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P92 · Packing and covering · Classic · Hard

Sum of radii in the unit circle · n = 49

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 = 49
ObjectiveMaximize the sum of the radii

Record comparison

the sum of the radii · Higher is better
RecordValue / intervalAuthor / holderSource
External best known[6.473305, 6.473315]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 = 49
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": [
    [
      "0.364597422",
      "1.535418905",
      "0.169090955"
    ],
    [
      "0.172651546",
      "1.076839729",
      "0.169090974"
    ],
    [
      "1.677498670",
      "1.489013551",
      "0.164453052"
    ],
    [
      "1.127156050",
      "0.174185281",
      "0.164453105"
    ],
    [
      "0.815895610",
      "0.166992489",
      "0.146890429"
    ],
    [
      "1.464166958",
      "1.715782837",
      "0.146890373"
    ],
    [
      "1.827946507",
      "1.220730090",
      "0.143135254"
    ],
    [
      "1.423853876",
      "0.255308772",
      "0.143135288"
    ],
    [
      "1.374682884",
      "1.440937356",
      "0.142155362"
    ],
    [
      "0.948873624",
      "0.423632975",
      "0.142155399"
    ],
    [
      "1.544203816",
      "1.214540281",
      "0.140674943"
    ],
    [
      "0.243370579",
      "0.592450394",
      "0.140590457"
    ],
    [
      "1.229120658",
      "0.461772417",
      "0.140674969"
    ],
    [
      "0.759277650",
      "1.825007628",
      "0.140590414"
    ],
    [
      "0.410518140",
      "0.370286540",
      "0.137429478"
    ],
    [
      "1.034843348",
      "1.861866540",
      "0.137429427"
    ],
    [
      "1.058098209",
      "0.680554563",
      "0.137019657"
    ],
    [
      "1.268327427",
      "1.182814905",
      "0.137019640"
    ],
    [
      "0.787220235",
      "0.649535826",
      "0.135628538"
    ],
    [
      "1.100309786",
      "1.397540517",
      "0.135628513"
    ],
    [
      "0.517784429",
      "0.621046160",
      "0.135309304"
    ],
    [
      "0.931502810",
      "1.609463824",
      "0.135309271"
    ],
    [
      "1.162586203",
      "0.931947023",
      "0.135222715"
    ],
    [
      "0.998791404",
      "1.146764392",
      "0.134916578"
    ],
    [
      "0.894606666",
      "0.897855820",
      "0.134916586"
    ],
    [
      "0.665791504",
      "1.567074282",
      "0.133762041"
    ],
    [
      "0.627718353",
      "0.866663846",
      "0.133788290"
    ],
    [
      "1.326209965",
      "0.718457261",
      "0.133757978"
    ],
    [
      "0.361492760",
      "0.840071995",
      "0.133762065"
    ],
    [
      "0.833697286",
      "1.358769651",
      "0.133788273"
    ],
    [
      "1.429499981",
      "0.965228319",
      "0.133757970"
    ],
    [
      "0.733221223",
      "1.111664731",
      "0.132963072"
    ],
    [
      "1.195109708",
      "1.646647466",
      "0.130907219"
    ],
    [
      "0.572524608",
      "1.320105644",
      "0.130230803"
    ],
    [
      "0.676307508",
      "0.407173084",
      "0.130907259"
    ],
    [
      "0.471958275",
      "1.079841903",
      "0.130230810"
    ],
    [
      "1.587347418",
      "0.754156349",
      "0.129808321"
    ],
    [
      "1.744533837",
      "0.548329658",
      "0.129174683"
    ],
    [
      "1.844284536",
      "0.786644973",
      "0.129174675"
    ],
    [
      "1.490716811",
      "0.516639072",
      "0.126613071"
    ],
    [
      "1.688719477",
      "0.989688999",
      "0.126613057"
    ],
    [
      "0.125480701",
      "0.807336282",
      "0.104509457"
    ],
    [
      "0.523467325",
      "1.758168824",
      "0.104509432"
    ],
    [
      "0.170838498",
      "1.347059265",
      "0.101134643"
    ],
    [
      "1.269160166",
      "1.862070200",
      "0.096887479"
    ],
    [
      "0.574826330",
      "0.203231838",
      "0.096887518"
    ],
    [
      "0.352197290",
      "1.271148552",
      "0.095470157"
    ],
    [
      "1.907566478",
      "0.999025191",
      "0.092432998"
    ],
    [
      "1.637662264",
      "0.354194555",
      "0.092433013"
    ]
  ]
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 49
}

