P92 · Packing and covering · Classic · Hard

Sum of radii in the unit circle · n = 35

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 = 35

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	[5.446245, 5.446255]	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

1234567891011121314151617181920212223242526272829303132333435

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.

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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": [
    [
      "1.183349882",
      "0.485513991",
      "0.194205246"
    ],
    [
      "1.558094557",
      "0.410228350",
      "0.188027011"
    ],
    [
      "1.769029911",
      "0.709111399",
      "0.177793710"
    ],
    [
      "0.317019665",
      "1.459925218",
      "0.176596487"
    ],
    [
      "0.186922739",
      "1.131889584",
      "0.176295262"
    ],
    [
      "0.771396305",
      "1.792369470",
      "0.175312770"
    ],
    [
      "0.288711020",
      "0.580459649",
      "0.174199709"
    ],
    [
      "1.119595728",
      "1.817505532",
      "0.173792742"
    ],
    [
      "0.521938899",
      "0.324264458",
      "0.172255510"
    ],
    [
      "1.609001937",
      "1.561549010",
      "0.171615638"
    ],
    [
      "1.784258855",
      "1.266241957",
      "0.171780987"
    ],
    [
      "0.521171305",
      "1.188832795",
      "0.162769087"
    ],
    [
      "0.967624612",
      "1.518716438",
      "0.161423818"
    ],
    [
      "1.034539198",
      "0.162316859",
      "0.161605103"
    ],
    [
      "0.617888185",
      "0.643333651",
      "0.160928245"
    ],
    [
      "1.433857464",
      "0.733464203",
      "0.158262278"
    ],
    [
      "1.454497710",
      "1.270380753",
      "0.158006129"
    ],
    [
      "0.650601423",
      "1.482019858",
      "0.157716184"
    ],
    [
      "0.839028295",
      "0.414373504",
      "0.157388729"
    ],
    [
      "0.413774073",
      "0.887391003",
      "0.157232973"
    ],
    [
      "0.721244195",
      "0.941912810",
      "0.155033744"
    ],
    [
      "0.836708633",
      "1.230139488",
      "0.155460465"
    ],
    [
      "1.283705148",
      "1.532639628",
      "0.154963223"
    ],
    [
      "1.602646325",
      "0.995979426",
      "0.153833734"
    ],
    [
      "1.143798626",
      "1.258374753",
      "0.152924834"
    ],
    [
      "0.922265359",
      "0.711247555",
      "0.150933522"
    ],
    [
      "1.304164587",
      "1.007449439",
      "0.144868306"
    ],
    [
      "1.016034565",
      "0.990617299",
      "0.143752951"
    ],
    [
      "1.324897109",
      "0.193384645",
      "0.130410175"
    ],
    [
      "0.484500238",
      "1.708910663",
      "0.123475994"
    ],
    [
      "0.138257468",
      "0.836663895",
      "0.122914556"
    ],
    [
      "1.878143448",
      "0.988102843",
      "0.121775963"
    ],
    [
      "1.411375741",
      "1.777045212",
      "0.120779173"
    ],
    [
      "0.755403754",
      "0.152292602",
      "0.117710276"
    ],
    [
      "1.171344085",
      "0.789686550",
      "0.110204156"
    ]
  ]
}

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

Instance parameters

{
  "n": 35
}

Starter answer

{
  "circles": [
    [
      "1.183349882",
      "0.485513991",
      "0.194205246"
    ],
    [
      "1.558094557",
      "0.410228350",
      "0.188027011"
    ],
    [
      "1.769029911",
      "0.709111399",
      "0.177793710"
    ],
    [
      "0.317019665",
      "1.459925218",
      "0.176596487"
    ],
    [
      "0.186922739",
      "1.131889584",
      "0.176295262"
    ],
    [
      "0.771396305",
      "1.792369470",
      "0.175312770"
    ],
    [
      "0.288711020",
      "0.580459649",
      "0.174199709"
    ],
    [
      "1.119595728",
      "1.817505532",
      "0.173792742"
    ],
    [
      "0.521938899",
      "0.324264458",
      "0.172255510"
    ],
    [
      "1.609001937",
      "1.561549010",
      "0.171615638"
    ],
    [
      "1.784258855",
      "1.266241957",
      "0.171780987"
    ],
    [
      "0.521171305",
      "1.188832795",
      "0.162769087"
    ],
    [
      "0.967624612",
      "1.518716438",
      "0.161423818"
    ],
    [
      "1.034539198",
      "0.162316859",
      "0.161605103"
    ],
    [
      "0.617888185",
      "0.643333651",
      "0.160928245"
    ],
    [
      "1.433857464",
      "0.733464203",
      "0.158262278"
    ],
    [
      "1.454497710",
      "1.270380753",
      "0.158006129"
    ],
    [
      "0.650601423",
      "1.482019858",
      "0.157716184"
    ],
    [
      "0.839028295",
      "0.414373504",
      "0.157388729"
    ],
    [
      "0.413774073",
      "0.887391003",
      "0.157232973"
    ],
    [
      "0.721244195",
      "0.941912810",
      "0.155033744"
    ],
    [
      "0.836708633",
      "1.230139488",
      "0.155460465"
    ],
    [
      "1.283705148",
      "1.532639628",
      "0.154963223"
    ],
    [
      "1.602646325",
      "0.995979426",
      "0.153833734"
    ],
    [
      "1.143798626",
      "1.258374753",
      "0.152924834"
    ],
    [
      "0.922265359",
      "0.711247555",
      "0.150933522"
    ],
    [
      "1.304164587",
      "1.007449439",
      "0.144868306"
    ],
    [
      "1.016034565",
      "0.990617299",
      "0.143752951"
    ],
    [
      "1.324897109",
      "0.193384645",
      "0.130410175"
    ],
    [
      "0.484500238",
      "1.708910663",
      "0.123475994"
    ],
    [
      "0.138257468",
      "0.836663895",
      "0.122914556"
    ],
    [
      "1.878143448",
      "0.988102843",
      "0.121775963"
    ],
    [
      "1.411375741",
      "1.777045212",
      "0.120779173"
    ],
    [
      "0.755403754",
      "0.152292602",
      "0.117710276"
    ],
    [
      "1.171344085",
      "0.789686550",
      "0.110204156"
    ]
  ]
}

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