P02 · Packing and covering · Classic · Weak baseline

Equal-circle packing in a unit circle · n = 22

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

Instancen = 22

ObjectiveMaximize common radius

Best known, unproven0.183833026value 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

12345678910111213141516171819202122

0x2

VERIFIED CONSTRUCTIONr = 0.183833026

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

0.183833026

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": [
    [
      "0.913887701",
      "0.188388516"
    ],
    [
      "1.27884402",
      "0.232944239"
    ],
    [
      "0.566406263",
      "0.308534166"
    ],
    [
      "1.587214159",
      "0.433159582"
    ],
    [
      "0.306914565",
      "0.568999874"
    ],
    [
      "1.161027359",
      "0.581222232"
    ],
    [
      "0.793722166",
      "0.597507879"
    ],
    [
      "1.776420168",
      "0.748404587"
    ],
    [
      "1.42072973",
      "0.841477886"
    ],
    [
      "0.553428045",
      "0.875783034"
    ],
    [
      "0.991478546",
      "0.907460683"
    ],
    [
      "0.188071594",
      "0.916928997"
    ],
    [
      "1.808066224",
      "1.114706175"
    ],
    [
      "1.405140878",
      "1.208813311"
    ],
    [
      "0.599350758",
      "1.240569864"
    ],
    [
      "1.042861454",
      "1.27151854"
    ],
    [
      "0.233994307",
      "1.281715827"
    ],
    [
      "1.675730352",
      "1.457730291"
    ],
    [
      "0.800720001",
      "1.548187735"
    ],
    [
      "0.43536355",
      "1.589333698"
    ],
    [
      "1.406267644",
      "1.707866604"
    ],
    [
      "1.054360516",
      "1.81435463"
    ]
  ],
  "radius": "0.183833026"
}

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

Instance parameters

{
  "n": 22
}

The current leader's answer

{
  "centers": [
    [
      "0.913887701",
      "0.188388516"
    ],
    [
      "1.27884402",
      "0.232944239"
    ],
    [
      "0.566406263",
      "0.308534166"
    ],
    [
      "1.587214159",
      "0.433159582"
    ],
    [
      "0.306914565",
      "0.568999874"
    ],
    [
      "1.161027359",
      "0.581222232"
    ],
    [
      "0.793722166",
      "0.597507879"
    ],
    [
      "1.776420168",
      "0.748404587"
    ],
    [
      "1.42072973",
      "0.841477886"
    ],
    [
      "0.553428045",
      "0.875783034"
    ],
    [
      "0.991478546",
      "0.907460683"
    ],
    [
      "0.188071594",
      "0.916928997"
    ],
    [
      "1.808066224",
      "1.114706175"
    ],
    [
      "1.405140878",
      "1.208813311"
    ],
    [
      "0.599350758",
      "1.240569864"
    ],
    [
      "1.042861454",
      "1.27151854"
    ],
    [
      "0.233994307",
      "1.281715827"
    ],
    [
      "1.675730352",
      "1.457730291"
    ],
    [
      "0.800720001",
      "1.548187735"
    ],
    [
      "0.43536355",
      "1.589333698"
    ],
    [
      "1.406267644",
      "1.707866604"
    ],
    [
      "1.054360516",
      "1.81435463"
    ]
  ],
  "radius": "0.183833026"
}

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