P92 · Packing and covering · Classic · Hard

Sum of radii in the unit circle · n = 13

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 = 13

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	[3.227, 3.228]	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

12345678910111213

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": [
    [
      "0.999841156",
      "0.738736245",
      "0.304612725"
    ],
    [
      "1.595232384",
      "0.632105606",
      "0.300251562"
    ],
    [
      "0.404320691",
      "0.632829707",
      "0.300251570"
    ],
    [
      "1.471955949",
      "1.530865098",
      "0.289676010"
    ],
    [
      "0.528689991",
      "1.531438647",
      "0.289676017"
    ],
    [
      "1.000173584",
      "1.285463023",
      "0.242114152"
    ],
    [
      "1.000464428",
      "1.763788473",
      "0.236211385"
    ],
    [
      "0.774088833",
      "0.259138807",
      "0.225460677"
    ],
    [
      "1.225010102",
      "0.258864641",
      "0.225460674"
    ],
    [
      "1.771863460",
      "1.120212795",
      "0.218831441"
    ],
    [
      "0.228283307",
      "1.121151346",
      "0.218831450"
    ],
    [
      "1.369035328",
      "1.064594802",
      "0.187818118"
    ],
    [
      "0.631043517",
      "1.065043526",
      "0.187818121"
    ]
  ]
}

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

Instance parameters

{
  "n": 13
}

Starter answer

{
  "circles": [
    [
      "0.999841156",
      "0.738736245",
      "0.304612725"
    ],
    [
      "1.595232384",
      "0.632105606",
      "0.300251562"
    ],
    [
      "0.404320691",
      "0.632829707",
      "0.300251570"
    ],
    [
      "1.471955949",
      "1.530865098",
      "0.289676010"
    ],
    [
      "0.528689991",
      "1.531438647",
      "0.289676017"
    ],
    [
      "1.000173584",
      "1.285463023",
      "0.242114152"
    ],
    [
      "1.000464428",
      "1.763788473",
      "0.236211385"
    ],
    [
      "0.774088833",
      "0.259138807",
      "0.225460677"
    ],
    [
      "1.225010102",
      "0.258864641",
      "0.225460674"
    ],
    [
      "1.771863460",
      "1.120212795",
      "0.218831441"
    ],
    [
      "0.228283307",
      "1.121151346",
      "0.218831450"
    ],
    [
      "1.369035328",
      "1.064594802",
      "0.187818118"
    ],
    [
      "0.631043517",
      "1.065043526",
      "0.187818121"
    ]
  ]
}

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