P92 · Packing and covering · Classic · Hard

Sum of radii in the unit circle · n = 39

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 = 39

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	[5.763365, 5.763375]	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

123456789101112131415161718192021222324252627282930313233343536373839

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": [
    [
      "1.332203895",
      "1.051695385",
      "0.193441828"
    ],
    [
      "0.979540466",
      "1.000116757",
      "0.162973447"
    ],
    [
      "1.693499471",
      "1.469485008",
      "0.162528992"
    ],
    [
      "1.168895297",
      "0.735789731",
      "0.162178868"
    ],
    [
      "1.205237813",
      "0.186839014",
      "0.161338358"
    ],
    [
      "1.464830008",
      "1.698414551",
      "0.161042419"
    ],
    [
      "1.085638703",
      "1.305916306",
      "0.160708803"
    ],
    [
      "0.969298163",
      "1.839203506",
      "0.160235075"
    ],
    [
      "0.883986983",
      "0.168490174",
      "0.160436058"
    ],
    [
      "1.642892224",
      "0.458375979",
      "0.159365125"
    ],
    [
      "0.657966457",
      "1.768208882",
      "0.159088690"
    ],
    [
      "1.489605382",
      "0.736876245",
      "0.158533056"
    ],
    [
      "0.422647165",
      "0.387257695",
      "0.158103670"
    ],
    [
      "0.277248339",
      "1.433631698",
      "0.157143895"
    ],
    [
      "0.236895430",
      "0.641798987",
      "0.157007383"
    ],
    [
      "0.168164326",
      "1.139520661",
      "0.156544841"
    ],
    [
      "1.028149568",
      "0.450306795",
      "0.156113241"
    ],
    [
      "0.852071290",
      "0.707857757",
      "0.155874030"
    ],
    [
      "0.775757174",
      "1.244473381",
      "0.155205417"
    ],
    [
      "0.872813023",
      "1.539100923",
      "0.154996494"
    ],
    [
      "0.665670910",
      "0.955469081",
      "0.154055751"
    ],
    [
      "1.801574629",
      "0.728083375",
      "0.153560080"
    ],
    [
      "1.386935537",
      "1.393984318",
      "0.153195253"
    ],
    [
      "0.545238580",
      "0.673102977",
      "0.152920734"
    ],
    [
      "0.472409650",
      "1.193332798",
      "0.152422734"
    ],
    [
      "1.333679883",
      "0.470543231",
      "0.150086508"
    ],
    [
      "1.169695772",
      "1.603862642",
      "0.148867697"
    ],
    [
      "0.724992789",
      "0.432296080",
      "0.147578079"
    ],
    [
      "0.578494750",
      "1.472969092",
      "0.146660038"
    ],
    [
      "0.369010603",
      "0.913930297",
      "0.145498597"
    ],
    [
      "1.834698740",
      "1.200048846",
      "0.141663511"
    ],
    [
      "1.657341553",
      "0.982574749",
      "0.138961780"
    ],
    [
      "1.471752739",
      "0.247725452",
      "0.112042994"
    ],
    [
      "1.241030215",
      "1.854725180",
      "0.111939924"
    ],
    [
      "0.614433125",
      "0.198754422",
      "0.110811442"
    ],
    [
      "0.409236394",
      "1.665783207",
      "0.109905107"
    ],
    [
      "1.584168442",
      "1.220201450",
      "0.109676015"
    ],
    [
      "0.117072852",
      "0.879105777",
      "0.108834604"
    ],
    [
      "1.897477354",
      "0.964785309",
      "0.101832044"
    ]
  ]
}

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

Instance parameters

{
  "n": 39
}

Starter answer

{
  "circles": [
    [
      "1.332203895",
      "1.051695385",
      "0.193441828"
    ],
    [
      "0.979540466",
      "1.000116757",
      "0.162973447"
    ],
    [
      "1.693499471",
      "1.469485008",
      "0.162528992"
    ],
    [
      "1.168895297",
      "0.735789731",
      "0.162178868"
    ],
    [
      "1.205237813",
      "0.186839014",
      "0.161338358"
    ],
    [
      "1.464830008",
      "1.698414551",
      "0.161042419"
    ],
    [
      "1.085638703",
      "1.305916306",
      "0.160708803"
    ],
    [
      "0.969298163",
      "1.839203506",
      "0.160235075"
    ],
    [
      "0.883986983",
      "0.168490174",
      "0.160436058"
    ],
    [
      "1.642892224",
      "0.458375979",
      "0.159365125"
    ],
    [
      "0.657966457",
      "1.768208882",
      "0.159088690"
    ],
    [
      "1.489605382",
      "0.736876245",
      "0.158533056"
    ],
    [
      "0.422647165",
      "0.387257695",
      "0.158103670"
    ],
    [
      "0.277248339",
      "1.433631698",
      "0.157143895"
    ],
    [
      "0.236895430",
      "0.641798987",
      "0.157007383"
    ],
    [
      "0.168164326",
      "1.139520661",
      "0.156544841"
    ],
    [
      "1.028149568",
      "0.450306795",
      "0.156113241"
    ],
    [
      "0.852071290",
      "0.707857757",
      "0.155874030"
    ],
    [
      "0.775757174",
      "1.244473381",
      "0.155205417"
    ],
    [
      "0.872813023",
      "1.539100923",
      "0.154996494"
    ],
    [
      "0.665670910",
      "0.955469081",
      "0.154055751"
    ],
    [
      "1.801574629",
      "0.728083375",
      "0.153560080"
    ],
    [
      "1.386935537",
      "1.393984318",
      "0.153195253"
    ],
    [
      "0.545238580",
      "0.673102977",
      "0.152920734"
    ],
    [
      "0.472409650",
      "1.193332798",
      "0.152422734"
    ],
    [
      "1.333679883",
      "0.470543231",
      "0.150086508"
    ],
    [
      "1.169695772",
      "1.603862642",
      "0.148867697"
    ],
    [
      "0.724992789",
      "0.432296080",
      "0.147578079"
    ],
    [
      "0.578494750",
      "1.472969092",
      "0.146660038"
    ],
    [
      "0.369010603",
      "0.913930297",
      "0.145498597"
    ],
    [
      "1.834698740",
      "1.200048846",
      "0.141663511"
    ],
    [
      "1.657341553",
      "0.982574749",
      "0.138961780"
    ],
    [
      "1.471752739",
      "0.247725452",
      "0.112042994"
    ],
    [
      "1.241030215",
      "1.854725180",
      "0.111939924"
    ],
    [
      "0.614433125",
      "0.198754422",
      "0.110811442"
    ],
    [
      "0.409236394",
      "1.665783207",
      "0.109905107"
    ],
    [
      "1.584168442",
      "1.220201450",
      "0.109676015"
    ],
    [
      "0.117072852",
      "0.879105777",
      "0.108834604"
    ],
    [
      "1.897477354",
      "0.964785309",
      "0.101832044"
    ]
  ]
}

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