P92 · Packing and covering · Classic · Hard

Sum of radii in the unit circle · n = 41

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 = 41

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	[5.910325, 5.910335]	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

1234567891011121314151617181920212223242526272829303132333435363738394041

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": [
    [
      "1.002660160",
      "1.015712102",
      "0.194305993"
    ],
    [
      "0.762148406",
      "1.542970231",
      "0.181232303"
    ],
    [
      "0.952599235",
      "1.832953051",
      "0.165699323"
    ],
    [
      "0.894001375",
      "0.679510299",
      "0.159018763"
    ],
    [
      "1.207562001",
      "0.728664314",
      "0.158371191"
    ],
    [
      "0.669186413",
      "0.902725824",
      "0.157788651"
    ],
    [
      "1.272899297",
      "1.798148883",
      "0.156486119"
    ],
    [
      "1.352652678",
      "1.007311526",
      "0.155787324"
    ],
    [
      "0.982948393",
      "0.154272037",
      "0.154100156"
    ],
    [
      "0.714414339",
      "1.211185216",
      "0.153968885"
    ],
    [
      "0.513286600",
      "0.307738570",
      "0.153763731"
    ],
    [
      "1.286700538",
      "0.203683323",
      "0.153644608"
    ],
    [
      "0.297914023",
      "0.526119511",
      "0.152953698"
    ],
    [
      "0.592848365",
      "0.602729405",
      "0.151768046"
    ],
    [
      "1.092801038",
      "0.440134835",
      "0.152143443"
    ],
    [
      "1.681313200",
      "0.495260283",
      "0.152090890"
    ],
    [
      "1.674855399",
      "1.515899031",
      "0.150540395"
    ],
    [
      "0.151102030",
      "0.991391468",
      "0.151058382"
    ],
    [
      "1.816318762",
      "0.766584526",
      "0.150965780"
    ],
    [
      "1.515613739",
      "0.748210818",
      "0.150300055"
    ],
    [
      "0.432162794",
      "1.098810645",
      "0.149830355"
    ],
    [
      "1.092894638",
      "1.550584758",
      "0.149601569"
    ],
    [
      "1.812843563",
      "1.249076951",
      "0.149850609"
    ],
    [
      "0.200046157",
      "1.288166781",
      "0.149725782"
    ],
    [
      "1.219719998",
      "1.280916382",
      "0.148401291"
    ],
    [
      "1.515481966",
      "1.263616222",
      "0.147866218"
    ],
    [
      "0.800062438",
      "0.390629935",
      "0.144751526"
    ],
    [
      "0.475681977",
      "1.389264256",
      "0.143865438"
    ],
    [
      "1.385525053",
      "0.484335899",
      "0.143898922"
    ],
    [
      "0.382804858",
      "0.809907281",
      "0.143258981"
    ],
    [
      "1.382669660",
      "1.520915207",
      "0.141688399"
    ],
    [
      "1.650048447",
      "1.007320605",
      "0.141608445"
    ],
    [
      "0.467714175",
      "1.674100519",
      "0.141082245"
    ],
    [
      "0.959516206",
      "1.322024287",
      "0.115029673"
    ],
    [
      "0.673479692",
      "1.824111632",
      "0.113560214"
    ],
    [
      "0.721906213",
      "0.151361434",
      "0.106958248"
    ],
    [
      "1.524651892",
      "1.723866018",
      "0.105996856"
    ],
    [
      "1.534012280",
      "0.283175889",
      "0.106128688"
    ],
    [
      "0.145648330",
      "0.734920934",
      "0.105470130"
    ],
    [
      "1.895802938",
      "1.009004104",
      "0.104151811"
    ],
    [
      "0.273527753",
      "1.528523936",
      "0.101612847"
    ]
  ]
}

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

Instance parameters

{
  "n": 41
}

Starter answer

{
  "circles": [
    [
      "1.002660160",
      "1.015712102",
      "0.194305993"
    ],
    [
      "0.762148406",
      "1.542970231",
      "0.181232303"
    ],
    [
      "0.952599235",
      "1.832953051",
      "0.165699323"
    ],
    [
      "0.894001375",
      "0.679510299",
      "0.159018763"
    ],
    [
      "1.207562001",
      "0.728664314",
      "0.158371191"
    ],
    [
      "0.669186413",
      "0.902725824",
      "0.157788651"
    ],
    [
      "1.272899297",
      "1.798148883",
      "0.156486119"
    ],
    [
      "1.352652678",
      "1.007311526",
      "0.155787324"
    ],
    [
      "0.982948393",
      "0.154272037",
      "0.154100156"
    ],
    [
      "0.714414339",
      "1.211185216",
      "0.153968885"
    ],
    [
      "0.513286600",
      "0.307738570",
      "0.153763731"
    ],
    [
      "1.286700538",
      "0.203683323",
      "0.153644608"
    ],
    [
      "0.297914023",
      "0.526119511",
      "0.152953698"
    ],
    [
      "0.592848365",
      "0.602729405",
      "0.151768046"
    ],
    [
      "1.092801038",
      "0.440134835",
      "0.152143443"
    ],
    [
      "1.681313200",
      "0.495260283",
      "0.152090890"
    ],
    [
      "1.674855399",
      "1.515899031",
      "0.150540395"
    ],
    [
      "0.151102030",
      "0.991391468",
      "0.151058382"
    ],
    [
      "1.816318762",
      "0.766584526",
      "0.150965780"
    ],
    [
      "1.515613739",
      "0.748210818",
      "0.150300055"
    ],
    [
      "0.432162794",
      "1.098810645",
      "0.149830355"
    ],
    [
      "1.092894638",
      "1.550584758",
      "0.149601569"
    ],
    [
      "1.812843563",
      "1.249076951",
      "0.149850609"
    ],
    [
      "0.200046157",
      "1.288166781",
      "0.149725782"
    ],
    [
      "1.219719998",
      "1.280916382",
      "0.148401291"
    ],
    [
      "1.515481966",
      "1.263616222",
      "0.147866218"
    ],
    [
      "0.800062438",
      "0.390629935",
      "0.144751526"
    ],
    [
      "0.475681977",
      "1.389264256",
      "0.143865438"
    ],
    [
      "1.385525053",
      "0.484335899",
      "0.143898922"
    ],
    [
      "0.382804858",
      "0.809907281",
      "0.143258981"
    ],
    [
      "1.382669660",
      "1.520915207",
      "0.141688399"
    ],
    [
      "1.650048447",
      "1.007320605",
      "0.141608445"
    ],
    [
      "0.467714175",
      "1.674100519",
      "0.141082245"
    ],
    [
      "0.959516206",
      "1.322024287",
      "0.115029673"
    ],
    [
      "0.673479692",
      "1.824111632",
      "0.113560214"
    ],
    [
      "0.721906213",
      "0.151361434",
      "0.106958248"
    ],
    [
      "1.524651892",
      "1.723866018",
      "0.105996856"
    ],
    [
      "1.534012280",
      "0.283175889",
      "0.106128688"
    ],
    [
      "0.145648330",
      "0.734920934",
      "0.105470130"
    ],
    [
      "1.895802938",
      "1.009004104",
      "0.104151811"
    ],
    [
      "0.273527753",
      "1.528523936",
      "0.101612847"
    ]
  ]
}

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

Talk strategy, share methods, ask why you are stuck. Every signed-in user can post. Posts carry your public byline, the same name your records use; the #number after it is the account's signup ordinal, so a name cannot be worn by someone else. New posts appear after an automated review.

对本周挑战的评价NUE_13#311 replies1h ago