P92 · Packing and covering · Classic · Hard

Sum of radii in the unit circle · n = 21

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 = 21

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	[4.181, 4.182]	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": [
    [
      "0.999994090",
      "0.976903721",
      "0.258930580"
    ],
    [
      "0.774823331",
      "0.258419226",
      "0.224986079"
    ],
    [
      "1.224795469",
      "0.258303578",
      "0.224986073"
    ],
    [
      "1.000199933",
      "1.777612861",
      "0.222387113"
    ],
    [
      "1.642384023",
      "1.444167891",
      "0.219011940"
    ],
    [
      "0.357844455",
      "1.444498120",
      "0.219011957"
    ],
    [
      "0.787224850",
      "1.398011905",
      "0.212877486"
    ],
    [
      "1.212979804",
      "1.397902453",
      "0.212877481"
    ],
    [
      "1.743772599",
      "0.743559768",
      "0.213260353"
    ],
    [
      "0.256095688",
      "0.743942160",
      "0.213260371"
    ],
    [
      "0.539168204",
      "1.057006893",
      "0.208805472"
    ],
    [
      "1.460861086",
      "1.056769965",
      "0.208805461"
    ],
    [
      "1.338121895",
      "0.667365985",
      "0.199484103"
    ],
    [
      "0.661707195",
      "0.667539847",
      "0.199484111"
    ],
    [
      "0.612110018",
      "1.736146860",
      "0.167911760"
    ],
    [
      "1.388268468",
      "1.735947312",
      "0.167911753"
    ],
    [
      "1.829226157",
      "1.110450293",
      "0.163450368"
    ],
    [
      "1.585388581",
      "0.402039690",
      "0.163198755"
    ],
    [
      "0.170830754",
      "1.110876601",
      "0.163450384"
    ],
    [
      "0.414304137",
      "0.402340682",
      "0.163198766"
    ],
    [
      "0.999887878",
      "0.563690625",
      "0.154282528"
    ]
  ]
}

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

Instance parameters

{
  "n": 21
}

Starter answer

{
  "circles": [
    [
      "0.999994090",
      "0.976903721",
      "0.258930580"
    ],
    [
      "0.774823331",
      "0.258419226",
      "0.224986079"
    ],
    [
      "1.224795469",
      "0.258303578",
      "0.224986073"
    ],
    [
      "1.000199933",
      "1.777612861",
      "0.222387113"
    ],
    [
      "1.642384023",
      "1.444167891",
      "0.219011940"
    ],
    [
      "0.357844455",
      "1.444498120",
      "0.219011957"
    ],
    [
      "0.787224850",
      "1.398011905",
      "0.212877486"
    ],
    [
      "1.212979804",
      "1.397902453",
      "0.212877481"
    ],
    [
      "1.743772599",
      "0.743559768",
      "0.213260353"
    ],
    [
      "0.256095688",
      "0.743942160",
      "0.213260371"
    ],
    [
      "0.539168204",
      "1.057006893",
      "0.208805472"
    ],
    [
      "1.460861086",
      "1.056769965",
      "0.208805461"
    ],
    [
      "1.338121895",
      "0.667365985",
      "0.199484103"
    ],
    [
      "0.661707195",
      "0.667539847",
      "0.199484111"
    ],
    [
      "0.612110018",
      "1.736146860",
      "0.167911760"
    ],
    [
      "1.388268468",
      "1.735947312",
      "0.167911753"
    ],
    [
      "1.829226157",
      "1.110450293",
      "0.163450368"
    ],
    [
      "1.585388581",
      "0.402039690",
      "0.163198755"
    ],
    [
      "0.170830754",
      "1.110876601",
      "0.163450384"
    ],
    [
      "0.414304137",
      "0.402340682",
      "0.163198766"
    ],
    [
      "0.999887878",
      "0.563690625",
      "0.154282528"
    ]
  ]
}

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