P92 · Packing and covering · Classic · Hard

Sum of radii in the unit circle · n = 19

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 = 19

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	[3.981, 3.982]	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": [
    [
      "0.337371919",
      "1.382494287",
      "0.234900102"
    ],
    [
      "0.337436215",
      "0.617394367",
      "0.234900111"
    ],
    [
      "0.999935713",
      "1.765099906",
      "0.234900090"
    ],
    [
      "1.662563789",
      "1.382605672",
      "0.234900088"
    ],
    [
      "1.662628136",
      "0.617505798",
      "0.234900097"
    ],
    [
      "1.000064356",
      "0.234900112",
      "0.234900108"
    ],
    [
      "1.213897220",
      "1.370552775",
      "0.213928344"
    ],
    [
      "0.786102820",
      "0.629447261",
      "0.213928357"
    ],
    [
      "1.000000022",
      "1.000000024",
      "0.213928350"
    ],
    [
      "1.213959529",
      "0.629483238",
      "0.213928352"
    ],
    [
      "0.572143316",
      "0.999964059",
      "0.213928355"
    ],
    [
      "1.427856718",
      "1.000035998",
      "0.213928347"
    ],
    [
      "0.786040526",
      "1.370516812",
      "0.213928349"
    ],
    [
      "0.589494016",
      "1.710879302",
      "0.179107470"
    ],
    [
      "1.410506072",
      "0.289120755",
      "0.179107475"
    ],
    [
      "1.410386511",
      "1.710948288",
      "0.179107463"
    ],
    [
      "0.179107483",
      "0.999931029",
      "0.179107479"
    ],
    [
      "0.589613519",
      "0.289051717",
      "0.179107482"
    ],
    [
      "1.820892531",
      "1.000069053",
      "0.179107466"
    ]
  ]
}

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

Instance parameters

{
  "n": 19
}

Starter answer

{
  "circles": [
    [
      "0.337371919",
      "1.382494287",
      "0.234900102"
    ],
    [
      "0.337436215",
      "0.617394367",
      "0.234900111"
    ],
    [
      "0.999935713",
      "1.765099906",
      "0.234900090"
    ],
    [
      "1.662563789",
      "1.382605672",
      "0.234900088"
    ],
    [
      "1.662628136",
      "0.617505798",
      "0.234900097"
    ],
    [
      "1.000064356",
      "0.234900112",
      "0.234900108"
    ],
    [
      "1.213897220",
      "1.370552775",
      "0.213928344"
    ],
    [
      "0.786102820",
      "0.629447261",
      "0.213928357"
    ],
    [
      "1.000000022",
      "1.000000024",
      "0.213928350"
    ],
    [
      "1.213959529",
      "0.629483238",
      "0.213928352"
    ],
    [
      "0.572143316",
      "0.999964059",
      "0.213928355"
    ],
    [
      "1.427856718",
      "1.000035998",
      "0.213928347"
    ],
    [
      "0.786040526",
      "1.370516812",
      "0.213928349"
    ],
    [
      "0.589494016",
      "1.710879302",
      "0.179107470"
    ],
    [
      "1.410506072",
      "0.289120755",
      "0.179107475"
    ],
    [
      "1.410386511",
      "1.710948288",
      "0.179107463"
    ],
    [
      "0.179107483",
      "0.999931029",
      "0.179107479"
    ],
    [
      "0.589613519",
      "0.289051717",
      "0.179107482"
    ],
    [
      "1.820892531",
      "1.000069053",
      "0.179107466"
    ]
  ]
}

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