MinMax Arena
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P92 · Packing and covering · Classic

Sum of radii in the unit circle

2y0
0x2
VERIFIED CONSTRUCTIONscored by the sum of the radii
n = 12Current record · open

Place n non-overlapping circles inside a circle of radius 1, each with its own radius, making the sum of the radii as large as possible.

Formal definition

  • ContainerThe closed disc of radius 1 centred at (1, 1); a circle may touch the boundary
  • SubmissionExactly n circles, each [x, y, r] with r > 0, every number to at most nine decimals
  • ConstraintsEvery circle lies inside the disc, (x − 1)² + (y − 1)² ≤ (1 − r)², and no two interiors meet: the distance between centres is at least the sum of the radii. All checked exactly in integers; touching is allowed
  • ScoreThe sum of the radii, an exact integer sum shown to nine decimals; larger is betterS=Σr
  • ObjectiveMaximise the sum of the radii over every legal arrangementmaxSΣr

Getting a feel for it

Where the room for improvement is

As in the square, equal circles are among the worst strategies: one large circle near the centre, or a ring of large ones, with small circles tucked into the gaps between them and the rim. A disc has no corners, so the big circle in a corner becomes a ring along the rim, and the trade of sizes plays out in a different geometry.

Where the frontier is

Erich Friedman's maximum-total-perimeter table runs to n = 50: n ≤ 4 and 6–10 are closed-form arrangements by Elser and Cantrell, n = 5 was improved by Haowei Lin in July 2026, and n = 33–50 were added by Jonathan Viquerat in September 2026. The table is still being rewritten and no row carries an optimality proof; the site exhibits the pictured arrangements, reconstructed and verified exactly.

