P92 · Packing and covering · Classic · Hard

Sum of radii in the unit circle · n = 26

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 = 26

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	[4.664, 4.665]	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": [
    [
      "1.354537306",
      "1.294714840",
      "0.229509709"
    ],
    [
      "0.646640125",
      "1.296125499",
      "0.229509716"
    ],
    [
      "0.781320310",
      "0.248345779",
      "0.217181454"
    ],
    [
      "1.215682352",
      "0.247480229",
      "0.217181449"
    ],
    [
      "0.673389634",
      "1.727423620",
      "0.202617121"
    ],
    [
      "1.329506950",
      "1.726116131",
      "0.202617114"
    ],
    [
      "1.772931752",
      "1.227771559",
      "0.194206368"
    ],
    [
      "0.999193125",
      "0.595076580",
      "0.192319231"
    ],
    [
      "0.227982179",
      "1.230850245",
      "0.194206383"
    ],
    [
      "1.001027849",
      "1.515783639",
      "0.187431953"
    ],
    [
      "0.455741968",
      "0.932899024",
      "0.180826217"
    ],
    [
      "0.818130037",
      "0.922364994",
      "0.181714922"
    ],
    [
      "1.181559157",
      "0.921640779",
      "0.181714919"
    ],
    [
      "1.371506946",
      "0.613296612",
      "0.180440141"
    ],
    [
      "0.626954869",
      "0.614780289",
      "0.180440148"
    ],
    [
      "1.543986317",
      "0.930730449",
      "0.180826207"
    ],
    [
      "1.730232200",
      "0.623520345",
      "0.178430771"
    ],
    [
      "0.268273190",
      "0.626433608",
      "0.178430785"
    ],
    [
      "1.641014391",
      "1.550881862",
      "0.154795719"
    ],
    [
      "0.361186254",
      "1.553432251",
      "0.154795729"
    ],
    [
      "1.001697066",
      "1.851606618",
      "0.148391691"
    ],
    [
      "1.560292663",
      "0.351081302",
      "0.142664974"
    ],
    [
      "0.437125611",
      "0.353319434",
      "0.142664982"
    ],
    [
      "1.000374578",
      "1.187959410",
      "0.140392926"
    ],
    [
      "1.859683365",
      "0.909602996",
      "0.135577009"
    ],
    [
      "0.139963203",
      "0.913029927",
      "0.135577021"
    ]
  ]
}

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

Instance parameters

{
  "n": 26
}

Starter answer

{
  "circles": [
    [
      "1.354537306",
      "1.294714840",
      "0.229509709"
    ],
    [
      "0.646640125",
      "1.296125499",
      "0.229509716"
    ],
    [
      "0.781320310",
      "0.248345779",
      "0.217181454"
    ],
    [
      "1.215682352",
      "0.247480229",
      "0.217181449"
    ],
    [
      "0.673389634",
      "1.727423620",
      "0.202617121"
    ],
    [
      "1.329506950",
      "1.726116131",
      "0.202617114"
    ],
    [
      "1.772931752",
      "1.227771559",
      "0.194206368"
    ],
    [
      "0.999193125",
      "0.595076580",
      "0.192319231"
    ],
    [
      "0.227982179",
      "1.230850245",
      "0.194206383"
    ],
    [
      "1.001027849",
      "1.515783639",
      "0.187431953"
    ],
    [
      "0.455741968",
      "0.932899024",
      "0.180826217"
    ],
    [
      "0.818130037",
      "0.922364994",
      "0.181714922"
    ],
    [
      "1.181559157",
      "0.921640779",
      "0.181714919"
    ],
    [
      "1.371506946",
      "0.613296612",
      "0.180440141"
    ],
    [
      "0.626954869",
      "0.614780289",
      "0.180440148"
    ],
    [
      "1.543986317",
      "0.930730449",
      "0.180826207"
    ],
    [
      "1.730232200",
      "0.623520345",
      "0.178430771"
    ],
    [
      "0.268273190",
      "0.626433608",
      "0.178430785"
    ],
    [
      "1.641014391",
      "1.550881862",
      "0.154795719"
    ],
    [
      "0.361186254",
      "1.553432251",
      "0.154795729"
    ],
    [
      "1.001697066",
      "1.851606618",
      "0.148391691"
    ],
    [
      "1.560292663",
      "0.351081302",
      "0.142664974"
    ],
    [
      "0.437125611",
      "0.353319434",
      "0.142664982"
    ],
    [
      "1.000374578",
      "1.187959410",
      "0.140392926"
    ],
    [
      "1.859683365",
      "0.909602996",
      "0.135577009"
    ],
    [
      "0.139963203",
      "0.913029927",
      "0.135577021"
    ]
  ]
}

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