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P92 · Packing and covering · Classic · Hard

Sum of radii in the unit circle · n = 45

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 = 45
ObjectiveMaximize the sum of the radii

Record comparison

the sum of the radii · Higher is better
RecordValue / intervalAuthor / holderSource
External best known[6.187995, 6.188005]See source attributionErich Friedman ↗
Site recordNo 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 betterS=Σr
  • ObjectiveMaximise the sum of the radii over every legal arrangementmaxSΣr
Open the full editor ↗

Layout shown: starter layout

2y0
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
The current arrangement for Sum of radii in the unit circle n = 45
LAYOUT & HISTORY

Starter layout

Inspect the layout, solver notes and record history.

Challenge this record ↗Submit a proof / idea ↓Share a proof or idea in the discussion. Accepted contributions can earn proof points.
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": [
    [
      "1.196550042",
      "0.750409060",
      "0.180371576"
    ],
    [
      "1.004938704",
      "1.826348637",
      "0.173636604"
    ],
    [
      "0.383029885",
      "0.991967408",
      "0.166063357"
    ],
    [
      "0.678947411",
      "0.851130974",
      "0.161659211"
    ],
    [
      "0.334415425",
      "0.481785074",
      "0.156466043"
    ],
    [
      "1.329963491",
      "1.777942660",
      "0.154972966"
    ],
    [
      "0.681771699",
      "1.785979260",
      "0.152042071"
    ],
    [
      "1.306820385",
      "0.208518294",
      "0.151128961"
    ],
    [
      "1.105374461",
      "0.432639634",
      "0.150219400"
    ],
    [
      "0.185176394",
      "1.239389227",
      "0.150738726"
    ],
    [
      "1.691024499",
      "0.505440809",
      "0.150233178"
    ],
    [
      "1.526839428",
      "0.756778729",
      "0.149979224"
    ],
    [
      "1.013332651",
      "0.148530969",
      "0.148426591"
    ],
    [
      "0.186108702",
      "0.747509545",
      "0.147843632"
    ],
    [
      "0.555973165",
      "0.272896057",
      "0.148037574"
    ],
    [
      "1.146056988",
      "1.538480900",
      "0.146960032"
    ],
    [
      "1.385269566",
      "1.016955697",
      "0.146220169"
    ],
    [
      "1.822596459",
      "0.770984198",
      "0.146118760"
    ],
    [
      "1.395352229",
      "0.491888955",
      "0.145749496"
    ],
    [
      "0.855322381",
      "1.546292514",
      "0.143879498"
    ],
    [
      "0.894325466",
      "0.635924312",
      "0.142810185"
    ],
    [
      "0.307244356",
      "1.505759065",
      "0.142268915"
    ],
    [
      "1.705446932",
      "1.487590927",
      "0.142445170"
    ],
    [
      "1.827041694",
      "1.231307231",
      "0.141221216"
    ],
    [
      "0.622815823",
      "0.553909494",
      "0.140816181"
    ],
    [
      "1.266904326",
      "1.277880548",
      "0.140297033"
    ],
    [
      "1.546444730",
      "1.253817045",
      "0.140277180"
    ],
    [
      "0.826178278",
      "0.364318241",
      "0.137214628"
    ],
    [
      "1.105188077",
      "1.053554986",
      "0.136242479"
    ],
    [
      "0.991223648",
      "1.301201380",
      "0.136368281"
    ],
    [
      "1.427138698",
      "1.503348342",
      "0.136308787"
    ],
    [
      "1.666807664",
      "1.005642266",
      "0.135545148"
    ],
    [
      "0.467208209",
      "1.280225259",
      "0.134234125"
    ],
    [
      "0.580868361",
      "1.520716775",
      "0.131763617"
    ],
    [
      "0.727494391",
      "1.306989657",
      "0.127424487"
    ],
    [
      "0.845821546",
      "1.084110217",
      "0.124917662"
    ],
    [
      "0.108502213",
      "0.991786083",
      "0.108464373"
    ],
    [
      "0.441001765",
      "0.723968187",
      "0.108134236"
    ],
    [
      "1.548763024",
      "0.293260166",
      "0.105225139"
    ],
    [
      "0.942729807",
      "0.877353337",
      "0.103423350"
    ],
    [
      "1.572119074",
      "1.692323038",
      "0.101873380"
    ],
    [
      "0.763468789",
      "0.133590109",
      "0.101883574"
    ],
    [
      "0.442581538",
      "1.706652790",
      "0.099959162"
    ],
    [
      "0.622668101",
      "1.105951380",
      "0.099302088"
    ],
    [
      "1.901166416",
      "1.002988476",
      "0.098828628"
    ]
  ]
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 45
}

