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P02 · Packing and covering · Classic · Weak baseline

Equal-circle packing in a unit circle · n = 25

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

Instancen = 25
ObjectiveMaximize common radius
Best known, unproven0.173827661value 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 = 0.173827661

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

0.173827661

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": [
    [
      "1.119223475",
      "0.182475385"
    ],
    [
      "0.772352169",
      "0.205810351"
    ],
    [
      "0.46579153",
      "0.369776196"
    ],
    [
      "1.57268885",
      "0.404526899"
    ],
    [
      "1.240840359",
      "0.508164742"
    ],
    [
      "0.893969053",
      "0.531499708"
    ],
    [
      "0.253825685",
      "0.645338718"
    ],
    [
      "1.766951902",
      "0.692842854"
    ],
    [
      "0.595468233",
      "0.709717227"
    ],
    [
      "1.106213449",
      "0.828695269"
    ],
    [
      "1.453560126",
      "0.843341419"
    ],
    [
      "0.368893535",
      "0.973399099"
    ],
    [
      "0.799050535",
      "0.991530055"
    ],
    [
      "1.825407192",
      "1.035548563"
    ],
    [
      "1.093651054",
      "1.176123549"
    ],
    [
      "1.440997731",
      "1.190769699"
    ],
    [
      "0.572475837",
      "1.255211927"
    ],
    [
      "0.226627618",
      "1.290612958"
    ],
    [
      "1.737703767",
      "1.371959521"
    ],
    [
      "0.85492095",
      "1.4579161"
    ],
    [
      "1.199033389",
      "1.507422186"
    ],
    [
      "0.414653128",
      "1.583034967"
    ],
    [
      "1.519371683",
      "1.642505866"
    ],
    [
      "0.706328748",
      "1.772216245"
    ],
    [
      "1.051632791",
      "1.824352301"
    ]
  ],
  "radius": "0.173827661"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 25
}

The current leader's answer

{
  "centers": [
    [
      "1.119223475",
      "0.182475385"
    ],
    [
      "0.772352169",
      "0.205810351"
    ],
    [
      "0.46579153",
      "0.369776196"
    ],
    [
      "1.57268885",
      "0.404526899"
    ],
    [
      "1.240840359",
      "0.508164742"
    ],
    [
      "0.893969053",
      "0.531499708"
    ],
    [
      "0.253825685",
      "0.645338718"
    ],
    [
      "1.766951902",
      "0.692842854"
    ],
    [
      "0.595468233",
      "0.709717227"
    ],
    [
      "1.106213449",
      "0.828695269"
    ],
    [
      "1.453560126",
      "0.843341419"
    ],
    [
      "0.368893535",
      "0.973399099"
    ],
    [
      "0.799050535",
      "0.991530055"
    ],
    [
      "1.825407192",
      "1.035548563"
    ],
    [
      "1.093651054",
      "1.176123549"
    ],
    [
      "1.440997731",
      "1.190769699"
    ],
    [
      "0.572475837",
      "1.255211927"
    ],
    [
      "0.226627618",
      "1.290612958"
    ],
    [
      "1.737703767",
      "1.371959521"
    ],
    [
      "0.85492095",
      "1.4579161"
    ],
    [
      "1.199033389",
      "1.507422186"
    ],
    [
      "0.414653128",
      "1.583034967"
    ],
    [
      "1.519371683",
      "1.642505866"
    ],
    [
      "0.706328748",
      "1.772216245"
    ],
    [
      "1.051632791",
      "1.824352301"
    ]
  ],
  "radius": "0.173827661"
}

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