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P92 · Packing and covering · Classic · Hard

Sum of radii in the unit circle · n = 42

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 = 42
ObjectiveMaximize the sum of the radii

Record comparison

the sum of the radii · Higher is better
RecordValue / intervalAuthor / holderSource
External best known[5.981655, 5.981665]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 = 42
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": [
    [
      "1.011640683",
      "1.012845727",
      "0.202341652"
    ],
    [
      "0.675745139",
      "0.879205572",
      "0.159162852"
    ],
    [
      "1.436269031",
      "1.718651330",
      "0.159291726"
    ],
    [
      "0.911627312",
      "0.665451380",
      "0.159162852"
    ],
    [
      "1.672342194",
      "1.504724059",
      "0.159291725"
    ],
    [
      "1.223097985",
      "0.721847544",
      "0.157372307"
    ],
    [
      "0.701286010",
      "1.194708618",
      "0.157372308"
    ],
    [
      "0.961299121",
      "1.365113231",
      "0.153504756"
    ],
    [
      "1.367148930",
      "0.997335972",
      "0.153504755"
    ],
    [
      "0.551467681",
      "0.279599743",
      "0.151378899"
    ],
    [
      "0.327121462",
      "0.482900161",
      "0.151378900"
    ],
    [
      "0.614995794",
      "0.575139887",
      "0.150912015"
    ],
    [
      "0.152793160",
      "0.934772364",
      "0.150285886"
    ],
    [
      "1.018273370",
      "0.150482395",
      "0.150285884"
    ],
    [
      "0.425361208",
      "1.060007835",
      "0.149676216"
    ],
    [
      "1.116140686",
      "0.434030011",
      "0.149676215"
    ],
    [
      "1.312026658",
      "0.208399085",
      "0.149122426"
    ],
    [
      "0.181586848",
      "1.232792939",
      "0.149122428"
    ],
    [
      "1.151959260",
      "1.837475395",
      "0.148849805"
    ],
    [
      "1.818507973",
      "1.233455248",
      "0.148849803"
    ],
    [
      "1.689342761",
      "0.497765307",
      "0.147103095"
    ],
    [
      "0.432506783",
      "1.636698181",
      "0.147103098"
    ],
    [
      "1.526702299",
      "0.742746029",
      "0.146950426"
    ],
    [
      "0.681498464",
      "1.792003535",
      "0.146353218"
    ],
    [
      "0.692273117",
      "1.498897862",
      "0.146950427"
    ],
    [
      "1.819449584",
      "0.760802988",
      "0.146353216"
    ],
    [
      "1.245472648",
      "1.270884136",
      "0.145884147"
    ],
    [
      "0.829758446",
      "0.374841731",
      "0.142758459"
    ],
    [
      "0.394578324",
      "0.769197837",
      "0.142758460"
    ],
    [
      "1.402059434",
      "0.483697851",
      "0.140524446"
    ],
    [
      "0.446715167",
      "1.349421785",
      "0.140524448"
    ],
    [
      "1.529812563",
      "1.241594637",
      "0.139960322"
    ],
    [
      "1.188405828",
      "1.550974193",
      "0.139960323"
    ],
    [
      "0.930653570",
      "1.654991638",
      "0.137989051"
    ],
    [
      "1.658635635",
      "0.995301172",
      "0.137989050"
    ],
    [
      "0.763716911",
      "0.136400325",
      "0.104659786"
    ],
    [
      "0.163773608",
      "0.680063262",
      "0.104659787"
    ],
    [
      "1.413386184",
      "1.456180192",
      "0.104175013"
    ],
    [
      "1.549853624",
      "0.290718125",
      "0.102548170"
    ],
    [
      "0.240156283",
      "1.477553461",
      "0.102548172"
    ],
    [
      "1.898307357",
      "0.995970411",
      "0.101683605"
    ],
    [
      "0.907786616",
      "1.893570945",
      "0.101683606"
    ]
  ]
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 42
}

Starter answer

{
  "circles": [
    [
      "1.011640683",
      "1.012845727",
      "0.202341652"
    ],
    [
      "0.675745139",
      "0.879205572",
      "0.159162852"
    ],
    [
      "1.436269031",
      "1.718651330",
      "0.159291726"
    ],
    [
      "0.911627312",
      "0.665451380",
      "0.159162852"
    ],
    [
      "1.672342194",
      "1.504724059",
      "0.159291725"
    ],
    [
      "1.223097985",
      "0.721847544",
      "0.157372307"
    ],
    [
      "0.701286010",
      "1.194708618",
      "0.157372308"
    ],
    [
      "0.961299121",
      "1.365113231",
      "0.153504756"
    ],
    [
      "1.367148930",
      "0.997335972",
      "0.153504755"
    ],
    [
      "0.551467681",
      "0.279599743",
      "0.151378899"
    ],
    [
      "0.327121462",
      "0.482900161",
      "0.151378900"
    ],
    [
      "0.614995794",
      "0.575139887",
      "0.150912015"
    ],
    [
      "0.152793160",
      "0.934772364",
      "0.150285886"
    ],
    [
      "1.018273370",
      "0.150482395",
      "0.150285884"
    ],
    [
      "0.425361208",
      "1.060007835",
      "0.149676216"
    ],
    [
      "1.116140686",
      "0.434030011",
      "0.149676215"
    ],
    [
      "1.312026658",
      "0.208399085",
      "0.149122426"
    ],
    [
      "0.181586848",
      "1.232792939",
      "0.149122428"
    ],
    [
      "1.151959260",
      "1.837475395",
      "0.148849805"
    ],
    [
      "1.818507973",
      "1.233455248",
      "0.148849803"
    ],
    [
      "1.689342761",
      "0.497765307",
      "0.147103095"
    ],
    [
      "0.432506783",
      "1.636698181",
      "0.147103098"
    ],
    [
      "1.526702299",
      "0.742746029",
      "0.146950426"
    ],
    [
      "0.681498464",
      "1.792003535",
      "0.146353218"
    ],
    [
      "0.692273117",
      "1.498897862",
      "0.146950427"
    ],
    [
      "1.819449584",
      "0.760802988",
      "0.146353216"
    ],
    [
      "1.245472648",
      "1.270884136",
      "0.145884147"
    ],
    [
      "0.829758446",
      "0.374841731",
      "0.142758459"
    ],
    [
      "0.394578324",
      "0.769197837",
      "0.142758460"
    ],
    [
      "1.402059434",
      "0.483697851",
      "0.140524446"
    ],
    [
      "0.446715167",
      "1.349421785",
      "0.140524448"
    ],
    [
      "1.529812563",
      "1.241594637",
      "0.139960322"
    ],
    [
      "1.188405828",
      "1.550974193",
      "0.139960323"
    ],
    [
      "0.930653570",
      "1.654991638",
      "0.137989051"
    ],
    [
      "1.658635635",
      "0.995301172",
      "0.137989050"
    ],
    [
      "0.763716911",
      "0.136400325",
      "0.104659786"
    ],
    [
      "0.163773608",
      "0.680063262",
      "0.104659787"
    ],
    [
      "1.413386184",
      "1.456180192",
      "0.104175013"
    ],
    [
      "1.549853624",
      "0.290718125",
      "0.102548170"
    ],
    [
      "0.240156283",
      "1.477553461",
      "0.102548172"
    ],
    [
      "1.898307357",
      "0.995970411",
      "0.101683605"
    ],
    [
      "0.907786616",
      "1.893570945",
      "0.101683606"
    ]
  ]
}

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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