Starter answer

{
  "circles": [
    [
      "0.364597422",
      "1.535418905",
      "0.169090955"
    ],
    [
      "0.172651546",
      "1.076839729",
      "0.169090974"
    ],
    [
      "1.677498670",
      "1.489013551",
      "0.164453052"
    ],
    [
      "1.127156050",
      "0.174185281",
      "0.164453105"
    ],
    [
      "0.815895610",
      "0.166992489",
      "0.146890429"
    ],
    [
      "1.464166958",
      "1.715782837",
      "0.146890373"
    ],
    [
      "1.827946507",
      "1.220730090",
      "0.143135254"
    ],
    [
      "1.423853876",
      "0.255308772",
      "0.143135288"
    ],
    [
      "1.374682884",
      "1.440937356",
      "0.142155362"
    ],
    [
      "0.948873624",
      "0.423632975",
      "0.142155399"
    ],
    [
      "1.544203816",
      "1.214540281",
      "0.140674943"
    ],
    [
      "0.243370579",
      "0.592450394",
      "0.140590457"
    ],
    [
      "1.229120658",
      "0.461772417",
      "0.140674969"
    ],
    [
      "0.759277650",
      "1.825007628",
      "0.140590414"
    ],
    [
      "0.410518140",
      "0.370286540",
      "0.137429478"
    ],
    [
      "1.034843348",
      "1.861866540",
      "0.137429427"
    ],
    [
      "1.058098209",
      "0.680554563",
      "0.137019657"
    ],
    [
      "1.268327427",
      "1.182814905",
      "0.137019640"
    ],
    [
      "0.787220235",
      "0.649535826",
      "0.135628538"
    ],
    [
      "1.100309786",
      "1.397540517",
      "0.135628513"
    ],
    [
      "0.517784429",
      "0.621046160",
      "0.135309304"
    ],
    [
      "0.931502810",
      "1.609463824",
      "0.135309271"
    ],
    [
      "1.162586203",
      "0.931947023",
      "0.135222715"
    ],
    [
      "0.998791404",
      "1.146764392",
      "0.134916578"
    ],
    [
      "0.894606666",
      "0.897855820",
      "0.134916586"
    ],
    [
      "0.665791504",
      "1.567074282",
      "0.133762041"
    ],
    [
      "0.627718353",
      "0.866663846",
      "0.133788290"
    ],
    [
      "1.326209965",
      "0.718457261",
      "0.133757978"
    ],
    [
      "0.361492760",
      "0.840071995",
      "0.133762065"
    ],
    [
      "0.833697286",
      "1.358769651",
      "0.133788273"
    ],
    [
      "1.429499981",
      "0.965228319",
      "0.133757970"
    ],
    [
      "0.733221223",
      "1.111664731",
      "0.132963072"
    ],
    [
      "1.195109708",
      "1.646647466",
      "0.130907219"
    ],
    [
      "0.572524608",
      "1.320105644",
      "0.130230803"
    ],
    [
      "0.676307508",
      "0.407173084",
      "0.130907259"
    ],
    [
      "0.471958275",
      "1.079841903",
      "0.130230810"
    ],
    [
      "1.587347418",
      "0.754156349",
      "0.129808321"
    ],
    [
      "1.744533837",
      "0.548329658",
      "0.129174683"
    ],
    [
      "1.844284536",
      "0.786644973",
      "0.129174675"
    ],
    [
      "1.490716811",
      "0.516639072",
      "0.126613071"
    ],
    [
      "1.688719477",
      "0.989688999",
      "0.126613057"
    ],
    [
      "0.125480701",
      "0.807336282",
      "0.104509457"
    ],
    [
      "0.523467325",
      "1.758168824",
      "0.104509432"
    ],
    [
      "0.170838498",
      "1.347059265",
      "0.101134643"
    ],
    [
      "1.269160166",
      "1.862070200",
      "0.096887479"
    ],
    [
      "0.574826330",
      "0.203231838",
      "0.096887518"
    ],
    [
      "0.352197290",
      "1.271148552",
      "0.095470157"
    ],
    [
      "1.907566478",
      "0.999025191",
      "0.092432998"
    ],
    [
      "1.637662264",
      "0.354194555",
      "0.092433013"
    ]
  ]
}

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