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P02 · Packing and covering · Classic · Weak baseline

Equal-circle packing in a unit circle · n = 28

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

Instancen = 28
ObjectiveMaximize common radius
Best known, unproven0.166252750value 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 = 0.166252749

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

0.166252749

common radius

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": [
    [
      "1.107395649",
      "0.173719509"
    ],
    [
      "0.552158256",
      "0.296741691"
    ],
    [
      "1.447841744",
      "0.296741691"
    ],
    [
      "0.865305921",
      "0.408538144"
    ],
    [
      "1.184906622",
      "0.500273087"
    ],
    [
      "0.312939573",
      "0.527683955"
    ],
    [
      "1.687060427",
      "0.527683955"
    ],
    [
      "0.625916022",
      "0.639958841"
    ],
    [
      "0.945661481",
      "0.731187942"
    ],
    [
      "1.424125305",
      "0.731215351"
    ],
    [
      "0.182996634",
      "0.83374725"
    ],
    [
      "1.817003366",
      "0.83374725"
    ],
    [
      "0.495973083",
      "0.946022136"
    ],
    [
      "1.184281458",
      "0.962748764"
    ],
    [
      "0.815718542",
      "1.037251236"
    ],
    [
      "1.504026917",
      "1.053977864"
    ],
    [
      "0.182996634",
      "1.16625275"
    ],
    [
      "1.817003366",
      "1.16625275"
    ],
    [
      "0.575874695",
      "1.268784649"
    ],
    [
      "1.054338519",
      "1.268812058"
    ],
    [
      "1.374083978",
      "1.360041159"
    ],
    [
      "0.312939573",
      "1.472316045"
    ],
    [
      "1.687060427",
      "1.472316045"
    ],
    [
      "0.815093378",
      "1.499726913"
    ],
    [
      "1.134694079",
      "1.591461856"
    ],
    [
      "0.552158256",
      "1.703258309"
    ],
    [
      "1.447841744",
      "1.703258309"
    ],
    [
      "0.892867304",
      "1.826632832"
    ]
  ],
  "radius": "0.166252749"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 28
}

The current leader's answer

{
  "centers": [
    [
      "1.107395649",
      "0.173719509"
    ],
    [
      "0.552158256",
      "0.296741691"
    ],
    [
      "1.447841744",
      "0.296741691"
    ],
    [
      "0.865305921",
      "0.408538144"
    ],
    [
      "1.184906622",
      "0.500273087"
    ],
    [
      "0.312939573",
      "0.527683955"
    ],
    [
      "1.687060427",
      "0.527683955"
    ],
    [
      "0.625916022",
      "0.639958841"
    ],
    [
      "0.945661481",
      "0.731187942"
    ],
    [
      "1.424125305",
      "0.731215351"
    ],
    [
      "0.182996634",
      "0.83374725"
    ],
    [
      "1.817003366",
      "0.83374725"
    ],
    [
      "0.495973083",
      "0.946022136"
    ],
    [
      "1.184281458",
      "0.962748764"
    ],
    [
      "0.815718542",
      "1.037251236"
    ],
    [
      "1.504026917",
      "1.053977864"
    ],
    [
      "0.182996634",
      "1.16625275"
    ],
    [
      "1.817003366",
      "1.16625275"
    ],
    [
      "0.575874695",
      "1.268784649"
    ],
    [
      "1.054338519",
      "1.268812058"
    ],
    [
      "1.374083978",
      "1.360041159"
    ],
    [
      "0.312939573",
      "1.472316045"
    ],
    [
      "1.687060427",
      "1.472316045"
    ],
    [
      "0.815093378",
      "1.499726913"
    ],
    [
      "1.134694079",
      "1.591461856"
    ],
    [
      "0.552158256",
      "1.703258309"
    ],
    [
      "1.447841744",
      "1.703258309"
    ],
    [
      "0.892867304",
      "1.826632832"
    ]
  ],
  "radius": "0.166252749"
}

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