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P92 · Packing and covering · Classic · Hard

Sum of radii in the unit circle · n = 47

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 = 47
ObjectiveMaximize the sum of the radii

Record comparison

the sum of the radii · Higher is better
RecordValue / intervalAuthor / holderSource
External best known[6.327785, 6.327795]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 = 47
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": [
    [
      "1.532043714",
      "1.281811778",
      "0.181126454"
    ],
    [
      "0.481919834",
      "1.177321697",
      "0.172006027"
    ],
    [
      "0.867102015",
      "0.406576433",
      "0.171529436"
    ],
    [
      "1.563276761",
      "1.622208535",
      "0.160700189"
    ],
    [
      "1.088308516",
      "0.162508855",
      "0.157865917"
    ],
    [
      "0.261483688",
      "1.415579847",
      "0.152584545"
    ],
    [
      "0.161703112",
      "1.127781534",
      "0.152020169"
    ],
    [
      "1.279055234",
      "1.495931719",
      "0.150310598"
    ],
    [
      "1.305588661",
      "1.794148913",
      "0.149084654"
    ],
    [
      "1.182998756",
      "0.453113800",
      "0.147776808"
    ],
    [
      "1.383349101",
      "0.237496272",
      "0.146554941"
    ],
    [
      "0.369063943",
      "0.425249389",
      "0.146525586"
    ],
    [
      "1.214093288",
      "1.207437846",
      "0.145406782"
    ],
    [
      "1.399443745",
      "0.983786362",
      "0.145066587"
    ],
    [
      "0.596466536",
      "0.244816911",
      "0.143763610"
    ],
    [
      "1.010114063",
      "0.684870007",
      "0.141359935"
    ],
    [
      "0.209023714",
      "0.663349894",
      "0.140362414"
    ],
    [
      "1.289334596",
      "0.720683141",
      "0.140147944"
    ],
    [
      "0.585921588",
      "1.753524629",
      "0.140197523"
    ],
    [
      "0.848669357",
      "1.847685160",
      "0.138912840"
    ],
    [
      "1.120409889",
      "0.940738266",
      "0.137268375"
    ],
    [
      "1.564815517",
      "0.754750700",
      "0.137431478"
    ],
    [
      "1.007816157",
      "1.400602112",
      "0.137193090"
    ],
    [
      "0.543577925",
      "1.479845884",
      "0.136737561"
    ],
    [
      "1.726354781",
      "0.533331440",
      "0.136651395"
    ],
    [
      "0.798776459",
      "1.577002469",
      "0.136329634"
    ],
    [
      "1.836499630",
      "0.782364452",
      "0.135652348"
    ],
    [
      "0.341670959",
      "0.904438370",
      "0.134808340"
    ],
    [
      "1.678707254",
      "1.002002441",
      "0.134790399"
    ],
    [
      "1.052975897",
      "1.667499965",
      "0.133498368"
    ],
    [
      "1.839421600",
      "1.216378287",
      "0.133138889"
    ],
    [
      "1.458347133",
      "0.506488855",
      "0.132697120"
    ],
    [
      "0.753686818",
      "1.312824501",
      "0.131668639"
    ],
    [
      "0.738060304",
      "0.679915786",
      "0.130738929"
    ],
    [
      "0.605298931",
      "0.902956647",
      "0.128823795"
    ],
    [
      "0.478143837",
      "0.678460832",
      "0.129181609"
    ],
    [
      "0.949035229",
      "1.142537711",
      "0.127481109"
    ],
    [
      "0.859803818",
      "0.905607666",
      "0.125694898"
    ],
    [
      "0.822284500",
      "0.128281761",
      "0.110350918"
    ],
    [
      "1.769741411",
      "1.448439430",
      "0.109157835"
    ],
    [
      "0.607786643",
      "0.488839538",
      "0.100521442"
    ],
    [
      "0.371698439",
      "1.643531653",
      "0.100613631"
    ],
    [
      "0.107911539",
      "0.881492650",
      "0.100074550"
    ],
    [
      "1.080744550",
      "1.897853625",
      "0.098522981"
    ],
    [
      "1.607746548",
      "0.332071494",
      "0.096958275"
    ],
    [
      "0.733385614",
      "1.086814181",
      "0.095251618"
    ],
    [
      "1.906745356",
      "1.000226648",
      "0.093254615"
    ]
  ]
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 47
}

