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P92 · Packing and covering · Classic · Hard

Sum of radii in the unit circle · n = 48

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 = 48
ObjectiveMaximize the sum of the radii

Record comparison

the sum of the radii · Higher is better
RecordValue / intervalAuthor / holderSource
External best known[6.408615, 6.408625]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 = 48
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": [
    [
      "1.235886324",
      "0.432198610",
      "0.169896351"
    ],
    [
      "0.240910992",
      "1.343430005",
      "0.166837176"
    ],
    [
      "1.675642639",
      "1.492076274",
      "0.164157888"
    ],
    [
      "1.065570463",
      "0.156614741",
      "0.154069636"
    ],
    [
      "0.148116721",
      "1.043624599",
      "0.147000453"
    ],
    [
      "0.403981640",
      "1.610958293",
      "0.146473245"
    ],
    [
      "0.920792883",
      "0.418853517",
      "0.145479564"
    ],
    [
      "1.825914695",
      "1.222228490",
      "0.144710233"
    ],
    [
      "1.465042472",
      "1.717906022",
      "0.144632501"
    ],
    [
      "0.432978057",
      "1.099851211",
      "0.143356933"
    ],
    [
      "0.550288570",
      "1.361126848",
      "0.143046129"
    ],
    [
      "1.044268904",
      "0.679666705",
      "0.143085587"
    ],
    [
      "0.769634940",
      "0.173887441",
      "0.142369531"
    ],
    [
      "1.716245093",
      "0.528323660",
      "0.142395427"
    ],
    [
      "1.373329755",
      "1.446849521",
      "0.141519290"
    ],
    [
      "1.540014452",
      "1.218475511",
      "0.141214641"
    ],
    [
      "1.525189604",
      "0.319654513",
      "0.140526845"
    ],
    [
      "0.713803514",
      "1.131072454",
      "0.139198730"
    ],
    [
      "0.877910525",
      "0.906954039",
      "0.138578472"
    ],
    [
      "0.764605142",
      "0.655020001",
      "0.137662128"
    ],
    [
      "1.262403084",
      "1.190497821",
      "0.137802961"
    ],
    [
      "1.151970961",
      "0.938537333",
      "0.137295823"
    ],
    [
      "0.988931064",
      "1.159623871",
      "0.137406316"
    ],
    [
      "0.603941488",
      "0.877722882",
      "0.136945561"
    ],
    [
      "0.828676848",
      "1.381222559",
      "0.136066519"
    ],
    [
      "1.098806540",
      "1.409474153",
      "0.135536504"
    ],
    [
      "1.836510571",
      "0.777683480",
      "0.134451289"
    ],
    [
      "1.421473416",
      "0.970051035",
      "0.134042870"
    ],
    [
      "0.229409815",
      "0.603544990",
      "0.133405626"
    ],
    [
      "0.494750772",
      "0.630515647",
      "0.133302527"
    ],
    [
      "0.385463896",
      "0.387996332",
      "0.132703560"
    ],
    [
      "1.316871760",
      "0.724204205",
      "0.133131548"
    ],
    [
      "0.785562049",
      "1.840479023",
      "0.132596620"
    ],
    [
      "0.940211944",
      "1.625360304",
      "0.132342340"
    ],
    [
      "1.049052810",
      "1.866442359",
      "0.132170212"
    ],
    [
      "0.339211091",
      "0.843027511",
      "0.130048729"
    ],
    [
      "0.679643993",
      "1.600875866",
      "0.129373430"
    ],
    [
      "0.646007873",
      "0.415822809",
      "0.129322157"
    ],
    [
      "1.199559495",
      "1.653569468",
      "0.128534850"
    ],
    [
      "1.682673032",
      "0.990241904",
      "0.127935966"
    ],
    [
      "1.575252130",
      "0.758478087",
      "0.127512104"
    ],
    [
      "1.327095696",
      "0.169915259",
      "0.107793593"
    ],
    [
      "1.475846212",
      "0.554474676",
      "0.099421647"
    ],
    [
      "0.117211045",
      "0.803085642",
      "0.095515835"
    ],
    [
      "0.563087076",
      "1.792573319",
      "0.094977696"
    ],
    [
      "1.905259349",
      "0.996313557",
      "0.094733144"
    ],
    [
      "1.275359428",
      "1.862970328",
      "0.094163038"
    ],
    [
      "0.538031226",
      "0.220482322",
      "0.093874755"
    ]
  ]
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 48
}