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 (1) ↓
n1
CURRENT RECORD1
Optimal
Answer sourceTrivial.
Solution methodPublished reference construction
n2
CURRENT RECORD1
Optimal
Answer sourceTrivial.
Solution methodPublished reference construction
n3
CURRENT RECORD1.392304844Source value unresolved
Hard
Answer sourceTrivial.
Solution methodPublished reference construction
n4
CURRENT RECORD1.666666666best known [1.666, 1.667]
Matches the best known
Record holderTmrg
Solution methodHuman
n5
CURRENT RECORD1.866843607best known [1.866835, 1.866845]
Matches the best known
Record holderNUE_13
Solution methodHuman
n6
CURRENT RECORD2.110575721best known [2.110, 2.111]
Matches the best known
Answer sourceVeit Elser
Solution methodPublished reference construction
n7
CURRENT RECORD2.333333331best known [2.333, 2.334]
Matches the best known
Record holderTmrg
Solution methodHuman
n8
CURRENT RECORD2.512966937best known [2.512, 2.513]
Matches the best known
Answer sourceVeit Elser
Solution methodPublished reference construction
n9
CURRENT RECORD2.66061192best known [2.660, 2.661]
Matches the best known
Record holderTmrg
Solution methodHuman
n10
CURRENT RECORD2.783982904best known [2.783, 2.784]
Matches the best known
Answer sourceDavid W. Cantrell
Solution methodPublished reference construction
n11
CURRENT RECORD2.947616238best known [2.947, 2.948]
Matches the best known
Answer sourceVeit Elser
Solution methodPublished reference construction
n12
CURRENT RECORD3.103853432best known [3.103, 3.104]
Matches the best known
Answer sourceVeit Elser
Solution methodPublished reference construction
n13
CURRENT RECORD3.227013902best known [3.227, 3.228]
Matches the best known
Answer sourceDavid W. Cantrell
Solution methodPublished reference construction
n14
CURRENT RECORD3.359761377best known [3.359, 3.360]
Matches the best known
Answer sourceVeit Elser
Solution methodPublished reference construction
n15
CURRENT RECORD3.48065231best known [3.480, 3.481]
Matches the best known
Answer sourceDavid W. Cantrell
Solution methodPublished reference construction
n16
CURRENT RECORD3.611063205best known [3.611, 3.612]
Matches the best known
Answer sourceDavid W. Cantrell
Solution methodPublished reference construction
n17
CURRENT RECORD3.730893245best known [3.730, 3.731]
Matches the best known
Answer sourceDavid W. Cantrell
Solution methodPublished reference construction
n18
CURRENT RECORD3.850913208best known [3.850, 3.851]
Matches the best known
Answer sourceDavid W. Cantrell
Solution methodPublished reference construction
n19
CURRENT RECORD3.981543885best known [3.981, 3.982]
Matches the best known
Answer sourceDavid W. Cantrell
Solution methodPublished reference construction
n20
CURRENT RECORD4.076548917best known [4.076, 4.077]
Matches the best known
Answer sourceDavid W. Cantrell
Solution methodPublished reference construction
n21
CURRENT RECORD4.181572894best known [4.181, 4.182]
Matches the best known
Answer sourceDavid W. Cantrell
Solution methodPublished reference construction
n22
CURRENT RECORD4.287688578best known [4.287, 4.288]
Matches the best known
Answer sourceDavid W. Cantrell
Solution methodPublished reference construction
n23
CURRENT RECORD4.372978434best known [4.372, 4.373]
Matches the best known
Answer sourceDavid W. Cantrell
Solution methodPublished reference construction
n24
CURRENT RECORD4.466438675best known [4.466, 4.467]
Matches the best known
Answer sourceDavid W. Cantrell
Solution methodPublished reference construction
n25
CURRENT RECORD4.564882207best known [4.564, 4.565]
Matches the best known
Answer sourceDavid W. Cantrell
Solution methodPublished reference construction
n26
CURRENT RECORD4.664464659best known [4.664, 4.665]
Matches the best known
Answer sourceDavid W. Cantrell
Solution methodPublished reference construction
n27
CURRENT RECORD4.762847432best known [4.762, 4.763]
Matches the best known
Answer sourceDavid W. Cantrell
Solution methodPublished reference construction
n28
CURRENT RECORD4.849872279best known [4.849, 4.850]
Matches the best known
Answer sourceDavid W. Cantrell
Solution methodPublished reference construction
n29
CURRENT RECORD4.934129701best known [4.934, 4.935]
Matches the best known
Answer sourceDavid W. Cantrell
Solution methodPublished reference construction
n30
CURRENT RECORD5.019820049best known [5.019, 5.020]
Matches the best known
Answer sourceDavid W. Cantrell
Solution methodPublished reference construction
n31
CURRENT RECORD5.104855721best known [5.104, 5.105]
Matches the best known
Answer sourceDavid W. Cantrell
Solution methodPublished reference construction
n32
CURRENT RECORD5.191687361best known [5.191, 5.192]
Matches the best known
Answer sourceDavid W. Cantrell
Solution methodPublished reference construction
n33
CURRENT RECORD5.277477052best known [5.277475, 5.277485]
Matches the best known
Answer sourceJonathan Viquerat
Solution methodPublished reference construction
n34
CURRENT RECORD5.360442286best known [5.360435, 5.360445]
Matches the best known
Answer sourceJonathan Viquerat
Solution methodPublished reference construction
n35
CURRENT RECORD5.44624869best known [5.446245, 5.446255]
Matches the best known
Answer sourceJonathan Viquerat
Solution methodPublished reference construction
n36
CURRENT RECORD5.530328043best known [5.530325, 5.530335]
Matches the best known
Answer sourceJonathan Viquerat
Solution methodPublished reference construction
n37
CURRENT RECORD5.621722703best known [5.621715, 5.621725]
Matches the best known
Answer sourceJonathan Viquerat
Solution methodPublished reference construction
n38
CURRENT RECORD5.692492719best known [5.692485, 5.692495]
Matches the best known
Answer sourceJonathan Viquerat
Solution methodPublished reference construction
n39
CURRENT RECORD5.763372582best known [5.763365, 5.763375]
Matches the best known
Answer sourceJonathan Viquerat
Solution methodPublished reference construction
n40
CURRENT RECORD5.836314155best known [5.836305, 5.836315]
Matches the best known
Answer sourceJonathan Viquerat
Solution methodPublished reference construction
n41
CURRENT RECORD5.910325983best known [5.910325, 5.910335]
Matches the best known
Answer sourceJonathan Viquerat
Solution methodPublished reference construction
n42
CURRENT RECORD5.981663735best known [5.981655, 5.981665]
Matches the best known
Answer sourceJonathan Viquerat
Solution methodPublished reference construction
n43
CURRENT RECORD6.047099045best known [6.047095, 6.047105]
Matches the best known
Answer sourceJonathan Viquerat
Solution methodPublished reference construction
n44
CURRENT RECORD6.11393115best known [6.113925, 6.113935]
Matches the best known
Answer sourceJonathan Viquerat
Solution methodPublished reference construction
n45
CURRENT RECORD6.187996093best known [6.187995, 6.188005]
Matches the best known
Answer sourceJonathan Viquerat
Solution methodPublished reference construction
n46
CURRENT RECORD6.260153261best known [6.260145, 6.260155]
Matches the best known
Answer sourceJonathan Viquerat
Solution methodPublished reference construction
n47
CURRENT RECORD6.3277948best known [6.327785, 6.327795]
Matches the best known
Answer sourceJonathan Viquerat
Solution methodPublished reference construction
n48
CURRENT RECORD6.40861798best known [6.408615, 6.408625]
Matches the best known
Answer sourceJonathan Viquerat
Solution methodPublished reference construction
n49
CURRENT RECORD6.473314965best known [6.473305, 6.473315]
Matches the best known
Answer sourceJonathan Viquerat
Solution methodPublished reference construction
n50
CURRENT RECORD6.534856438best known [6.534855, 6.534865]
Matches the best known
Answer sourceJonathan Viquerat
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-in-disc-radii.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-in-disc-radii.json
BibTeX (click to copy)
@misc{minmaxarena-circles-in-disc-radii-2026-08,
  title  = {{Sum of radii in the unit circle} (P92)},
  author = {{MinMax Arena}},
  year   = {2026},
  note   = {Machine-verified records, 2026-08 edition},
  url    = {https://minmaxarena.com/data/editions/2026-08/circles-in-disc-radii.json},
  license = {CC BY 4.0}
}
DISCUSSION

Discussion

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对本周挑战的评价NUE_13#311 replies