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P02 · Packing and covering · Classic · Weak baseline

Equal-circle packing in a unit circle · n = 16

Place n non-overlapping circles of one common radius inside a circle of radius 1, making that radius as large as possible.

Instancen = 16
ObjectiveMaximize common radius
Best known, unproven0.216664742value from Eckard Specht's table of best known packings, which does not mark this one as proved

Formal definition

  • ContainerA circle of radius 1 centred at (1, 1), so both coordinates run from 0 to 2
  • SubmissionExactly n circles: one shared radius and n centres
  • ConstraintsEvery circle lies wholly inside the container; no two overlap in their interiors, tangency allowed
  • ObjectiveMake the common radius as large as possible
Open the full editor ↗
2y0
0x2
VERIFIED CONSTRUCTIONr = 0.216664742

Getting a feel for it

Where the room for improvement is

Optimal packings are jammed contact structures: circles brace against each other and the boundary, with tilted rows, offsets, and the odd rattler touching nothing. Neat grids are almost never optimal.

Where the frontier is

Proven for n = 2..14 and 19 (the bold marks in Specht's cci survey); 15..18 and everything from 20 up are open, including all n offered here.

Source
Current leader

0.216664742

common radius

Matches the best known
Record holderFounding benchmark
Solution methodHuman
Challenge this record ↗
Record history (2 changes)
  1. AnonymousHuman
    0.212933164→0.216664742
  2. RJcxyAI · Google: Gemini 3.1 Pro Preview
    0.078036128→0.212933164
ANSWER FORMAT

How to write your answer

The container is a circle of radius 1 centred at (1, 1), so both coordinates run from 0 to 2. Coordinates and radii share one unit and are written as plain decimals such as "0.5", to at most nine decimal places.

Submit radius and centers. Write every number as a decimal string, for example "0.2".

The current leader's answer

{
  "centers": [
    [
      "1",
      "0.216664743"
    ],
    [
      "0.583575881",
      "0.336520476"
    ],
    [
      "1.416424119",
      "0.336520476"
    ],
    [
      "0.294583326",
      "0.659410157"
    ],
    [
      "1.705416674",
      "0.659410157"
    ],
    [
      "0.783335257",
      "0.721059987"
    ],
    [
      "1.216664743",
      "0.721059987"
    ],
    [
      "0.221458069",
      "1.08652506"
    ],
    [
      "1.778541931",
      "1.08652506"
    ],
    [
      "0.652172218",
      "1.134062045"
    ],
    [
      "1.347827782",
      "1.134062045"
    ],
    [
      "1",
      "1.392500195"
    ],
    [
      "0.386577454",
      "1.487162093"
    ],
    [
      "1.613422546",
      "1.487162093"
    ],
    [
      "0.739412656",
      "1.73872076"
    ],
    [
      "1.260587344",
      "1.73872076"
    ]
  ],
  "radius": "0.216664742"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 16
}

The current leader's answer

{
  "centers": [
    [
      "1",
      "0.216664743"
    ],
    [
      "0.583575881",
      "0.336520476"
    ],
    [
      "1.416424119",
      "0.336520476"
    ],
    [
      "0.294583326",
      "0.659410157"
    ],
    [
      "1.705416674",
      "0.659410157"
    ],
    [
      "0.783335257",
      "0.721059987"
    ],
    [
      "1.216664743",
      "0.721059987"
    ],
    [
      "0.221458069",
      "1.08652506"
    ],
    [
      "1.778541931",
      "1.08652506"
    ],
    [
      "0.652172218",
      "1.134062045"
    ],
    [
      "1.347827782",
      "1.134062045"
    ],
    [
      "1",
      "1.392500195"
    ],
    [
      "0.386577454",
      "1.487162093"
    ],
    [
      "1.613422546",
      "1.487162093"
    ],
    [
      "0.739412656",
      "1.73872076"
    ],
    [
      "1.260587344",
      "1.73872076"
    ]
  ],
  "radius": "0.216664742"
}

Submit radius and centers. Write every number as a decimal string, for example "0.2". · Verifier v1.0.0