P92 · Packing and covering · Classic · Hard

Sum of radii in the unit circle · n = 27

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 = 27

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	[4.762, 4.763]	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

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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

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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": [
    [
      "0.792023039",
      "1.764233961",
      "0.207972245"
    ],
    [
      "1.207967525",
      "1.764236536",
      "0.207972238"
    ],
    [
      "1.765834544",
      "0.797996502",
      "0.207972246"
    ],
    [
      "1.557864575",
      "0.437776678",
      "0.207972255"
    ],
    [
      "0.442142530",
      "0.437769655",
      "0.207972274"
    ],
    [
      "0.234168014",
      "0.797986903",
      "0.207972271"
    ],
    [
      "0.630036529",
      "0.786398428",
      "0.188065825"
    ],
    [
      "1.369966210",
      "0.786403066",
      "0.188065814"
    ],
    [
      "0.999997375",
      "1.427198625",
      "0.188065810"
    ],
    [
      "1.000001353",
      "0.789941580",
      "0.181915963"
    ],
    [
      "0.818083422",
      "1.105028134",
      "0.181915961"
    ],
    [
      "1.181915340",
      "1.105030405",
      "0.181915956"
    ],
    [
      "1.539830720",
      "1.108429930",
      "0.176015567"
    ],
    [
      "0.460167979",
      "1.108423190",
      "0.176015582"
    ],
    [
      "0.636181670",
      "1.413292206",
      "0.176015575"
    ],
    [
      "1.363813243",
      "1.413296731",
      "0.176015565"
    ],
    [
      "0.823987725",
      "0.478277983",
      "0.176015586"
    ],
    [
      "1.176018893",
      "0.478280198",
      "0.176015581"
    ],
    [
      "1.000005223",
      "0.174807237",
      "0.174807236"
    ],
    [
      "0.285359564",
      "1.412591997",
      "0.174807229"
    ],
    [
      "1.714635328",
      "1.412600885",
      "0.174807208"
    ],
    [
      "1.540815583",
      "1.671977059",
      "0.137425561"
    ],
    [
      "0.688468680",
      "0.195647556",
      "0.137425587"
    ],
    [
      "0.459176106",
      "1.671970355",
      "0.137425572"
    ],
    [
      "1.852355330",
      "1.132382209",
      "0.137425563"
    ],
    [
      "1.311541496",
      "0.195651490",
      "0.137425580"
    ],
    [
      "0.147643037",
      "1.132371571",
      "0.137425582"
    ]
  ]
}

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

Instance parameters

{
  "n": 27
}

Starter answer

{
  "circles": [
    [
      "0.792023039",
      "1.764233961",
      "0.207972245"
    ],
    [
      "1.207967525",
      "1.764236536",
      "0.207972238"
    ],
    [
      "1.765834544",
      "0.797996502",
      "0.207972246"
    ],
    [
      "1.557864575",
      "0.437776678",
      "0.207972255"
    ],
    [
      "0.442142530",
      "0.437769655",
      "0.207972274"
    ],
    [
      "0.234168014",
      "0.797986903",
      "0.207972271"
    ],
    [
      "0.630036529",
      "0.786398428",
      "0.188065825"
    ],
    [
      "1.369966210",
      "0.786403066",
      "0.188065814"
    ],
    [
      "0.999997375",
      "1.427198625",
      "0.188065810"
    ],
    [
      "1.000001353",
      "0.789941580",
      "0.181915963"
    ],
    [
      "0.818083422",
      "1.105028134",
      "0.181915961"
    ],
    [
      "1.181915340",
      "1.105030405",
      "0.181915956"
    ],
    [
      "1.539830720",
      "1.108429930",
      "0.176015567"
    ],
    [
      "0.460167979",
      "1.108423190",
      "0.176015582"
    ],
    [
      "0.636181670",
      "1.413292206",
      "0.176015575"
    ],
    [
      "1.363813243",
      "1.413296731",
      "0.176015565"
    ],
    [
      "0.823987725",
      "0.478277983",
      "0.176015586"
    ],
    [
      "1.176018893",
      "0.478280198",
      "0.176015581"
    ],
    [
      "1.000005223",
      "0.174807237",
      "0.174807236"
    ],
    [
      "0.285359564",
      "1.412591997",
      "0.174807229"
    ],
    [
      "1.714635328",
      "1.412600885",
      "0.174807208"
    ],
    [
      "1.540815583",
      "1.671977059",
      "0.137425561"
    ],
    [
      "0.688468680",
      "0.195647556",
      "0.137425587"
    ],
    [
      "0.459176106",
      "1.671970355",
      "0.137425572"
    ],
    [
      "1.852355330",
      "1.132382209",
      "0.137425563"
    ],
    [
      "1.311541496",
      "0.195651490",
      "0.137425580"
    ],
    [
      "0.147643037",
      "1.132371571",
      "0.137425582"
    ]
  ]
}

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