P92 · Packing and covering · Classic · Hard

Sum of radii in the unit circle · n = 38

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.

Instancen = 38

ObjectiveMaximize the sum of the radii

Record comparison

the sum of the radii · Higher is better

Record	Value / interval	Author / holder	Source
External best known	[5.692485, 5.692495]	See source attribution	Erich Friedman ↗
Site record	No player record yet	—	—

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 better $S = Σ r$
ObjectiveMaximise the sum of the radii over every legal arrangement $\max_{S} Σ r$

Open the full editor ↗

Layout shown: starter layout

2y0

1234567891011121314151617181920212223242526272829303132333435363738

0x2

VERIFIED CONSTRUCTIONscored by the sum of the radii

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

LAYOUT & HISTORY

Starter layout

Inspect the layout, solver notes and record history.

Challenge this record ↗Submit a proof / idea ↓Share a proof or idea in the discussion. Accepted contributions can earn proof points.

ANSWER FORMAT

How to write your answer

The container is the disc of radius 1 centred at (1, 1), so every coordinate lies in [0, 2] and none is negative. Coordinates and radii share one unit and are written as plain decimals, to at most nine places.

Submit circles: exactly n triples [x, y, r], every number a decimal string of at most nine places such as "0.25". The container is the closed disc of radius 1 centred at (1, 1).

Starter answer

{
  "circles": [
    [
      "1.477826183",
      "1.326065238",
      "0.194222612"
    ],
    [
      "0.726464333",
      "1.790079087",
      "0.163909858"
    ],
    [
      "1.635978148",
      "0.457249120",
      "0.163909859"
    ],
    [
      "1.050641420",
      "1.835088938",
      "0.163376973"
    ],
    [
      "1.307176743",
      "1.011597262",
      "0.163564109"
    ],
    [
      "1.796076336",
      "0.742705321",
      "0.163376975"
    ],
    [
      "1.122787358",
      "1.281807230",
      "0.163564109"
    ],
    [
      "0.309647248",
      "1.474613703",
      "0.162238047"
    ],
    [
      "1.190264529",
      "0.184129608",
      "0.162238048"
    ],
    [
      "0.180181009",
      "1.178454070",
      "0.160983292"
    ],
    [
      "0.867288244",
      "0.171545643",
      "0.160983294"
    ],
    [
      "0.228559497",
      "0.666966420",
      "0.159743006"
    ],
    [
      "0.408641267",
      "0.403068962",
      "0.159743006"
    ],
    [
      "0.927239699",
      "1.537762073",
      "0.158541104"
    ],
    [
      "1.474224112",
      "0.736194020",
      "0.158541105"
    ],
    [
      "0.986196936",
      "0.990580890",
      "0.158102992"
    ],
    [
      "1.159187752",
      "0.727676571",
      "0.156610374"
    ],
    [
      "0.804454781",
      "1.247513356",
      "0.156610373"
    ],
    [
      "0.494158204",
      "1.216234296",
      "0.155258741"
    ],
    [
      "1.016935787",
      "0.450139683",
      "0.155258742"
    ],
    [
      "0.848793919",
      "0.710034441",
      "0.154284425"
    ],
    [
      "0.674865725",
      "0.964914253",
      "0.154284425"
    ],
    [
      "1.237354733",
      "1.578198486",
      "0.154199103"
    ],
    [
      "1.624936988",
      "1.010223340",
      "0.154199104"
    ],
    [
      "0.541507890",
      "0.687128208",
      "0.153854178"
    ],
    [
      "1.321967901",
      "0.467527936",
      "0.150268574"
    ],
    [
      "0.621556117",
      "1.493933256",
      "0.150268573"
    ],
    [
      "1.528226440",
      "1.666296323",
      "0.149721244"
    ],
    [
      "1.813015391",
      "1.248957698",
      "0.149721244"
    ],
    [
      "0.714030344",
      "0.439644003",
      "0.147828484"
    ],
    [
      "0.373953806",
      "0.938002794",
      "0.147828483"
    ],
    [
      "0.462320770",
      "1.704520780",
      "0.113744684"
    ],
    [
      "1.460008842",
      "0.242477495",
      "0.113744686"
    ],
    [
      "1.732130786",
      "1.499600915",
      "0.113649864"
    ],
    [
      "0.597231229",
      "0.208070880",
      "0.111532547"
    ],
    [
      "0.115736950",
      "0.913667654",
      "0.111532546"
    ],
    [
      "1.324458319",
      "1.828184202",
      "0.110526968"
    ],
    [
      "1.889473010",
      "1.000194024",
      "0.110526968"
    ]
  ]
}

Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 38
}

Starter answer

{
  "circles": [
    [
      "1.477826183",
      "1.326065238",
      "0.194222612"
    ],
    [
      "0.726464333",
      "1.790079087",
      "0.163909858"
    ],
    [
      "1.635978148",
      "0.457249120",
      "0.163909859"
    ],
    [
      "1.050641420",
      "1.835088938",
      "0.163376973"
    ],
    [
      "1.307176743",
      "1.011597262",
      "0.163564109"
    ],
    [
      "1.796076336",
      "0.742705321",
      "0.163376975"
    ],
    [
      "1.122787358",
      "1.281807230",
      "0.163564109"
    ],
    [
      "0.309647248",
      "1.474613703",
      "0.162238047"
    ],
    [
      "1.190264529",
      "0.184129608",
      "0.162238048"
    ],
    [
      "0.180181009",
      "1.178454070",
      "0.160983292"
    ],
    [
      "0.867288244",
      "0.171545643",
      "0.160983294"
    ],
    [
      "0.228559497",
      "0.666966420",
      "0.159743006"
    ],
    [
      "0.408641267",
      "0.403068962",
      "0.159743006"
    ],
    [
      "0.927239699",
      "1.537762073",
      "0.158541104"
    ],
    [
      "1.474224112",
      "0.736194020",
      "0.158541105"
    ],
    [
      "0.986196936",
      "0.990580890",
      "0.158102992"
    ],
    [
      "1.159187752",
      "0.727676571",
      "0.156610374"
    ],
    [
      "0.804454781",
      "1.247513356",
      "0.156610373"
    ],
    [
      "0.494158204",
      "1.216234296",
      "0.155258741"
    ],
    [
      "1.016935787",
      "0.450139683",
      "0.155258742"
    ],
    [
      "0.848793919",
      "0.710034441",
      "0.154284425"
    ],
    [
      "0.674865725",
      "0.964914253",
      "0.154284425"
    ],
    [
      "1.237354733",
      "1.578198486",
      "0.154199103"
    ],
    [
      "1.624936988",
      "1.010223340",
      "0.154199104"
    ],
    [
      "0.541507890",
      "0.687128208",
      "0.153854178"
    ],
    [
      "1.321967901",
      "0.467527936",
      "0.150268574"
    ],
    [
      "0.621556117",
      "1.493933256",
      "0.150268573"
    ],
    [
      "1.528226440",
      "1.666296323",
      "0.149721244"
    ],
    [
      "1.813015391",
      "1.248957698",
      "0.149721244"
    ],
    [
      "0.714030344",
      "0.439644003",
      "0.147828484"
    ],
    [
      "0.373953806",
      "0.938002794",
      "0.147828483"
    ],
    [
      "0.462320770",
      "1.704520780",
      "0.113744684"
    ],
    [
      "1.460008842",
      "0.242477495",
      "0.113744686"
    ],
    [
      "1.732130786",
      "1.499600915",
      "0.113649864"
    ],
    [
      "0.597231229",
      "0.208070880",
      "0.111532547"
    ],
    [
      "0.115736950",
      "0.913667654",
      "0.111532546"
    ],
    [
      "1.324458319",
      "1.828184202",
      "0.110526968"
    ],
    [
      "1.889473010",
      "1.000194024",
      "0.110526968"
    ]
  ]
}

Submit circles: exactly n triples [x, y, r], every number a decimal string of at most nine places such as "0.25". The container is the closed disc of radius 1 centred at (1, 1). · Verifier v1.0.0

DISCUSSION

Discussion

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对本周挑战的评价NUE_13#311 replies1h ago