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P92 · Packing and covering · Classic · Hard

Sum of radii in the unit circle · n = 10

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 = 10
ObjectiveMaximize the sum of the radii

Record comparison

the sum of the radii · Higher is better
RecordValue / intervalAuthor / holderSource
External best known[2.783, 2.784]See source attributionErich Friedman ↗
Site recordNo 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 betterS=Σr
  • ObjectiveMaximise the sum of the radii over every legal arrangementmaxSΣr
Open the full editor ↗

Layout shown: starter layout

2y0
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
The current arrangement for Sum of radii in the unit circle n = 10
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": [
    [
      "1.000000005",
      "1.000000006",
      "0.490290596"
    ],
    [
      "1.479278619",
      "1.570555454",
      "0.254854697"
    ],
    [
      "1.645113299",
      "0.627078629",
      "0.254854700"
    ],
    [
      "1.000402756",
      "1.745145192",
      "0.254854698"
    ],
    [
      "1.254476219",
      "0.299654823",
      "0.254854703"
    ],
    [
      "0.521338437",
      "1.571073222",
      "0.254854700"
    ],
    [
      "0.266245181",
      "1.129789744",
      "0.254854703"
    ],
    [
      "0.744766884",
      "0.299930313",
      "0.254854705"
    ],
    [
      "1.733894697",
      "1.128996486",
      "0.254854698"
    ],
    [
      "0.354483964",
      "0.627776199",
      "0.254854704"
    ]
  ]
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 10
}

Starter answer

{
  "circles": [
    [
      "1.000000005",
      "1.000000006",
      "0.490290596"
    ],
    [
      "1.479278619",
      "1.570555454",
      "0.254854697"
    ],
    [
      "1.645113299",
      "0.627078629",
      "0.254854700"
    ],
    [
      "1.000402756",
      "1.745145192",
      "0.254854698"
    ],
    [
      "1.254476219",
      "0.299654823",
      "0.254854703"
    ],
    [
      "0.521338437",
      "1.571073222",
      "0.254854700"
    ],
    [
      "0.266245181",
      "1.129789744",
      "0.254854703"
    ],
    [
      "0.744766884",
      "0.299930313",
      "0.254854705"
    ],
    [
      "1.733894697",
      "1.128996486",
      "0.254854698"
    ],
    [
      "0.354483964",
      "0.627776199",
      "0.254854704"
    ]
  ]
}

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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