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P92 · Packing and covering · Classic · Hard

Sum of radii in the unit circle · n = 43

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 = 43
ObjectiveMaximize the sum of the radii

Record comparison

the sum of the radii · Higher is better
RecordValue / intervalAuthor / holderSource
External best known[6.047095, 6.047105]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 = 43
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": [
    [
      "1.215414152",
      "1.785611256",
      "0.185390705"
    ],
    [
      "0.907961006",
      "0.718640428",
      "0.184098584"
    ],
    [
      "0.752256227",
      "1.479992528",
      "0.180719771"
    ],
    [
      "1.008853250",
      "0.157457050",
      "0.157410537"
    ],
    [
      "1.677715058",
      "1.502355906",
      "0.156401069"
    ],
    [
      "0.460319877",
      "1.648464725",
      "0.156340628"
    ],
    [
      "1.116801677",
      "0.451016384",
      "0.155367252"
    ],
    [
      "0.525608369",
      "0.301271086",
      "0.155447150"
    ],
    [
      "1.314425265",
      "0.213874230",
      "0.153325934"
    ],
    [
      "0.592895198",
      "0.601356023",
      "0.152089004"
    ],
    [
      "1.242569779",
      "0.730054911",
      "0.150704822"
    ],
    [
      "0.815922026",
      "0.397164075",
      "0.150293728"
    ],
    [
      "0.304247017",
      "0.511892782",
      "0.150105376"
    ],
    [
      "1.819354406",
      "1.232615116",
      "0.148265631"
    ],
    [
      "1.373881834",
      "1.492763439",
      "0.147583541"
    ],
    [
      "0.184006440",
      "1.246821323",
      "0.147494131"
    ],
    [
      "0.719363687",
      "1.805792647",
      "0.146736540"
    ],
    [
      "1.079723002",
      "1.482445505",
      "0.146756190"
    ],
    [
      "1.695786067",
      "0.504560913",
      "0.145846536"
    ],
    [
      "1.525243655",
      "1.241239705",
      "0.145971546"
    ],
    [
      "0.146162599",
      "0.956718804",
      "0.145066336"
    ],
    [
      "1.234810791",
      "1.235901413",
      "0.144510373"
    ],
    [
      "1.383164967",
      "0.988761932",
      "0.143737582"
    ],
    [
      "1.823064186",
      "0.765023861",
      "0.144051146"
    ],
    [
      "1.536252548",
      "0.745835772",
      "0.143401628"
    ],
    [
      "0.456069407",
      "1.349022591",
      "0.143131670"
    ],
    [
      "1.097290812",
      "0.984357827",
      "0.142170494"
    ],
    [
      "0.411305635",
      "1.067236430",
      "0.142187862"
    ],
    [
      "0.949505605",
      "1.225871136",
      "0.140971071"
    ],
    [
      "0.632914380",
      "0.891401926",
      "0.140704712"
    ],
    [
      "1.409790772",
      "0.491726430",
      "0.140436596"
    ],
    [
      "0.674444573",
      "1.168937041",
      "0.139920477"
    ],
    [
      "0.374118413",
      "0.790452193",
      "0.137083337"
    ],
    [
      "1.662995640",
      "0.995019637",
      "0.136163050"
    ],
    [
      "1.510969244",
      "1.719587832",
      "0.117449028"
    ],
    [
      "0.741913933",
      "0.147927376",
      "0.109698829"
    ],
    [
      "0.251744057",
      "1.490421863",
      "0.105348917"
    ],
    [
      "0.851645392",
      "1.001236490",
      "0.104054122"
    ],
    [
      "0.942140787",
      "1.691607680",
      "0.103598920"
    ],
    [
      "1.557130700",
      "0.297473013",
      "0.103373665"
    ],
    [
      "0.950630101",
      "1.896836621",
      "0.101805526"
    ],
    [
      "0.149473545",
      "0.710213288",
      "0.101461414"
    ],
    [
      "1.899572085",
      "0.997218716",
      "0.100423615"
    ]
  ]
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 43
}

