P92 · Packing and covering · Classic · Hard

Sum of radii in the unit circle · n = 20

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 = 20

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	[4.076, 4.077]	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": [
    [
      "0.999806532",
      "0.547516772",
      "0.260815327"
    ],
    [
      "1.766847703",
      "1.072190083",
      "0.229761849"
    ],
    [
      "1.461748794",
      "0.383174241",
      "0.229489801"
    ],
    [
      "0.537723878",
      "0.383569361",
      "0.229489814"
    ],
    [
      "0.233214342",
      "1.072845936",
      "0.229761869"
    ],
    [
      "0.638198432",
      "1.688694291",
      "0.222053856"
    ],
    [
      "1.362390541",
      "1.688384567",
      "0.222053847"
    ],
    [
      "1.000012411",
      "1.028957317",
      "0.220625260"
    ],
    [
      "0.609086078",
      "0.826820198",
      "0.219468834"
    ],
    [
      "1.390765709",
      "0.826485927",
      "0.219468824"
    ],
    [
      "1.366667327",
      "1.255815410",
      "0.210536450"
    ],
    [
      "0.633551653",
      "1.256128932",
      "0.210536459"
    ],
    [
      "1.000195557",
      "1.457210650",
      "0.207628111"
    ],
    [
      "0.246893562",
      "0.666827264",
      "0.176487170"
    ],
    [
      "1.752821235",
      "0.666183294",
      "0.176487155"
    ],
    [
      "1.686051289",
      "1.464960709",
      "0.171232944"
    ],
    [
      "0.314346677",
      "1.465547345",
      "0.171232958"
    ],
    [
      "1.000356026",
      "1.832419326",
      "0.167580597"
    ],
    [
      "1.150561609",
      "0.164374531",
      "0.150918895"
    ],
    [
      "0.848723844",
      "0.164503595",
      "0.150918897"
    ]
  ]
}

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

Instance parameters

{
  "n": 20
}

Starter answer

{
  "circles": [
    [
      "0.999806532",
      "0.547516772",
      "0.260815327"
    ],
    [
      "1.766847703",
      "1.072190083",
      "0.229761849"
    ],
    [
      "1.461748794",
      "0.383174241",
      "0.229489801"
    ],
    [
      "0.537723878",
      "0.383569361",
      "0.229489814"
    ],
    [
      "0.233214342",
      "1.072845936",
      "0.229761869"
    ],
    [
      "0.638198432",
      "1.688694291",
      "0.222053856"
    ],
    [
      "1.362390541",
      "1.688384567",
      "0.222053847"
    ],
    [
      "1.000012411",
      "1.028957317",
      "0.220625260"
    ],
    [
      "0.609086078",
      "0.826820198",
      "0.219468834"
    ],
    [
      "1.390765709",
      "0.826485927",
      "0.219468824"
    ],
    [
      "1.366667327",
      "1.255815410",
      "0.210536450"
    ],
    [
      "0.633551653",
      "1.256128932",
      "0.210536459"
    ],
    [
      "1.000195557",
      "1.457210650",
      "0.207628111"
    ],
    [
      "0.246893562",
      "0.666827264",
      "0.176487170"
    ],
    [
      "1.752821235",
      "0.666183294",
      "0.176487155"
    ],
    [
      "1.686051289",
      "1.464960709",
      "0.171232944"
    ],
    [
      "0.314346677",
      "1.465547345",
      "0.171232958"
    ],
    [
      "1.000356026",
      "1.832419326",
      "0.167580597"
    ],
    [
      "1.150561609",
      "0.164374531",
      "0.150918895"
    ],
    [
      "0.848723844",
      "0.164503595",
      "0.150918897"
    ]
  ]
}

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