P92 · Packing and covering · Classic · Hard

Sum of radii in the unit circle · n = 33

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 = 33

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	[5.277475, 5.277485]	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

123456789101112131415161718192021222324252627282930313233

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.

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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": [
    [
      "1.419809311",
      "1.336226869",
      "0.205669672"
    ],
    [
      "0.465651781",
      "0.938679091",
      "0.205669701"
    ],
    [
      "0.523676447",
      "1.660513299",
      "0.185652441"
    ],
    [
      "0.866421697",
      "1.803317371",
      "0.185652431"
    ],
    [
      "1.563001853",
      "0.407518609",
      "0.182684097"
    ],
    [
      "1.024237764",
      "0.183043580",
      "0.182684111"
    ],
    [
      "0.666715657",
      "0.252439836",
      "0.181510750"
    ],
    [
      "1.765482307",
      "0.710238830",
      "0.181510721"
    ],
    [
      "0.811523795",
      "1.452363397",
      "0.169569288"
    ],
    [
      "1.430813286",
      "1.710675485",
      "0.168940595"
    ],
    [
      "0.192030877",
      "1.194539401",
      "0.168940626"
    ],
    [
      "1.787110653",
      "1.272080347",
      "0.167190960"
    ],
    [
      "0.252557513",
      "0.632711880",
      "0.167190997"
    ],
    [
      "1.127600541",
      "1.566916545",
      "0.166625495"
    ],
    [
      "0.507616754",
      "1.308601604",
      "0.166625509"
    ],
    [
      "0.900776096",
      "0.508514049",
      "0.165416174"
    ],
    [
      "1.418843165",
      "0.724365676",
      "0.165416161"
    ],
    [
      "1.588294793",
      "1.007024071",
      "0.164143577"
    ],
    [
      "0.580756367",
      "0.587235194",
      "0.164143601"
    ],
    [
      "1.223867963",
      "0.462693009",
      "0.160908698"
    ],
    [
      "1.067305622",
      "1.248315365",
      "0.157630868"
    ],
    [
      "0.776292848",
      "1.127065492",
      "0.157630874"
    ],
    [
      "1.268458140",
      "1.008084933",
      "0.155694834"
    ],
    [
      "0.805220789",
      "0.815078023",
      "0.155694844"
    ],
    [
      "1.104184198",
      "0.749946888",
      "0.150280946"
    ],
    [
      "1.335177186",
      "0.195538962",
      "0.128506277"
    ],
    [
      "0.133636938",
      "0.903151694",
      "0.128240543"
    ],
    [
      "1.678828145",
      "1.546952419",
      "0.128240515"
    ],
    [
      "1.173890212",
      "1.856190960",
      "0.126329143"
    ],
    [
      "0.269625746",
      "1.479431105",
      "0.126329160"
    ],
    [
      "0.386699426",
      "0.374370399",
      "0.123900695"
    ],
    [
      "1.876084602",
      "0.994919792",
      "0.123900668"
    ],
    [
      "1.004482996",
      "0.989240499",
      "0.108952080"
    ]
  ]
}

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

Instance parameters

{
  "n": 33
}

Starter answer

{
  "circles": [
    [
      "1.419809311",
      "1.336226869",
      "0.205669672"
    ],
    [
      "0.465651781",
      "0.938679091",
      "0.205669701"
    ],
    [
      "0.523676447",
      "1.660513299",
      "0.185652441"
    ],
    [
      "0.866421697",
      "1.803317371",
      "0.185652431"
    ],
    [
      "1.563001853",
      "0.407518609",
      "0.182684097"
    ],
    [
      "1.024237764",
      "0.183043580",
      "0.182684111"
    ],
    [
      "0.666715657",
      "0.252439836",
      "0.181510750"
    ],
    [
      "1.765482307",
      "0.710238830",
      "0.181510721"
    ],
    [
      "0.811523795",
      "1.452363397",
      "0.169569288"
    ],
    [
      "1.430813286",
      "1.710675485",
      "0.168940595"
    ],
    [
      "0.192030877",
      "1.194539401",
      "0.168940626"
    ],
    [
      "1.787110653",
      "1.272080347",
      "0.167190960"
    ],
    [
      "0.252557513",
      "0.632711880",
      "0.167190997"
    ],
    [
      "1.127600541",
      "1.566916545",
      "0.166625495"
    ],
    [
      "0.507616754",
      "1.308601604",
      "0.166625509"
    ],
    [
      "0.900776096",
      "0.508514049",
      "0.165416174"
    ],
    [
      "1.418843165",
      "0.724365676",
      "0.165416161"
    ],
    [
      "1.588294793",
      "1.007024071",
      "0.164143577"
    ],
    [
      "0.580756367",
      "0.587235194",
      "0.164143601"
    ],
    [
      "1.223867963",
      "0.462693009",
      "0.160908698"
    ],
    [
      "1.067305622",
      "1.248315365",
      "0.157630868"
    ],
    [
      "0.776292848",
      "1.127065492",
      "0.157630874"
    ],
    [
      "1.268458140",
      "1.008084933",
      "0.155694834"
    ],
    [
      "0.805220789",
      "0.815078023",
      "0.155694844"
    ],
    [
      "1.104184198",
      "0.749946888",
      "0.150280946"
    ],
    [
      "1.335177186",
      "0.195538962",
      "0.128506277"
    ],
    [
      "0.133636938",
      "0.903151694",
      "0.128240543"
    ],
    [
      "1.678828145",
      "1.546952419",
      "0.128240515"
    ],
    [
      "1.173890212",
      "1.856190960",
      "0.126329143"
    ],
    [
      "0.269625746",
      "1.479431105",
      "0.126329160"
    ],
    [
      "0.386699426",
      "0.374370399",
      "0.123900695"
    ],
    [
      "1.876084602",
      "0.994919792",
      "0.123900668"
    ],
    [
      "1.004482996",
      "0.989240499",
      "0.108952080"
    ]
  ]
}

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