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P92 · Packing and covering · Classic · Hard

Sum of radii in the unit circle · n = 46

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 = 46
ObjectiveMaximize the sum of the radii

Record comparison

the sum of the radii · Higher is better
RecordValue / intervalAuthor / holderSource
External best known[6.260145, 6.260155]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 = 46
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": [
    [
      "1.003302083",
      "1.340622806",
      "0.175565189"
    ],
    [
      "0.670523398",
      "0.913513379",
      "0.175565196"
    ],
    [
      "1.655282044",
      "0.489442648",
      "0.169299471"
    ],
    [
      "1.493422010",
      "1.690403852",
      "0.151399528"
    ],
    [
      "0.451197074",
      "0.352745740",
      "0.151399547"
    ],
    [
      "1.817219657",
      "0.763879017",
      "0.149352548"
    ],
    [
      "1.428770509",
      "0.265318488",
      "0.149352555"
    ],
    [
      "0.427215892",
      "1.126559908",
      "0.147834503"
    ],
    [
      "1.704991545",
      "1.479073227",
      "0.147636089"
    ],
    [
      "0.707843633",
      "0.199269794",
      "0.147636106"
    ],
    [
      "0.737231584",
      "1.524453837",
      "0.147834498"
    ],
    [
      "1.226945736",
      "1.823027788",
      "0.146255830"
    ],
    [
      "0.257443160",
      "0.578706256",
      "0.146255846"
    ],
    [
      "0.756439970",
      "1.818087088",
      "0.146426352"
    ],
    [
      "0.147191743",
      "1.036139041",
      "0.146426362"
    ],
    [
      "0.494637064",
      "1.689171192",
      "0.145395630"
    ],
    [
      "0.700576192",
      "1.233293562",
      "0.145624056"
    ],
    [
      "0.208181973",
      "1.321516458",
      "0.145395634"
    ],
    [
      "1.524266198",
      "0.773777205",
      "0.143768080"
    ],
    [
      "1.347543416",
      "0.546959870",
      "0.143768083"
    ],
    [
      "0.993519409",
      "1.656512588",
      "0.140476033"
    ],
    [
      "0.361829733",
      "0.845761720",
      "0.140476044"
    ],
    [
      "0.537443766",
      "0.629284218",
      "0.138276170"
    ],
    [
      "0.987835745",
      "0.138880365",
      "0.138794452"
    ],
    [
      "1.832008163",
      "1.222345303",
      "0.138794439"
    ],
    [
      "1.246365177",
      "1.539159264",
      "0.138276159"
    ],
    [
      "1.244827153",
      "0.809245056",
      "0.137912852"
    ],
    [
      "0.477487560",
      "1.407111184",
      "0.137185249"
    ],
    [
      "1.140743586",
      "1.061823732",
      "0.135270925"
    ],
    [
      "1.408702933",
      "1.027350236",
      "0.134896862"
    ],
    [
      "1.073410891",
      "0.597014973",
      "0.134896867"
    ],
    [
      "0.974465681",
      "0.848412035",
      "0.135270927"
    ],
    [
      "1.304639326",
      "1.274212896",
      "0.133003132"
    ],
    [
      "0.808596809",
      "0.637560208",
      "0.133003140"
    ],
    [
      "1.160465704",
      "0.345971462",
      "0.130812343"
    ],
    [
      "1.673412163",
      "1.004319795",
      "0.130812335"
    ],
    [
      "0.904888563",
      "0.393696953",
      "0.129182651"
    ],
    [
      "1.564644736",
      "1.240470245",
      "0.129182641"
    ],
    [
      "0.899492836",
      "1.078309347",
      "0.106542432"
    ],
    [
      "0.990964580",
      "1.898464851",
      "0.101489717"
    ],
    [
      "0.126596650",
      "0.789079878",
      "0.101489727"
    ],
    [
      "0.290220534",
      "1.553018710",
      "0.100213034"
    ],
    [
      "1.462111601",
      "1.443082357",
      "0.097896005"
    ],
    [
      "0.683356116",
      "0.443577764",
      "0.097896014"
    ],
    [
      "1.224250400",
      "0.126275146",
      "0.097956008"
    ],
    [
      "1.902035947",
      "0.996188543",
      "0.097956000"
    ]
  ]
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 46
}

