P92 · Packing and covering · Classic · Hard

Sum of radii in the unit circle · n = 25

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 = 25

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	[4.564, 4.565]	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

12345678910111213141516171819202122232425

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.

Challenge this record ↗Submit a proof / idea ↓Share a proof or idea in the discussion. Accepted contributions can earn proof points.

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": [
    [
      "1.296740062",
      "1.336162884",
      "0.230873910"
    ],
    [
      "1.231078797",
      "0.262038136",
      "0.226705021"
    ],
    [
      "0.780887983",
      "0.255096561",
      "0.223539305"
    ],
    [
      "0.533370381",
      "1.624998322",
      "0.220021727"
    ],
    [
      "0.266320032",
      "1.283602475",
      "0.213414556"
    ],
    [
      "1.204080958",
      "1.765198981",
      "0.208053967"
    ],
    [
      "1.730000955",
      "1.320694331",
      "0.202663027"
    ],
    [
      "0.997593621",
      "0.611666954",
      "0.193718143"
    ],
    [
      "1.748162569",
      "0.685619946",
      "0.188469317"
    ],
    [
      "1.527483917",
      "0.988570578",
      "0.186334835"
    ],
    [
      "1.375514352",
      "0.648652231",
      "0.186008059"
    ],
    [
      "0.916594179",
      "1.499996994",
      "0.183073559"
    ],
    [
      "0.261586473",
      "0.648384483",
      "0.182144261"
    ],
    [
      "0.623661497",
      "0.627335211",
      "0.180542096"
    ],
    [
      "1.163080396",
      "0.947204289",
      "0.180409074"
    ],
    [
      "0.658104782",
      "1.244598015",
      "0.180306966"
    ],
    [
      "0.462126986",
      "0.946077286",
      "0.176795151"
    ],
    [
      "0.810825974",
      "0.926618215",
      "0.172446369"
    ],
    [
      "1.548886308",
      "1.635346970",
      "0.160391786"
    ],
    [
      "0.846921977",
      "1.830916335",
      "0.155100694"
    ],
    [
      "1.587415472",
      "0.386801439",
      "0.150841939"
    ],
    [
      "0.428249308",
      "0.365678504",
      "0.146031256"
    ],
    [
      "1.856827241",
      "0.998953686",
      "0.143172120"
    ],
    [
      "0.143403179",
      "0.950167477",
      "0.141954899"
    ],
    [
      "0.965261390",
      "1.188836175",
      "0.131870170"
    ]
  ]
}

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

Instance parameters

{
  "n": 25
}

Starter answer

{
  "circles": [
    [
      "1.296740062",
      "1.336162884",
      "0.230873910"
    ],
    [
      "1.231078797",
      "0.262038136",
      "0.226705021"
    ],
    [
      "0.780887983",
      "0.255096561",
      "0.223539305"
    ],
    [
      "0.533370381",
      "1.624998322",
      "0.220021727"
    ],
    [
      "0.266320032",
      "1.283602475",
      "0.213414556"
    ],
    [
      "1.204080958",
      "1.765198981",
      "0.208053967"
    ],
    [
      "1.730000955",
      "1.320694331",
      "0.202663027"
    ],
    [
      "0.997593621",
      "0.611666954",
      "0.193718143"
    ],
    [
      "1.748162569",
      "0.685619946",
      "0.188469317"
    ],
    [
      "1.527483917",
      "0.988570578",
      "0.186334835"
    ],
    [
      "1.375514352",
      "0.648652231",
      "0.186008059"
    ],
    [
      "0.916594179",
      "1.499996994",
      "0.183073559"
    ],
    [
      "0.261586473",
      "0.648384483",
      "0.182144261"
    ],
    [
      "0.623661497",
      "0.627335211",
      "0.180542096"
    ],
    [
      "1.163080396",
      "0.947204289",
      "0.180409074"
    ],
    [
      "0.658104782",
      "1.244598015",
      "0.180306966"
    ],
    [
      "0.462126986",
      "0.946077286",
      "0.176795151"
    ],
    [
      "0.810825974",
      "0.926618215",
      "0.172446369"
    ],
    [
      "1.548886308",
      "1.635346970",
      "0.160391786"
    ],
    [
      "0.846921977",
      "1.830916335",
      "0.155100694"
    ],
    [
      "1.587415472",
      "0.386801439",
      "0.150841939"
    ],
    [
      "0.428249308",
      "0.365678504",
      "0.146031256"
    ],
    [
      "1.856827241",
      "0.998953686",
      "0.143172120"
    ],
    [
      "0.143403179",
      "0.950167477",
      "0.141954899"
    ],
    [
      "0.965261390",
      "1.188836175",
      "0.131870170"
    ]
  ]
}

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

Talk strategy, share methods, ask why you are stuck. Every signed-in user can post. Posts carry your public byline, the same name your records use; the #number after it is the account's signup ordinal, so a name cannot be worn by someone else. New posts appear after an automated review.

对本周挑战的评价NUE_13#311 replies1h ago