P92 · Packing and covering · Classic · Hard

Sum of radii in the unit circle · n = 16

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 = 16

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	[3.611, 3.612]	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 $\max_{S} Σ r$

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Layout shown: starter layout

2y0

12345678910111213141516

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": [
    [
      "0.569442453",
      "1.592162239",
      "0.267855260"
    ],
    [
      "1.430130103",
      "1.592472809",
      "0.267855250"
    ],
    [
      "0.303770670",
      "0.773503628",
      "0.267855283"
    ],
    [
      "1.696392641",
      "0.774006192",
      "0.267855265"
    ],
    [
      "1.000264240",
      "0.267855334",
      "0.267855286"
    ],
    [
      "0.648938640",
      "1.113926762",
      "0.216942445"
    ],
    [
      "1.350979083",
      "1.114180100",
      "0.216942437"
    ],
    [
      "0.999866835",
      "1.369084542",
      "0.216942436"
    ],
    [
      "1.217050211",
      "0.701482692",
      "0.216942446"
    ],
    [
      "0.783165341",
      "0.701326112",
      "0.216942451"
    ],
    [
      "0.999713865",
      "1.793013450",
      "0.206986498"
    ],
    [
      "1.754112158",
      "1.245326849",
      "0.206986500"
    ],
    [
      "1.466353172",
      "0.358606908",
      "0.206986519"
    ],
    [
      "0.534109916",
      "0.358270466",
      "0.206986528"
    ],
    [
      "0.245710999",
      "1.244782535",
      "0.206986515"
    ],
    [
      "1.000000023",
      "1.000000042",
      "0.152142086"
    ]
  ]
}

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

Instance parameters

{
  "n": 16
}

Starter answer

{
  "circles": [
    [
      "0.569442453",
      "1.592162239",
      "0.267855260"
    ],
    [
      "1.430130103",
      "1.592472809",
      "0.267855250"
    ],
    [
      "0.303770670",
      "0.773503628",
      "0.267855283"
    ],
    [
      "1.696392641",
      "0.774006192",
      "0.267855265"
    ],
    [
      "1.000264240",
      "0.267855334",
      "0.267855286"
    ],
    [
      "0.648938640",
      "1.113926762",
      "0.216942445"
    ],
    [
      "1.350979083",
      "1.114180100",
      "0.216942437"
    ],
    [
      "0.999866835",
      "1.369084542",
      "0.216942436"
    ],
    [
      "1.217050211",
      "0.701482692",
      "0.216942446"
    ],
    [
      "0.783165341",
      "0.701326112",
      "0.216942451"
    ],
    [
      "0.999713865",
      "1.793013450",
      "0.206986498"
    ],
    [
      "1.754112158",
      "1.245326849",
      "0.206986500"
    ],
    [
      "1.466353172",
      "0.358606908",
      "0.206986519"
    ],
    [
      "0.534109916",
      "0.358270466",
      "0.206986528"
    ],
    [
      "0.245710999",
      "1.244782535",
      "0.206986515"
    ],
    [
      "1.000000023",
      "1.000000042",
      "0.152142086"
    ]
  ]
}

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