P92 · Packing and covering · Classic · Hard

Sum of radii in the unit circle · n = 36

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 = 36

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	[5.530325, 5.530335]	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

123456789101112131415161718192021222324252627282930313233343536

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": [
    [
      "1.596489141",
      "0.437267415",
      "0.179958990"
    ],
    [
      "1.298072620",
      "0.236049770",
      "0.179959000"
    ],
    [
      "0.794045332",
      "1.800918557",
      "0.173024873"
    ],
    [
      "0.334704725",
      "1.491192459",
      "0.173024888"
    ],
    [
      "0.191115336",
      "1.177643845",
      "0.171838339"
    ],
    [
      "1.138556849",
      "1.816488694",
      "0.171838308"
    ],
    [
      "0.955507137",
      "0.170090560",
      "0.168898745"
    ],
    [
      "1.617697739",
      "1.556093063",
      "0.168862230"
    ],
    [
      "1.786062916",
      "0.730121450",
      "0.168898718"
    ],
    [
      "0.252893603",
      "0.635827034",
      "0.168862275"
    ],
    [
      "0.456952545",
      "0.368711870",
      "0.167278412"
    ],
    [
      "1.788831589",
      "1.266777131",
      "0.167278369"
    ],
    [
      "0.980336663",
      "1.523563227",
      "0.161086543"
    ],
    [
      "0.521992853",
      "1.214509218",
      "0.161086557"
    ],
    [
      "1.463859583",
      "1.268040707",
      "0.157696092"
    ],
    [
      "0.577610644",
      "0.670456912",
      "0.157696119"
    ],
    [
      "0.839041332",
      "1.238710525",
      "0.156884261"
    ],
    [
      "0.663951034",
      "1.498378906",
      "0.156299839"
    ],
    [
      "0.705309765",
      "0.956418217",
      "0.155482637"
    ],
    [
      "1.150880525",
      "1.256859593",
      "0.155482623"
    ],
    [
      "1.295588382",
      "1.530705351",
      "0.154246069"
    ],
    [
      "0.397186120",
      "0.924926826",
      "0.154246096"
    ],
    [
      "1.011784897",
      "0.982522504",
      "0.152102213"
    ],
    [
      "1.314654194",
      "0.997633881",
      "0.151143833"
    ],
    [
      "0.884232250",
      "0.707407083",
      "0.151143846"
    ],
    [
      "0.768761769",
      "0.428958473",
      "0.150297790"
    ],
    [
      "1.616084050",
      "1.000294692",
      "0.150297765"
    ],
    [
      "1.468494835",
      "0.740116266",
      "0.148826606"
    ],
    [
      "1.065295051",
      "0.468244917",
      "0.148826617"
    ],
    [
      "1.176734147",
      "0.737893898",
      "0.142942545"
    ],
    [
      "0.508752117",
      "1.728547319",
      "0.121304558"
    ],
    [
      "1.425966348",
      "1.771008427",
      "0.119147387"
    ],
    [
      "0.125530982",
      "0.894145338",
      "0.119147417"
    ],
    [
      "0.669997651",
      "0.180779843",
      "0.116810770"
    ],
    [
      "1.883188463",
      "0.998815559",
      "0.116810742"
    ],
    [
      "1.319037777",
      "0.526849984",
      "0.111595971"
    ]
  ]
}

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

Instance parameters

{
  "n": 36
}

Starter answer

{
  "circles": [
    [
      "1.596489141",
      "0.437267415",
      "0.179958990"
    ],
    [
      "1.298072620",
      "0.236049770",
      "0.179959000"
    ],
    [
      "0.794045332",
      "1.800918557",
      "0.173024873"
    ],
    [
      "0.334704725",
      "1.491192459",
      "0.173024888"
    ],
    [
      "0.191115336",
      "1.177643845",
      "0.171838339"
    ],
    [
      "1.138556849",
      "1.816488694",
      "0.171838308"
    ],
    [
      "0.955507137",
      "0.170090560",
      "0.168898745"
    ],
    [
      "1.617697739",
      "1.556093063",
      "0.168862230"
    ],
    [
      "1.786062916",
      "0.730121450",
      "0.168898718"
    ],
    [
      "0.252893603",
      "0.635827034",
      "0.168862275"
    ],
    [
      "0.456952545",
      "0.368711870",
      "0.167278412"
    ],
    [
      "1.788831589",
      "1.266777131",
      "0.167278369"
    ],
    [
      "0.980336663",
      "1.523563227",
      "0.161086543"
    ],
    [
      "0.521992853",
      "1.214509218",
      "0.161086557"
    ],
    [
      "1.463859583",
      "1.268040707",
      "0.157696092"
    ],
    [
      "0.577610644",
      "0.670456912",
      "0.157696119"
    ],
    [
      "0.839041332",
      "1.238710525",
      "0.156884261"
    ],
    [
      "0.663951034",
      "1.498378906",
      "0.156299839"
    ],
    [
      "0.705309765",
      "0.956418217",
      "0.155482637"
    ],
    [
      "1.150880525",
      "1.256859593",
      "0.155482623"
    ],
    [
      "1.295588382",
      "1.530705351",
      "0.154246069"
    ],
    [
      "0.397186120",
      "0.924926826",
      "0.154246096"
    ],
    [
      "1.011784897",
      "0.982522504",
      "0.152102213"
    ],
    [
      "1.314654194",
      "0.997633881",
      "0.151143833"
    ],
    [
      "0.884232250",
      "0.707407083",
      "0.151143846"
    ],
    [
      "0.768761769",
      "0.428958473",
      "0.150297790"
    ],
    [
      "1.616084050",
      "1.000294692",
      "0.150297765"
    ],
    [
      "1.468494835",
      "0.740116266",
      "0.148826606"
    ],
    [
      "1.065295051",
      "0.468244917",
      "0.148826617"
    ],
    [
      "1.176734147",
      "0.737893898",
      "0.142942545"
    ],
    [
      "0.508752117",
      "1.728547319",
      "0.121304558"
    ],
    [
      "1.425966348",
      "1.771008427",
      "0.119147387"
    ],
    [
      "0.125530982",
      "0.894145338",
      "0.119147417"
    ],
    [
      "0.669997651",
      "0.180779843",
      "0.116810770"
    ],
    [
      "1.883188463",
      "0.998815559",
      "0.116810742"
    ],
    [
      "1.319037777",
      "0.526849984",
      "0.111595971"
    ]
  ]
}

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