P92 · Packing and covering · Classic · Hard

Sum of radii in the unit circle · n = 29

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 = 29

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	[4.934, 4.935]	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

1234567891011121314151617181920212223242526272829

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": [
    [
      "0.670209874",
      "0.617405927",
      "0.214647629"
    ],
    [
      "1.329898038",
      "0.617498918",
      "0.214647617"
    ],
    [
      "1.741142226",
      "1.286218990",
      "0.205510787"
    ],
    [
      "0.258777173",
      "1.286010119",
      "0.205510813"
    ],
    [
      "1.517717382",
      "1.617571433",
      "0.194130429"
    ],
    [
      "0.482108666",
      "1.617425543",
      "0.194130447"
    ],
    [
      "1.731785335",
      "0.649232684",
      "0.188490611"
    ],
    [
      "0.268313609",
      "0.649026393",
      "0.188490635"
    ],
    [
      "1.530002360",
      "0.960616885",
      "0.182557214"
    ],
    [
      "0.470008822",
      "0.960467497",
      "0.182557231"
    ],
    [
      "1.354195517",
      "1.279406187",
      "0.181495892"
    ],
    [
      "0.645725826",
      "1.279306360",
      "0.181495903"
    ],
    [
      "1.286282259",
      "0.228340574",
      "0.176947266"
    ],
    [
      "0.713935381",
      "0.228259875",
      "0.176947275"
    ],
    [
      "0.826171087",
      "0.973205156",
      "0.173832731"
    ],
    [
      "1.173836542",
      "0.973254153",
      "0.173832726"
    ],
    [
      "0.999961556",
      "1.273116635",
      "0.172793900"
    ],
    [
      "1.000081496",
      "0.422130155",
      "0.168690228"
    ],
    [
      "1.163543652",
      "1.567085293",
      "0.163623510"
    ],
    [
      "0.836296630",
      "1.567039191",
      "0.163623514"
    ],
    [
      "0.999881285",
      "1.842903723",
      "0.157096268"
    ],
    [
      "1.855726339",
      "0.956884159",
      "0.143188151"
    ],
    [
      "0.144285870",
      "0.956642960",
      "0.143188173"
    ],
    [
      "1.574635527",
      "0.360491349",
      "0.140245788"
    ],
    [
      "0.425544840",
      "0.360329343",
      "0.140245802"
    ],
    [
      "1.000039187",
      "0.722271128",
      "0.131450746"
    ],
    [
      "1.000123148",
      "0.126719969",
      "0.126719960"
    ],
    [
      "1.280690841",
      "1.829792125",
      "0.124019224"
    ],
    [
      "0.719075428",
      "1.829713018",
      "0.124019231"
    ]
  ]
}

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

Instance parameters

{
  "n": 29
}

Starter answer

{
  "circles": [
    [
      "0.670209874",
      "0.617405927",
      "0.214647629"
    ],
    [
      "1.329898038",
      "0.617498918",
      "0.214647617"
    ],
    [
      "1.741142226",
      "1.286218990",
      "0.205510787"
    ],
    [
      "0.258777173",
      "1.286010119",
      "0.205510813"
    ],
    [
      "1.517717382",
      "1.617571433",
      "0.194130429"
    ],
    [
      "0.482108666",
      "1.617425543",
      "0.194130447"
    ],
    [
      "1.731785335",
      "0.649232684",
      "0.188490611"
    ],
    [
      "0.268313609",
      "0.649026393",
      "0.188490635"
    ],
    [
      "1.530002360",
      "0.960616885",
      "0.182557214"
    ],
    [
      "0.470008822",
      "0.960467497",
      "0.182557231"
    ],
    [
      "1.354195517",
      "1.279406187",
      "0.181495892"
    ],
    [
      "0.645725826",
      "1.279306360",
      "0.181495903"
    ],
    [
      "1.286282259",
      "0.228340574",
      "0.176947266"
    ],
    [
      "0.713935381",
      "0.228259875",
      "0.176947275"
    ],
    [
      "0.826171087",
      "0.973205156",
      "0.173832731"
    ],
    [
      "1.173836542",
      "0.973254153",
      "0.173832726"
    ],
    [
      "0.999961556",
      "1.273116635",
      "0.172793900"
    ],
    [
      "1.000081496",
      "0.422130155",
      "0.168690228"
    ],
    [
      "1.163543652",
      "1.567085293",
      "0.163623510"
    ],
    [
      "0.836296630",
      "1.567039191",
      "0.163623514"
    ],
    [
      "0.999881285",
      "1.842903723",
      "0.157096268"
    ],
    [
      "1.855726339",
      "0.956884159",
      "0.143188151"
    ],
    [
      "0.144285870",
      "0.956642960",
      "0.143188173"
    ],
    [
      "1.574635527",
      "0.360491349",
      "0.140245788"
    ],
    [
      "0.425544840",
      "0.360329343",
      "0.140245802"
    ],
    [
      "1.000039187",
      "0.722271128",
      "0.131450746"
    ],
    [
      "1.000123148",
      "0.126719969",
      "0.126719960"
    ],
    [
      "1.280690841",
      "1.829792125",
      "0.124019224"
    ],
    [
      "0.719075428",
      "1.829713018",
      "0.124019231"
    ]
  ]
}

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