P92 · Packing and covering · Classic · Hard

Sum of radii in the unit circle · n = 37

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 = 37

ObjectiveMaximize the sum of the radii

Record comparison

the sum of the radii · Higher is better

Record	Value / interval	Author / holder	Source
External best known	[5.621715, 5.621725]	See source attribution	Erich Friedman ↗
Site record	No 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 better $S = Σ r$
ObjectiveMaximise the sum of the radii over every legal arrangement $\max_{S} Σ r$

Open the full editor ↗

Layout shown: starter layout

2y0

12345678910111213141516171819202122232425262728293031323334353637

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

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": [
    [
      "1.617246323",
      "1.561313239",
      "0.165694554"
    ],
    [
      "0.382753699",
      "0.438686788",
      "0.165694589"
    ],
    [
      "1.787069046",
      "1.276745171",
      "0.165694556"
    ],
    [
      "0.212930955",
      "0.723254922",
      "0.165694588"
    ],
    [
      "1.153866278",
      "1.819994353",
      "0.165694558"
    ],
    [
      "0.846133848",
      "0.180005652",
      "0.165694586"
    ],
    [
      "0.822511729",
      "1.815207627",
      "0.165694563"
    ],
    [
      "1.177488443",
      "0.184792429",
      "0.165694581"
    ],
    [
      "0.366797239",
      "1.543249305",
      "0.165694573"
    ],
    [
      "0.205265367",
      "1.253894459",
      "0.165694580"
    ],
    [
      "1.794734705",
      "0.746105712",
      "0.165694563"
    ],
    [
      "1.633202901",
      "0.456750853",
      "0.165694570"
    ],
    [
      "0.992194475",
      "1.540328054",
      "0.157339512"
    ],
    [
      "1.007805642",
      "0.459672006",
      "0.157339526"
    ],
    [
      "0.535965024",
      "0.723076191",
      "0.157339529"
    ],
    [
      "1.464035030",
      "1.276923882",
      "0.157339508"
    ],
    [
      "0.528159462",
      "1.263404239",
      "0.157339522"
    ],
    [
      "1.471840635",
      "0.736595882",
      "0.157339516"
    ],
    [
      "1.000000045",
      "1.000000043",
      "0.155324629"
    ],
    [
      "0.840805619",
      "1.266758540",
      "0.155324627"
    ],
    [
      "1.159194482",
      "0.733241549",
      "0.155324630"
    ],
    [
      "1.151422462",
      "1.271245706",
      "0.155324623"
    ],
    [
      "0.848577625",
      "0.728754367",
      "0.155324635"
    ],
    [
      "1.310616890",
      "1.004487221",
      "0.155324624"
    ],
    [
      "0.689383191",
      "0.995512873",
      "0.155324633"
    ],
    [
      "0.683621943",
      "1.530147694",
      "0.151400908"
    ],
    [
      "1.316378193",
      "0.469852406",
      "0.151400913"
    ],
    [
      "1.617310417",
      "1.008917730",
      "0.151400902"
    ],
    [
      "1.300932305",
      "1.539065335",
      "0.151400899"
    ],
    [
      "0.382689637",
      "0.991082387",
      "0.151400919"
    ],
    [
      "0.699067775",
      "0.460934703",
      "0.151400922"
    ],
    [
      "1.431084752",
      "1.772209711",
      "0.115612131"
    ],
    [
      "1.884295598",
      "1.012774618",
      "0.115612135"
    ],
    [
      "0.115704419",
      "0.987225534",
      "0.115612154"
    ],
    [
      "0.568915318",
      "0.227790279",
      "0.115612157"
    ],
    [
      "0.546789220",
      "1.759435232",
      "0.115612141"
    ],
    [
      "1.453210965",
      "0.240564885",
      "0.115612147"
    ]
  ]
}

Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 37
}

Starter answer

{
  "circles": [
    [
      "1.617246323",
      "1.561313239",
      "0.165694554"
    ],
    [
      "0.382753699",
      "0.438686788",
      "0.165694589"
    ],
    [
      "1.787069046",
      "1.276745171",
      "0.165694556"
    ],
    [
      "0.212930955",
      "0.723254922",
      "0.165694588"
    ],
    [
      "1.153866278",
      "1.819994353",
      "0.165694558"
    ],
    [
      "0.846133848",
      "0.180005652",
      "0.165694586"
    ],
    [
      "0.822511729",
      "1.815207627",
      "0.165694563"
    ],
    [
      "1.177488443",
      "0.184792429",
      "0.165694581"
    ],
    [
      "0.366797239",
      "1.543249305",
      "0.165694573"
    ],
    [
      "0.205265367",
      "1.253894459",
      "0.165694580"
    ],
    [
      "1.794734705",
      "0.746105712",
      "0.165694563"
    ],
    [
      "1.633202901",
      "0.456750853",
      "0.165694570"
    ],
    [
      "0.992194475",
      "1.540328054",
      "0.157339512"
    ],
    [
      "1.007805642",
      "0.459672006",
      "0.157339526"
    ],
    [
      "0.535965024",
      "0.723076191",
      "0.157339529"
    ],
    [
      "1.464035030",
      "1.276923882",
      "0.157339508"
    ],
    [
      "0.528159462",
      "1.263404239",
      "0.157339522"
    ],
    [
      "1.471840635",
      "0.736595882",
      "0.157339516"
    ],
    [
      "1.000000045",
      "1.000000043",
      "0.155324629"
    ],
    [
      "0.840805619",
      "1.266758540",
      "0.155324627"
    ],
    [
      "1.159194482",
      "0.733241549",
      "0.155324630"
    ],
    [
      "1.151422462",
      "1.271245706",
      "0.155324623"
    ],
    [
      "0.848577625",
      "0.728754367",
      "0.155324635"
    ],
    [
      "1.310616890",
      "1.004487221",
      "0.155324624"
    ],
    [
      "0.689383191",
      "0.995512873",
      "0.155324633"
    ],
    [
      "0.683621943",
      "1.530147694",
      "0.151400908"
    ],
    [
      "1.316378193",
      "0.469852406",
      "0.151400913"
    ],
    [
      "1.617310417",
      "1.008917730",
      "0.151400902"
    ],
    [
      "1.300932305",
      "1.539065335",
      "0.151400899"
    ],
    [
      "0.382689637",
      "0.991082387",
      "0.151400919"
    ],
    [
      "0.699067775",
      "0.460934703",
      "0.151400922"
    ],
    [
      "1.431084752",
      "1.772209711",
      "0.115612131"
    ],
    [
      "1.884295598",
      "1.012774618",
      "0.115612135"
    ],
    [
      "0.115704419",
      "0.987225534",
      "0.115612154"
    ],
    [
      "0.568915318",
      "0.227790279",
      "0.115612157"
    ],
    [
      "0.546789220",
      "1.759435232",
      "0.115612141"
    ],
    [
      "1.453210965",
      "0.240564885",
      "0.115612147"
    ]
  ]
}

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 replies1h ago