P92 · Packing and covering · Classic · Hard

Sum of radii in the unit circle · n = 23

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 = 23

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	[4.372, 4.373]	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

1234567891011121314151617181920212223

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": [
    [
      "1.382109109",
      "0.893103882",
      "0.249849608"
    ],
    [
      "0.617227461",
      "0.895504565",
      "0.249849620"
    ],
    [
      "1.747594989",
      "1.198574001",
      "0.226482125"
    ],
    [
      "0.253666260",
      "1.203262924",
      "0.226482147"
    ],
    [
      "0.416287357",
      "0.475812065",
      "0.215466099"
    ],
    [
      "1.580410759",
      "0.472158341",
      "0.215466082"
    ],
    [
      "1.000422183",
      "1.134501329",
      "0.201767084"
    ],
    [
      "0.656933038",
      "1.344356228",
      "0.200754805"
    ],
    [
      "1.345221864",
      "1.342195922",
      "0.200754796"
    ],
    [
      "0.684030056",
      "1.739765035",
      "0.195581381"
    ],
    [
      "1.320607475",
      "1.737767018",
      "0.195581373"
    ],
    [
      "1.001646008",
      "1.524420761",
      "0.188154268"
    ],
    [
      "0.997447251",
      "0.186653474",
      "0.186649468"
    ],
    [
      "0.814415017",
      "0.509007562",
      "0.184043067"
    ],
    [
      "1.182499336",
      "0.507852290",
      "0.184043063"
    ],
    [
      "1.798138878",
      "0.798462474",
      "0.176809230"
    ],
    [
      "0.200611776",
      "0.803476516",
      "0.176809248"
    ],
    [
      "0.376430071",
      "1.566862345",
      "0.157282743"
    ],
    [
      "1.627116020",
      "1.562936847",
      "0.157282730"
    ],
    [
      "0.999324487",
      "0.784764706",
      "0.147971260"
    ],
    [
      "0.665845635",
      "0.214190146",
      "0.146093525"
    ],
    [
      "1.329215178",
      "0.212108101",
      "0.146093519"
    ],
    [
      "1.002687624",
      "1.856284588",
      "0.143711193"
    ]
  ]
}

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

Instance parameters

{
  "n": 23
}

Starter answer

{
  "circles": [
    [
      "1.382109109",
      "0.893103882",
      "0.249849608"
    ],
    [
      "0.617227461",
      "0.895504565",
      "0.249849620"
    ],
    [
      "1.747594989",
      "1.198574001",
      "0.226482125"
    ],
    [
      "0.253666260",
      "1.203262924",
      "0.226482147"
    ],
    [
      "0.416287357",
      "0.475812065",
      "0.215466099"
    ],
    [
      "1.580410759",
      "0.472158341",
      "0.215466082"
    ],
    [
      "1.000422183",
      "1.134501329",
      "0.201767084"
    ],
    [
      "0.656933038",
      "1.344356228",
      "0.200754805"
    ],
    [
      "1.345221864",
      "1.342195922",
      "0.200754796"
    ],
    [
      "0.684030056",
      "1.739765035",
      "0.195581381"
    ],
    [
      "1.320607475",
      "1.737767018",
      "0.195581373"
    ],
    [
      "1.001646008",
      "1.524420761",
      "0.188154268"
    ],
    [
      "0.997447251",
      "0.186653474",
      "0.186649468"
    ],
    [
      "0.814415017",
      "0.509007562",
      "0.184043067"
    ],
    [
      "1.182499336",
      "0.507852290",
      "0.184043063"
    ],
    [
      "1.798138878",
      "0.798462474",
      "0.176809230"
    ],
    [
      "0.200611776",
      "0.803476516",
      "0.176809248"
    ],
    [
      "0.376430071",
      "1.566862345",
      "0.157282743"
    ],
    [
      "1.627116020",
      "1.562936847",
      "0.157282730"
    ],
    [
      "0.999324487",
      "0.784764706",
      "0.147971260"
    ],
    [
      "0.665845635",
      "0.214190146",
      "0.146093525"
    ],
    [
      "1.329215178",
      "0.212108101",
      "0.146093519"
    ],
    [
      "1.002687624",
      "1.856284588",
      "0.143711193"
    ]
  ]
}

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