Starter answer

{
  "circles": [
    [
      "1.196550042",
      "0.750409060",
      "0.180371576"
    ],
    [
      "1.004938704",
      "1.826348637",
      "0.173636604"
    ],
    [
      "0.383029885",
      "0.991967408",
      "0.166063357"
    ],
    [
      "0.678947411",
      "0.851130974",
      "0.161659211"
    ],
    [
      "0.334415425",
      "0.481785074",
      "0.156466043"
    ],
    [
      "1.329963491",
      "1.777942660",
      "0.154972966"
    ],
    [
      "0.681771699",
      "1.785979260",
      "0.152042071"
    ],
    [
      "1.306820385",
      "0.208518294",
      "0.151128961"
    ],
    [
      "1.105374461",
      "0.432639634",
      "0.150219400"
    ],
    [
      "0.185176394",
      "1.239389227",
      "0.150738726"
    ],
    [
      "1.691024499",
      "0.505440809",
      "0.150233178"
    ],
    [
      "1.526839428",
      "0.756778729",
      "0.149979224"
    ],
    [
      "1.013332651",
      "0.148530969",
      "0.148426591"
    ],
    [
      "0.186108702",
      "0.747509545",
      "0.147843632"
    ],
    [
      "0.555973165",
      "0.272896057",
      "0.148037574"
    ],
    [
      "1.146056988",
      "1.538480900",
      "0.146960032"
    ],
    [
      "1.385269566",
      "1.016955697",
      "0.146220169"
    ],
    [
      "1.822596459",
      "0.770984198",
      "0.146118760"
    ],
    [
      "1.395352229",
      "0.491888955",
      "0.145749496"
    ],
    [
      "0.855322381",
      "1.546292514",
      "0.143879498"
    ],
    [
      "0.894325466",
      "0.635924312",
      "0.142810185"
    ],
    [
      "0.307244356",
      "1.505759065",
      "0.142268915"
    ],
    [
      "1.705446932",
      "1.487590927",
      "0.142445170"
    ],
    [
      "1.827041694",
      "1.231307231",
      "0.141221216"
    ],
    [
      "0.622815823",
      "0.553909494",
      "0.140816181"
    ],
    [
      "1.266904326",
      "1.277880548",
      "0.140297033"
    ],
    [
      "1.546444730",
      "1.253817045",
      "0.140277180"
    ],
    [
      "0.826178278",
      "0.364318241",
      "0.137214628"
    ],
    [
      "1.105188077",
      "1.053554986",
      "0.136242479"
    ],
    [
      "0.991223648",
      "1.301201380",
      "0.136368281"
    ],
    [
      "1.427138698",
      "1.503348342",
      "0.136308787"
    ],
    [
      "1.666807664",
      "1.005642266",
      "0.135545148"
    ],
    [
      "0.467208209",
      "1.280225259",
      "0.134234125"
    ],
    [
      "0.580868361",
      "1.520716775",
      "0.131763617"
    ],
    [
      "0.727494391",
      "1.306989657",
      "0.127424487"
    ],
    [
      "0.845821546",
      "1.084110217",
      "0.124917662"
    ],
    [
      "0.108502213",
      "0.991786083",
      "0.108464373"
    ],
    [
      "0.441001765",
      "0.723968187",
      "0.108134236"
    ],
    [
      "1.548763024",
      "0.293260166",
      "0.105225139"
    ],
    [
      "0.942729807",
      "0.877353337",
      "0.103423350"
    ],
    [
      "1.572119074",
      "1.692323038",
      "0.101873380"
    ],
    [
      "0.763468789",
      "0.133590109",
      "0.101883574"
    ],
    [
      "0.442581538",
      "1.706652790",
      "0.099959162"
    ],
    [
      "0.622668101",
      "1.105951380",
      "0.099302088"
    ],
    [
      "1.901166416",
      "1.002988476",
      "0.098828628"
    ]
  ]
}

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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