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P92 · Packing and covering · Classic · Hard

Sum of radii in the unit circle · n = 50

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 = 50
ObjectiveMaximize the sum of the radii

Record comparison

the sum of the radii · Higher is better
RecordValue / intervalAuthor / holderSource
External best known[6.534855, 6.534865]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 = 50
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": [
    [
      "0.655503953",
      "0.881279390",
      "0.181224401"
    ],
    [
      "1.395095977",
      "0.520690051",
      "0.168360364"
    ],
    [
      "1.154047332",
      "1.827299484",
      "0.158480530"
    ],
    [
      "1.713027896",
      "0.537310531",
      "0.150005691"
    ],
    [
      "0.981937567",
      "0.845289069",
      "0.147187240"
    ],
    [
      "1.271381596",
      "0.808187187",
      "0.144625020"
    ],
    [
      "0.879050457",
      "1.118385203",
      "0.144647054"
    ],
    [
      "1.558073072",
      "0.785908191",
      "0.142930810"
    ],
    [
      "0.170778635",
      "1.218704583",
      "0.142422151"
    ],
    [
      "0.822113648",
      "0.605174439",
      "0.141254643"
    ],
    [
      "1.160529047",
      "1.071167820",
      "0.140764364"
    ],
    [
      "0.856323190",
      "1.847954641",
      "0.139959245"
    ],
    [
      "1.092252545",
      "0.144215369",
      "0.139257374"
    ],
    [
      "1.438733649",
      "1.740750399",
      "0.139071217"
    ],
    [
      "0.542358487",
      "0.581142735",
      "0.139530814"
    ],
    [
      "0.283078502",
      "1.476770998",
      "0.139019733"
    ],
    [
      "1.242983173",
      "1.543922612",
      "0.138524579"
    ],
    [
      "0.970362070",
      "1.593787094",
      "0.138619303"
    ],
    [
      "1.063620943",
      "1.332954874",
      "0.138383700"
    ],
    [
      "1.359772135",
      "0.216514223",
      "0.137859667"
    ],
    [
      "1.838175572",
      "0.795986071",
      "0.137352926"
    ],
    [
      "0.138692357",
      "0.941429562",
      "0.136703207"
    ],
    [
      "1.434955527",
      "1.036081082",
      "0.135896026"
    ],
    [
      "1.098491754",
      "0.587545185",
      "0.135685148"
    ],
    [
      "1.332299633",
      "1.286074986",
      "0.134354199"
    ],
    [
      "0.387706687",
      "1.047466340",
      "0.133947738"
    ],
    [
      "0.274345275",
      "0.527130832",
      "0.133870661"
    ],
    [
      "0.796342097",
      "1.384233279",
      "0.133769669"
    ],
    [
      "0.356792936",
      "0.781645306",
      "0.133664820"
    ],
    [
      "0.611694772",
      "1.192122570",
      "0.132690757"
    ],
    [
      "0.448281204",
      "0.327998696",
      "0.130529251"
    ],
    [
      "0.459098474",
      "1.680823901",
      "0.130462395"
    ],
    [
      "0.823504991",
      "0.147591683",
      "0.129511386"
    ],
    [
      "1.592705160",
      "1.248559688",
      "0.128739769"
    ],
    [
      "0.951215993",
      "0.370040022",
      "0.126990784"
    ],
    [
      "1.748152122",
      "1.450945266",
      "0.126453647"
    ],
    [
      "1.846000918",
      "1.217769443",
      "0.126420533"
    ],
    [
      "1.695301568",
      "1.016264717",
      "0.125203090"
    ],
    [
      "0.708968198",
      "1.627779790",
      "0.124975574"
    ],
    [
      "0.544160317",
      "1.440377729",
      "0.124586381"
    ],
    [
      "1.499253332",
      "1.484076905",
      "0.124640583"
    ],
    [
      "0.699830876",
      "0.369342261",
      "0.124395300"
    ],
    [
      "1.580307111",
      "0.319123181",
      "0.105377405"
    ],
    [
      "1.176911140",
      "0.366635980",
      "0.098730031"
    ],
    [
      "0.400564206",
      "1.275276306",
      "0.094224776"
    ],
    [
      "0.140783507",
      "0.711192123",
      "0.093543728"
    ],
    [
      "0.625656449",
      "1.827686068",
      "0.091596278"
    ],
    [
      "1.645292615",
      "1.641503903",
      "0.090093511"
    ],
    [
      "1.910188895",
      "1.011358735",
      "0.089740231"
    ],
    [
      "0.607466948",
      "0.177439044",
      "0.088578734"
    ]
  ]
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 50
}