Starter answer

{
  "circles": [
    [
      "1.235886324",
      "0.432198610",
      "0.169896351"
    ],
    [
      "0.240910992",
      "1.343430005",
      "0.166837176"
    ],
    [
      "1.675642639",
      "1.492076274",
      "0.164157888"
    ],
    [
      "1.065570463",
      "0.156614741",
      "0.154069636"
    ],
    [
      "0.148116721",
      "1.043624599",
      "0.147000453"
    ],
    [
      "0.403981640",
      "1.610958293",
      "0.146473245"
    ],
    [
      "0.920792883",
      "0.418853517",
      "0.145479564"
    ],
    [
      "1.825914695",
      "1.222228490",
      "0.144710233"
    ],
    [
      "1.465042472",
      "1.717906022",
      "0.144632501"
    ],
    [
      "0.432978057",
      "1.099851211",
      "0.143356933"
    ],
    [
      "0.550288570",
      "1.361126848",
      "0.143046129"
    ],
    [
      "1.044268904",
      "0.679666705",
      "0.143085587"
    ],
    [
      "0.769634940",
      "0.173887441",
      "0.142369531"
    ],
    [
      "1.716245093",
      "0.528323660",
      "0.142395427"
    ],
    [
      "1.373329755",
      "1.446849521",
      "0.141519290"
    ],
    [
      "1.540014452",
      "1.218475511",
      "0.141214641"
    ],
    [
      "1.525189604",
      "0.319654513",
      "0.140526845"
    ],
    [
      "0.713803514",
      "1.131072454",
      "0.139198730"
    ],
    [
      "0.877910525",
      "0.906954039",
      "0.138578472"
    ],
    [
      "0.764605142",
      "0.655020001",
      "0.137662128"
    ],
    [
      "1.262403084",
      "1.190497821",
      "0.137802961"
    ],
    [
      "1.151970961",
      "0.938537333",
      "0.137295823"
    ],
    [
      "0.988931064",
      "1.159623871",
      "0.137406316"
    ],
    [
      "0.603941488",
      "0.877722882",
      "0.136945561"
    ],
    [
      "0.828676848",
      "1.381222559",
      "0.136066519"
    ],
    [
      "1.098806540",
      "1.409474153",
      "0.135536504"
    ],
    [
      "1.836510571",
      "0.777683480",
      "0.134451289"
    ],
    [
      "1.421473416",
      "0.970051035",
      "0.134042870"
    ],
    [
      "0.229409815",
      "0.603544990",
      "0.133405626"
    ],
    [
      "0.494750772",
      "0.630515647",
      "0.133302527"
    ],
    [
      "0.385463896",
      "0.387996332",
      "0.132703560"
    ],
    [
      "1.316871760",
      "0.724204205",
      "0.133131548"
    ],
    [
      "0.785562049",
      "1.840479023",
      "0.132596620"
    ],
    [
      "0.940211944",
      "1.625360304",
      "0.132342340"
    ],
    [
      "1.049052810",
      "1.866442359",
      "0.132170212"
    ],
    [
      "0.339211091",
      "0.843027511",
      "0.130048729"
    ],
    [
      "0.679643993",
      "1.600875866",
      "0.129373430"
    ],
    [
      "0.646007873",
      "0.415822809",
      "0.129322157"
    ],
    [
      "1.199559495",
      "1.653569468",
      "0.128534850"
    ],
    [
      "1.682673032",
      "0.990241904",
      "0.127935966"
    ],
    [
      "1.575252130",
      "0.758478087",
      "0.127512104"
    ],
    [
      "1.327095696",
      "0.169915259",
      "0.107793593"
    ],
    [
      "1.475846212",
      "0.554474676",
      "0.099421647"
    ],
    [
      "0.117211045",
      "0.803085642",
      "0.095515835"
    ],
    [
      "0.563087076",
      "1.792573319",
      "0.094977696"
    ],
    [
      "1.905259349",
      "0.996313557",
      "0.094733144"
    ],
    [
      "1.275359428",
      "1.862970328",
      "0.094163038"
    ],
    [
      "0.538031226",
      "0.220482322",
      "0.093874755"
    ]
  ]
}

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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