P92 · Packing and covering · Classic · Hard

Sum of radii in the unit circle · n = 28

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 = 28

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	[4.849, 4.850]	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

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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": [
    [
      "0.999972038",
      "0.532483327",
      "0.234828277"
    ],
    [
      "1.756908782",
      "1.252536654",
      "0.202074147"
    ],
    [
      "0.243121525",
      "1.252627391",
      "0.202074173"
    ],
    [
      "1.539252607",
      "1.590759661",
      "0.200131039"
    ],
    [
      "0.460818280",
      "1.590824335",
      "0.200131059"
    ],
    [
      "0.809891975",
      "0.912530711",
      "0.190102827"
    ],
    [
      "1.190097624",
      "0.912507932",
      "0.190102821"
    ],
    [
      "0.679201257",
      "0.255279965",
      "0.189124075"
    ],
    [
      "1.320709620",
      "0.255241569",
      "0.189124064"
    ],
    [
      "0.631470324",
      "1.244038179",
      "0.186369596"
    ],
    [
      "1.368558996",
      "1.243993998",
      "0.186369585"
    ],
    [
      "1.000013993",
      "1.232798291",
      "0.182345429"
    ],
    [
      "0.594371984",
      "0.613218221",
      "0.178728885"
    ],
    [
      "1.405581746",
      "0.613169643",
      "0.178728872"
    ],
    [
      "0.445284145",
      "0.934154359",
      "0.175145651"
    ],
    [
      "1.554708019",
      "0.934087890",
      "0.175145634"
    ],
    [
      "1.171179192",
      "1.542086391",
      "0.171146654"
    ],
    [
      "0.828885879",
      "1.542106916",
      "0.171146659"
    ],
    [
      "1.752619854",
      "0.650891114",
      "0.170353292"
    ],
    [
      "0.247338361",
      "0.650981262",
      "0.170353314"
    ],
    [
      "1.000050005",
      "1.833339580",
      "0.166660418"
    ],
    [
      "0.999949074",
      "0.148827526",
      "0.148827524"
    ],
    [
      "1.863670217",
      "0.934064416",
      "0.133816563"
    ],
    [
      "0.136321909",
      "0.934167906",
      "0.133816584"
    ],
    [
      "0.702901945",
      "1.816684545",
      "0.130954028"
    ],
    [
      "1.297196057",
      "1.816648894",
      "0.130954021"
    ],
    [
      "1.613426937",
      "0.383995154",
      "0.130658537"
    ],
    [
      "0.386499334",
      "0.384068603",
      "0.130658551"
    ]
  ]
}

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

Instance parameters

{
  "n": 28
}

Starter answer

{
  "circles": [
    [
      "0.999972038",
      "0.532483327",
      "0.234828277"
    ],
    [
      "1.756908782",
      "1.252536654",
      "0.202074147"
    ],
    [
      "0.243121525",
      "1.252627391",
      "0.202074173"
    ],
    [
      "1.539252607",
      "1.590759661",
      "0.200131039"
    ],
    [
      "0.460818280",
      "1.590824335",
      "0.200131059"
    ],
    [
      "0.809891975",
      "0.912530711",
      "0.190102827"
    ],
    [
      "1.190097624",
      "0.912507932",
      "0.190102821"
    ],
    [
      "0.679201257",
      "0.255279965",
      "0.189124075"
    ],
    [
      "1.320709620",
      "0.255241569",
      "0.189124064"
    ],
    [
      "0.631470324",
      "1.244038179",
      "0.186369596"
    ],
    [
      "1.368558996",
      "1.243993998",
      "0.186369585"
    ],
    [
      "1.000013993",
      "1.232798291",
      "0.182345429"
    ],
    [
      "0.594371984",
      "0.613218221",
      "0.178728885"
    ],
    [
      "1.405581746",
      "0.613169643",
      "0.178728872"
    ],
    [
      "0.445284145",
      "0.934154359",
      "0.175145651"
    ],
    [
      "1.554708019",
      "0.934087890",
      "0.175145634"
    ],
    [
      "1.171179192",
      "1.542086391",
      "0.171146654"
    ],
    [
      "0.828885879",
      "1.542106916",
      "0.171146659"
    ],
    [
      "1.752619854",
      "0.650891114",
      "0.170353292"
    ],
    [
      "0.247338361",
      "0.650981262",
      "0.170353314"
    ],
    [
      "1.000050005",
      "1.833339580",
      "0.166660418"
    ],
    [
      "0.999949074",
      "0.148827526",
      "0.148827524"
    ],
    [
      "1.863670217",
      "0.934064416",
      "0.133816563"
    ],
    [
      "0.136321909",
      "0.934167906",
      "0.133816584"
    ],
    [
      "0.702901945",
      "1.816684545",
      "0.130954028"
    ],
    [
      "1.297196057",
      "1.816648894",
      "0.130954021"
    ],
    [
      "1.613426937",
      "0.383995154",
      "0.130658537"
    ],
    [
      "0.386499334",
      "0.384068603",
      "0.130658551"
    ]
  ]
}

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