P92 · Packing and covering · Classic · Hard

Sum of radii in the unit circle · n = 24

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 = 24

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	[4.466, 4.467]	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

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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": [
    [
      "1.414179996",
      "0.898055357",
      "0.236934194"
    ],
    [
      "0.585345863",
      "0.900001599",
      "0.236934207"
    ],
    [
      "1.001080343",
      "1.460059785",
      "0.221680789"
    ],
    [
      "1.764497615",
      "1.182152187",
      "0.214101772"
    ],
    [
      "1.594873743",
      "0.485473683",
      "0.213481022"
    ],
    [
      "0.236366290",
      "1.185740528",
      "0.214101796"
    ],
    [
      "0.402716495",
      "0.488273034",
      "0.213481040"
    ],
    [
      "0.658392031",
      "1.716431662",
      "0.206293296"
    ],
    [
      "1.344968896",
      "1.714819423",
      "0.206293286"
    ],
    [
      "0.998109810",
      "0.195014326",
      "0.195012106"
    ],
    [
      "1.191420014",
      "0.530890890",
      "0.192520990"
    ],
    [
      "0.806379090",
      "0.531795021",
      "0.192520995"
    ],
    [
      "0.614199339",
      "1.324288257",
      "0.188332404"
    ],
    [
      "1.387319436",
      "1.322472818",
      "0.188332393"
    ],
    [
      "0.999649246",
      "0.850611832",
      "0.180302705"
    ],
    [
      "1.807889529",
      "0.802258074",
      "0.168262444"
    ],
    [
      "0.191190745",
      "0.806054344",
      "0.168262464"
    ],
    [
      "1.001974568",
      "1.840868603",
      "0.159129078"
    ],
    [
      "0.346108443",
      "1.536977392",
      "0.153880098"
    ],
    [
      "1.656406227",
      "1.533900532",
      "0.153880083"
    ],
    [
      "0.653701529",
      "0.224455713",
      "0.150652266"
    ],
    [
      "1.342652548",
      "0.222837973",
      "0.150652259"
    ],
    [
      "1.131010045",
      "1.132509130",
      "0.130698493"
    ],
    [
      "0.869613777",
      "1.133122937",
      "0.130698495"
    ]
  ]
}

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

Instance parameters

{
  "n": 24
}

Starter answer

{
  "circles": [
    [
      "1.414179996",
      "0.898055357",
      "0.236934194"
    ],
    [
      "0.585345863",
      "0.900001599",
      "0.236934207"
    ],
    [
      "1.001080343",
      "1.460059785",
      "0.221680789"
    ],
    [
      "1.764497615",
      "1.182152187",
      "0.214101772"
    ],
    [
      "1.594873743",
      "0.485473683",
      "0.213481022"
    ],
    [
      "0.236366290",
      "1.185740528",
      "0.214101796"
    ],
    [
      "0.402716495",
      "0.488273034",
      "0.213481040"
    ],
    [
      "0.658392031",
      "1.716431662",
      "0.206293296"
    ],
    [
      "1.344968896",
      "1.714819423",
      "0.206293286"
    ],
    [
      "0.998109810",
      "0.195014326",
      "0.195012106"
    ],
    [
      "1.191420014",
      "0.530890890",
      "0.192520990"
    ],
    [
      "0.806379090",
      "0.531795021",
      "0.192520995"
    ],
    [
      "0.614199339",
      "1.324288257",
      "0.188332404"
    ],
    [
      "1.387319436",
      "1.322472818",
      "0.188332393"
    ],
    [
      "0.999649246",
      "0.850611832",
      "0.180302705"
    ],
    [
      "1.807889529",
      "0.802258074",
      "0.168262444"
    ],
    [
      "0.191190745",
      "0.806054344",
      "0.168262464"
    ],
    [
      "1.001974568",
      "1.840868603",
      "0.159129078"
    ],
    [
      "0.346108443",
      "1.536977392",
      "0.153880098"
    ],
    [
      "1.656406227",
      "1.533900532",
      "0.153880083"
    ],
    [
      "0.653701529",
      "0.224455713",
      "0.150652266"
    ],
    [
      "1.342652548",
      "0.222837973",
      "0.150652259"
    ],
    [
      "1.131010045",
      "1.132509130",
      "0.130698493"
    ],
    [
      "0.869613777",
      "1.133122937",
      "0.130698495"
    ]
  ]
}

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