P92 · Packing and covering · Classic · Hard

Sum of radii in the unit circle · n = 17

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 = 17

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	[3.730, 3.731]	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

1234567891011121314151617

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

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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": [
    [
      "0.999922820",
      "0.607129478",
      "0.270922900"
    ],
    [
      "0.483140444",
      "0.474180070",
      "0.262687040"
    ],
    [
      "1.516652900",
      "0.473976999",
      "0.262687029"
    ],
    [
      "1.679456159",
      "1.300555927",
      "0.257036651"
    ],
    [
      "0.320662039",
      "1.300822957",
      "0.257036665"
    ],
    [
      "1.000147616",
      "1.751041068",
      "0.248958917"
    ],
    [
      "0.632017213",
      "0.936028832",
      "0.222563999"
    ],
    [
      "1.367957650",
      "0.935884216",
      "0.222563992"
    ],
    [
      "0.787989879",
      "1.341720313",
      "0.212077290"
    ],
    [
      "1.212144447",
      "1.341636958",
      "0.212077286"
    ],
    [
      "1.784483803",
      "0.853704458",
      "0.201991715"
    ],
    [
      "0.215458776",
      "0.854012774",
      "0.201991727"
    ],
    [
      "0.564125914",
      "1.679240517",
      "0.192935009"
    ],
    [
      "1.436141054",
      "1.679069136",
      "0.192935002"
    ],
    [
      "1.176528057",
      "0.195833529",
      "0.176686044"
    ],
    [
      "0.823155972",
      "0.195902957",
      "0.176686047"
    ],
    [
      "1.000007312",
      "1.037108303",
      "0.159055932"
    ]
  ]
}

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

Instance parameters

{
  "n": 17
}

Starter answer

{
  "circles": [
    [
      "0.999922820",
      "0.607129478",
      "0.270922900"
    ],
    [
      "0.483140444",
      "0.474180070",
      "0.262687040"
    ],
    [
      "1.516652900",
      "0.473976999",
      "0.262687029"
    ],
    [
      "1.679456159",
      "1.300555927",
      "0.257036651"
    ],
    [
      "0.320662039",
      "1.300822957",
      "0.257036665"
    ],
    [
      "1.000147616",
      "1.751041068",
      "0.248958917"
    ],
    [
      "0.632017213",
      "0.936028832",
      "0.222563999"
    ],
    [
      "1.367957650",
      "0.935884216",
      "0.222563992"
    ],
    [
      "0.787989879",
      "1.341720313",
      "0.212077290"
    ],
    [
      "1.212144447",
      "1.341636958",
      "0.212077286"
    ],
    [
      "1.784483803",
      "0.853704458",
      "0.201991715"
    ],
    [
      "0.215458776",
      "0.854012774",
      "0.201991727"
    ],
    [
      "0.564125914",
      "1.679240517",
      "0.192935009"
    ],
    [
      "1.436141054",
      "1.679069136",
      "0.192935002"
    ],
    [
      "1.176528057",
      "0.195833529",
      "0.176686044"
    ],
    [
      "0.823155972",
      "0.195902957",
      "0.176686047"
    ],
    [
      "1.000007312",
      "1.037108303",
      "0.159055932"
    ]
  ]
}

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