tan
Solution methodPublished reference construction

Data and citation

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-cover-disc.json

Cite the frozen 2026-08 edition: records move, a frozen edition never does, so the citation is still checkable years later.

GET https://minmaxarena.com/data/editions/2026-08/circles-cover-disc.json
BibTeX (click to copy)
@misc{minmaxarena-circles-cover-disc-2026-08,
  title  = {{Equal discs covering a disc} (P87)},
  author = {{MinMax Arena}},
  year   = {2026},
  note   = {Machine-verified records, 2026-08 edition},
  url    = {https://minmaxarena.com/data/editions/2026-08/circles-cover-disc.json},
  license = {CC BY 4.0}
}
DISCUSSION

Discussion

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