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P02 · Packing and covering · Classic · Weak baseline

Equal-circle packing in a unit circle · n = 24

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

Instancen = 24
ObjectiveMaximize common radius
Best known, unproven0.176939130value 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 = 0.176939130

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

0.176939130

common radius

Matches the best known
Record holderFounding benchmark
Solution methodHuman
Challenge this record ↗
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": [
    [
      "1.11770021",
      "0.185398346"
    ],
    [
      "0.764768916",
      "0.211269671"
    ],
    [
      "1.448873398",
      "0.310114598"
    ],
    [
      "0.455322532",
      "0.382945991"
    ],
    [
      "1.023343536",
      "0.526465262"
    ],
    [
      "1.697067675",
      "0.562363334"
    ],
    [
      "0.690697098",
      "0.647197078"
    ],
    [
      "1.354516723",
      "0.651181515"
    ],
    [
      "0.246565435",
      "0.668691154"
    ],
    [
      "1.816401751",
      "0.895513753"
    ],
    [
      "1.155987809",
      "0.944125356"
    ],
    [
      "0.526680514",
      "0.960770648"
    ],
    [
      "0.177088545",
      "1.015682199"
    ],
    [
      "1.495364291",
      "1.044392136"
    ],
    [
      "0.845610213",
      "1.11411233"
    ],
    [
      "1.784815483",
      "1.247979541"
    ],
    [
      "1.307188142",
      "1.357180569"
    ],
    [
      "0.259735385",
      "1.359774227"
    ],
    [
      "0.612989037",
      "1.380790437"
    ],
    [
      "0.965647553",
      "1.463375673"
    ],
    [
      "1.60814792",
      "1.554603734"
    ],
    [
      "0.628278926",
      "1.734338231"
    ],
    [
      "1.319057897",
      "1.758703666"
    ],
    [
      "0.970986707",
      "1.822549344"
    ]
  ],
  "radius": "0.17693913"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 24
}

The current leader's answer

{
  "centers": [
    [
      "1.11770021",
      "0.185398346"
    ],
    [
      "0.764768916",
      "0.211269671"
    ],
    [
      "1.448873398",
      "0.310114598"
    ],
    [
      "0.455322532",
      "0.382945991"
    ],
    [
      "1.023343536",
      "0.526465262"
    ],
    [
      "1.697067675",
      "0.562363334"
    ],
    [
      "0.690697098",
      "0.647197078"
    ],
    [
      "1.354516723",
      "0.651181515"
    ],
    [
      "0.246565435",
      "0.668691154"
    ],
    [
      "1.816401751",
      "0.895513753"
    ],
    [
      "1.155987809",
      "0.944125356"
    ],
    [
      "0.526680514",
      "0.960770648"
    ],
    [
      "0.177088545",
      "1.015682199"
    ],
    [
      "1.495364291",
      "1.044392136"
    ],
    [
      "0.845610213",
      "1.11411233"
    ],
    [
      "1.784815483",
      "1.247979541"
    ],
    [
      "1.307188142",
      "1.357180569"
    ],
    [
      "0.259735385",
      "1.359774227"
    ],
    [
      "0.612989037",
      "1.380790437"
    ],
    [
      "0.965647553",
      "1.463375673"
    ],
    [
      "1.60814792",
      "1.554603734"
    ],
    [
      "0.628278926",
      "1.734338231"
    ],
    [
      "1.319057897",
      "1.758703666"
    ],
    [
      "0.970986707",
      "1.822549344"
    ]
  ],
  "radius": "0.17693913"
}

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