Starter answer

{
  "circles": [
    [
      "1.215414152",
      "1.785611256",
      "0.185390705"
    ],
    [
      "0.907961006",
      "0.718640428",
      "0.184098584"
    ],
    [
      "0.752256227",
      "1.479992528",
      "0.180719771"
    ],
    [
      "1.008853250",
      "0.157457050",
      "0.157410537"
    ],
    [
      "1.677715058",
      "1.502355906",
      "0.156401069"
    ],
    [
      "0.460319877",
      "1.648464725",
      "0.156340628"
    ],
    [
      "1.116801677",
      "0.451016384",
      "0.155367252"
    ],
    [
      "0.525608369",
      "0.301271086",
      "0.155447150"
    ],
    [
      "1.314425265",
      "0.213874230",
      "0.153325934"
    ],
    [
      "0.592895198",
      "0.601356023",
      "0.152089004"
    ],
    [
      "1.242569779",
      "0.730054911",
      "0.150704822"
    ],
    [
      "0.815922026",
      "0.397164075",
      "0.150293728"
    ],
    [
      "0.304247017",
      "0.511892782",
      "0.150105376"
    ],
    [
      "1.819354406",
      "1.232615116",
      "0.148265631"
    ],
    [
      "1.373881834",
      "1.492763439",
      "0.147583541"
    ],
    [
      "0.184006440",
      "1.246821323",
      "0.147494131"
    ],
    [
      "0.719363687",
      "1.805792647",
      "0.146736540"
    ],
    [
      "1.079723002",
      "1.482445505",
      "0.146756190"
    ],
    [
      "1.695786067",
      "0.504560913",
      "0.145846536"
    ],
    [
      "1.525243655",
      "1.241239705",
      "0.145971546"
    ],
    [
      "0.146162599",
      "0.956718804",
      "0.145066336"
    ],
    [
      "1.234810791",
      "1.235901413",
      "0.144510373"
    ],
    [
      "1.383164967",
      "0.988761932",
      "0.143737582"
    ],
    [
      "1.823064186",
      "0.765023861",
      "0.144051146"
    ],
    [
      "1.536252548",
      "0.745835772",
      "0.143401628"
    ],
    [
      "0.456069407",
      "1.349022591",
      "0.143131670"
    ],
    [
      "1.097290812",
      "0.984357827",
      "0.142170494"
    ],
    [
      "0.411305635",
      "1.067236430",
      "0.142187862"
    ],
    [
      "0.949505605",
      "1.225871136",
      "0.140971071"
    ],
    [
      "0.632914380",
      "0.891401926",
      "0.140704712"
    ],
    [
      "1.409790772",
      "0.491726430",
      "0.140436596"
    ],
    [
      "0.674444573",
      "1.168937041",
      "0.139920477"
    ],
    [
      "0.374118413",
      "0.790452193",
      "0.137083337"
    ],
    [
      "1.662995640",
      "0.995019637",
      "0.136163050"
    ],
    [
      "1.510969244",
      "1.719587832",
      "0.117449028"
    ],
    [
      "0.741913933",
      "0.147927376",
      "0.109698829"
    ],
    [
      "0.251744057",
      "1.490421863",
      "0.105348917"
    ],
    [
      "0.851645392",
      "1.001236490",
      "0.104054122"
    ],
    [
      "0.942140787",
      "1.691607680",
      "0.103598920"
    ],
    [
      "1.557130700",
      "0.297473013",
      "0.103373665"
    ],
    [
      "0.950630101",
      "1.896836621",
      "0.101805526"
    ],
    [
      "0.149473545",
      "0.710213288",
      "0.101461414"
    ],
    [
      "1.899572085",
      "0.997218716",
      "0.100423615"
    ]
  ]
}

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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