Starter answer

{
  "circles": [
    [
      "1.003302083",
      "1.340622806",
      "0.175565189"
    ],
    [
      "0.670523398",
      "0.913513379",
      "0.175565196"
    ],
    [
      "1.655282044",
      "0.489442648",
      "0.169299471"
    ],
    [
      "1.493422010",
      "1.690403852",
      "0.151399528"
    ],
    [
      "0.451197074",
      "0.352745740",
      "0.151399547"
    ],
    [
      "1.817219657",
      "0.763879017",
      "0.149352548"
    ],
    [
      "1.428770509",
      "0.265318488",
      "0.149352555"
    ],
    [
      "0.427215892",
      "1.126559908",
      "0.147834503"
    ],
    [
      "1.704991545",
      "1.479073227",
      "0.147636089"
    ],
    [
      "0.707843633",
      "0.199269794",
      "0.147636106"
    ],
    [
      "0.737231584",
      "1.524453837",
      "0.147834498"
    ],
    [
      "1.226945736",
      "1.823027788",
      "0.146255830"
    ],
    [
      "0.257443160",
      "0.578706256",
      "0.146255846"
    ],
    [
      "0.756439970",
      "1.818087088",
      "0.146426352"
    ],
    [
      "0.147191743",
      "1.036139041",
      "0.146426362"
    ],
    [
      "0.494637064",
      "1.689171192",
      "0.145395630"
    ],
    [
      "0.700576192",
      "1.233293562",
      "0.145624056"
    ],
    [
      "0.208181973",
      "1.321516458",
      "0.145395634"
    ],
    [
      "1.524266198",
      "0.773777205",
      "0.143768080"
    ],
    [
      "1.347543416",
      "0.546959870",
      "0.143768083"
    ],
    [
      "0.993519409",
      "1.656512588",
      "0.140476033"
    ],
    [
      "0.361829733",
      "0.845761720",
      "0.140476044"
    ],
    [
      "0.537443766",
      "0.629284218",
      "0.138276170"
    ],
    [
      "0.987835745",
      "0.138880365",
      "0.138794452"
    ],
    [
      "1.832008163",
      "1.222345303",
      "0.138794439"
    ],
    [
      "1.246365177",
      "1.539159264",
      "0.138276159"
    ],
    [
      "1.244827153",
      "0.809245056",
      "0.137912852"
    ],
    [
      "0.477487560",
      "1.407111184",
      "0.137185249"
    ],
    [
      "1.140743586",
      "1.061823732",
      "0.135270925"
    ],
    [
      "1.408702933",
      "1.027350236",
      "0.134896862"
    ],
    [
      "1.073410891",
      "0.597014973",
      "0.134896867"
    ],
    [
      "0.974465681",
      "0.848412035",
      "0.135270927"
    ],
    [
      "1.304639326",
      "1.274212896",
      "0.133003132"
    ],
    [
      "0.808596809",
      "0.637560208",
      "0.133003140"
    ],
    [
      "1.160465704",
      "0.345971462",
      "0.130812343"
    ],
    [
      "1.673412163",
      "1.004319795",
      "0.130812335"
    ],
    [
      "0.904888563",
      "0.393696953",
      "0.129182651"
    ],
    [
      "1.564644736",
      "1.240470245",
      "0.129182641"
    ],
    [
      "0.899492836",
      "1.078309347",
      "0.106542432"
    ],
    [
      "0.990964580",
      "1.898464851",
      "0.101489717"
    ],
    [
      "0.126596650",
      "0.789079878",
      "0.101489727"
    ],
    [
      "0.290220534",
      "1.553018710",
      "0.100213034"
    ],
    [
      "1.462111601",
      "1.443082357",
      "0.097896005"
    ],
    [
      "0.683356116",
      "0.443577764",
      "0.097896014"
    ],
    [
      "1.224250400",
      "0.126275146",
      "0.097956008"
    ],
    [
      "1.902035947",
      "0.996188543",
      "0.097956000"
    ]
  ]
}

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