P92 · Packing and covering · Classic · Hard

Sum of radii in the unit circle · n = 22

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 = 22

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	[4.287, 4.288]	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

12345678910111213141516171819202122

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": [
    [
      "1.000000029",
      "1.000000026",
      "0.270210210"
    ],
    [
      "1.771199758",
      "1.175737197",
      "0.209030575"
    ],
    [
      "0.657060900",
      "1.712758999",
      "0.209030584"
    ],
    [
      "1.343438363",
      "1.712518576",
      "0.209030575"
    ],
    [
      "0.381422537",
      "0.507055266",
      "0.209030601"
    ],
    [
      "1.618232070",
      "0.506622131",
      "0.209030585"
    ],
    [
      "0.228923553",
      "1.176277370",
      "0.209030595"
    ],
    [
      "0.999723025",
      "0.209030646",
      "0.209030597"
    ],
    [
      "1.000167205",
      "1.477305147",
      "0.207094940"
    ],
    [
      "0.534624731",
      "0.893952624",
      "0.207094952"
    ],
    [
      "0.626932087",
      "1.297725611",
      "0.207094947"
    ],
    [
      "0.792754482",
      "0.570035489",
      "0.207094953"
    ],
    [
      "1.206944359",
      "0.569890436",
      "0.207094948"
    ],
    [
      "1.465300919",
      "0.893626676",
      "0.207094941"
    ],
    [
      "1.373276425",
      "1.297464203",
      "0.207094937"
    ],
    [
      "1.000295019",
      "1.842200007",
      "0.157799941"
    ],
    [
      "1.658642389",
      "1.524872507",
      "0.157799937"
    ],
    [
      "1.821018670",
      "0.812305352",
      "0.157799942"
    ],
    [
      "1.365151188",
      "0.241076070",
      "0.157799951"
    ],
    [
      "0.341725479",
      "1.525333778",
      "0.157799950"
    ],
    [
      "0.634317388",
      "0.241331999",
      "0.157799959"
    ],
    [
      "0.178850078",
      "0.812880476",
      "0.157799958"
    ]
  ]
}

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

Instance parameters

{
  "n": 22
}

Starter answer

{
  "circles": [
    [
      "1.000000029",
      "1.000000026",
      "0.270210210"
    ],
    [
      "1.771199758",
      "1.175737197",
      "0.209030575"
    ],
    [
      "0.657060900",
      "1.712758999",
      "0.209030584"
    ],
    [
      "1.343438363",
      "1.712518576",
      "0.209030575"
    ],
    [
      "0.381422537",
      "0.507055266",
      "0.209030601"
    ],
    [
      "1.618232070",
      "0.506622131",
      "0.209030585"
    ],
    [
      "0.228923553",
      "1.176277370",
      "0.209030595"
    ],
    [
      "0.999723025",
      "0.209030646",
      "0.209030597"
    ],
    [
      "1.000167205",
      "1.477305147",
      "0.207094940"
    ],
    [
      "0.534624731",
      "0.893952624",
      "0.207094952"
    ],
    [
      "0.626932087",
      "1.297725611",
      "0.207094947"
    ],
    [
      "0.792754482",
      "0.570035489",
      "0.207094953"
    ],
    [
      "1.206944359",
      "0.569890436",
      "0.207094948"
    ],
    [
      "1.465300919",
      "0.893626676",
      "0.207094941"
    ],
    [
      "1.373276425",
      "1.297464203",
      "0.207094937"
    ],
    [
      "1.000295019",
      "1.842200007",
      "0.157799941"
    ],
    [
      "1.658642389",
      "1.524872507",
      "0.157799937"
    ],
    [
      "1.821018670",
      "0.812305352",
      "0.157799942"
    ],
    [
      "1.365151188",
      "0.241076070",
      "0.157799951"
    ],
    [
      "0.341725479",
      "1.525333778",
      "0.157799950"
    ],
    [
      "0.634317388",
      "0.241331999",
      "0.157799959"
    ],
    [
      "0.178850078",
      "0.812880476",
      "0.157799958"
    ]
  ]
}

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