Starter answer

{
  "circles": [
    [
      "1.532043714",
      "1.281811778",
      "0.181126454"
    ],
    [
      "0.481919834",
      "1.177321697",
      "0.172006027"
    ],
    [
      "0.867102015",
      "0.406576433",
      "0.171529436"
    ],
    [
      "1.563276761",
      "1.622208535",
      "0.160700189"
    ],
    [
      "1.088308516",
      "0.162508855",
      "0.157865917"
    ],
    [
      "0.261483688",
      "1.415579847",
      "0.152584545"
    ],
    [
      "0.161703112",
      "1.127781534",
      "0.152020169"
    ],
    [
      "1.279055234",
      "1.495931719",
      "0.150310598"
    ],
    [
      "1.305588661",
      "1.794148913",
      "0.149084654"
    ],
    [
      "1.182998756",
      "0.453113800",
      "0.147776808"
    ],
    [
      "1.383349101",
      "0.237496272",
      "0.146554941"
    ],
    [
      "0.369063943",
      "0.425249389",
      "0.146525586"
    ],
    [
      "1.214093288",
      "1.207437846",
      "0.145406782"
    ],
    [
      "1.399443745",
      "0.983786362",
      "0.145066587"
    ],
    [
      "0.596466536",
      "0.244816911",
      "0.143763610"
    ],
    [
      "1.010114063",
      "0.684870007",
      "0.141359935"
    ],
    [
      "0.209023714",
      "0.663349894",
      "0.140362414"
    ],
    [
      "1.289334596",
      "0.720683141",
      "0.140147944"
    ],
    [
      "0.585921588",
      "1.753524629",
      "0.140197523"
    ],
    [
      "0.848669357",
      "1.847685160",
      "0.138912840"
    ],
    [
      "1.120409889",
      "0.940738266",
      "0.137268375"
    ],
    [
      "1.564815517",
      "0.754750700",
      "0.137431478"
    ],
    [
      "1.007816157",
      "1.400602112",
      "0.137193090"
    ],
    [
      "0.543577925",
      "1.479845884",
      "0.136737561"
    ],
    [
      "1.726354781",
      "0.533331440",
      "0.136651395"
    ],
    [
      "0.798776459",
      "1.577002469",
      "0.136329634"
    ],
    [
      "1.836499630",
      "0.782364452",
      "0.135652348"
    ],
    [
      "0.341670959",
      "0.904438370",
      "0.134808340"
    ],
    [
      "1.678707254",
      "1.002002441",
      "0.134790399"
    ],
    [
      "1.052975897",
      "1.667499965",
      "0.133498368"
    ],
    [
      "1.839421600",
      "1.216378287",
      "0.133138889"
    ],
    [
      "1.458347133",
      "0.506488855",
      "0.132697120"
    ],
    [
      "0.753686818",
      "1.312824501",
      "0.131668639"
    ],
    [
      "0.738060304",
      "0.679915786",
      "0.130738929"
    ],
    [
      "0.605298931",
      "0.902956647",
      "0.128823795"
    ],
    [
      "0.478143837",
      "0.678460832",
      "0.129181609"
    ],
    [
      "0.949035229",
      "1.142537711",
      "0.127481109"
    ],
    [
      "0.859803818",
      "0.905607666",
      "0.125694898"
    ],
    [
      "0.822284500",
      "0.128281761",
      "0.110350918"
    ],
    [
      "1.769741411",
      "1.448439430",
      "0.109157835"
    ],
    [
      "0.607786643",
      "0.488839538",
      "0.100521442"
    ],
    [
      "0.371698439",
      "1.643531653",
      "0.100613631"
    ],
    [
      "0.107911539",
      "0.881492650",
      "0.100074550"
    ],
    [
      "1.080744550",
      "1.897853625",
      "0.098522981"
    ],
    [
      "1.607746548",
      "0.332071494",
      "0.096958275"
    ],
    [
      "0.733385614",
      "1.086814181",
      "0.095251618"
    ],
    [
      "1.906745356",
      "1.000226648",
      "0.093254615"
    ]
  ]
}

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 replies