Starter answer

{
  "circles": [
    [
      "0.655503953",
      "0.881279390",
      "0.181224401"
    ],
    [
      "1.395095977",
      "0.520690051",
      "0.168360364"
    ],
    [
      "1.154047332",
      "1.827299484",
      "0.158480530"
    ],
    [
      "1.713027896",
      "0.537310531",
      "0.150005691"
    ],
    [
      "0.981937567",
      "0.845289069",
      "0.147187240"
    ],
    [
      "1.271381596",
      "0.808187187",
      "0.144625020"
    ],
    [
      "0.879050457",
      "1.118385203",
      "0.144647054"
    ],
    [
      "1.558073072",
      "0.785908191",
      "0.142930810"
    ],
    [
      "0.170778635",
      "1.218704583",
      "0.142422151"
    ],
    [
      "0.822113648",
      "0.605174439",
      "0.141254643"
    ],
    [
      "1.160529047",
      "1.071167820",
      "0.140764364"
    ],
    [
      "0.856323190",
      "1.847954641",
      "0.139959245"
    ],
    [
      "1.092252545",
      "0.144215369",
      "0.139257374"
    ],
    [
      "1.438733649",
      "1.740750399",
      "0.139071217"
    ],
    [
      "0.542358487",
      "0.581142735",
      "0.139530814"
    ],
    [
      "0.283078502",
      "1.476770998",
      "0.139019733"
    ],
    [
      "1.242983173",
      "1.543922612",
      "0.138524579"
    ],
    [
      "0.970362070",
      "1.593787094",
      "0.138619303"
    ],
    [
      "1.063620943",
      "1.332954874",
      "0.138383700"
    ],
    [
      "1.359772135",
      "0.216514223",
      "0.137859667"
    ],
    [
      "1.838175572",
      "0.795986071",
      "0.137352926"
    ],
    [
      "0.138692357",
      "0.941429562",
      "0.136703207"
    ],
    [
      "1.434955527",
      "1.036081082",
      "0.135896026"
    ],
    [
      "1.098491754",
      "0.587545185",
      "0.135685148"
    ],
    [
      "1.332299633",
      "1.286074986",
      "0.134354199"
    ],
    [
      "0.387706687",
      "1.047466340",
      "0.133947738"
    ],
    [
      "0.274345275",
      "0.527130832",
      "0.133870661"
    ],
    [
      "0.796342097",
      "1.384233279",
      "0.133769669"
    ],
    [
      "0.356792936",
      "0.781645306",
      "0.133664820"
    ],
    [
      "0.611694772",
      "1.192122570",
      "0.132690757"
    ],
    [
      "0.448281204",
      "0.327998696",
      "0.130529251"
    ],
    [
      "0.459098474",
      "1.680823901",
      "0.130462395"
    ],
    [
      "0.823504991",
      "0.147591683",
      "0.129511386"
    ],
    [
      "1.592705160",
      "1.248559688",
      "0.128739769"
    ],
    [
      "0.951215993",
      "0.370040022",
      "0.126990784"
    ],
    [
      "1.748152122",
      "1.450945266",
      "0.126453647"
    ],
    [
      "1.846000918",
      "1.217769443",
      "0.126420533"
    ],
    [
      "1.695301568",
      "1.016264717",
      "0.125203090"
    ],
    [
      "0.708968198",
      "1.627779790",
      "0.124975574"
    ],
    [
      "0.544160317",
      "1.440377729",
      "0.124586381"
    ],
    [
      "1.499253332",
      "1.484076905",
      "0.124640583"
    ],
    [
      "0.699830876",
      "0.369342261",
      "0.124395300"
    ],
    [
      "1.580307111",
      "0.319123181",
      "0.105377405"
    ],
    [
      "1.176911140",
      "0.366635980",
      "0.098730031"
    ],
    [
      "0.400564206",
      "1.275276306",
      "0.094224776"
    ],
    [
      "0.140783507",
      "0.711192123",
      "0.093543728"
    ],
    [
      "0.625656449",
      "1.827686068",
      "0.091596278"
    ],
    [
      "1.645292615",
      "1.641503903",
      "0.090093511"
    ],
    [
      "1.910188895",
      "1.011358735",
      "0.089740231"
    ],
    [
      "0.607466948",
      "0.177439044",
      "0.088578734"
    ]
  ]
}

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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