P92 · Packing and covering · Classic · Hard

Sum of radii in the unit circle · n = 12

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 = 12

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	[3.103, 3.104]	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

123456789101112

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": [
    [
      "1.000189398",
      "1.678403930",
      "0.321596043"
    ],
    [
      "1.587420358",
      "0.660634054",
      "0.321596047"
    ],
    [
      "0.412390283",
      "0.660962054",
      "0.321596058"
    ],
    [
      "0.733296694",
      "1.154080532",
      "0.266746322"
    ],
    [
      "1.266789314",
      "1.153931604",
      "0.266746318"
    ],
    [
      "0.999914033",
      "0.691987902",
      "0.266746324"
    ],
    [
      "1.756051384",
      "1.178609390",
      "0.223137714"
    ],
    [
      "1.222930014",
      "0.255811042",
      "0.223137723"
    ],
    [
      "1.533021564",
      "1.565157521",
      "0.223137713"
    ],
    [
      "0.467294078",
      "1.565455039",
      "0.223137720"
    ],
    [
      "0.244048464",
      "1.179031477",
      "0.223137724"
    ],
    [
      "0.776654580",
      "0.255935611",
      "0.223137726"
    ]
  ]
}

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

Instance parameters

{
  "n": 12
}

Starter answer

{
  "circles": [
    [
      "1.000189398",
      "1.678403930",
      "0.321596043"
    ],
    [
      "1.587420358",
      "0.660634054",
      "0.321596047"
    ],
    [
      "0.412390283",
      "0.660962054",
      "0.321596058"
    ],
    [
      "0.733296694",
      "1.154080532",
      "0.266746322"
    ],
    [
      "1.266789314",
      "1.153931604",
      "0.266746318"
    ],
    [
      "0.999914033",
      "0.691987902",
      "0.266746324"
    ],
    [
      "1.756051384",
      "1.178609390",
      "0.223137714"
    ],
    [
      "1.222930014",
      "0.255811042",
      "0.223137723"
    ],
    [
      "1.533021564",
      "1.565157521",
      "0.223137713"
    ],
    [
      "0.467294078",
      "1.565455039",
      "0.223137720"
    ],
    [
      "0.244048464",
      "1.179031477",
      "0.223137724"
    ],
    [
      "0.776654580",
      "0.255935611",
      "0.223137726"
    ]
  ]
}

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