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P02 · Packing and covering · Classic · Weak baseline

Equal-circle packing in a unit circle · n = 21

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

Instancen = 21
ObjectiveMaximize common radius
Best known, unproven0.190392146value 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 = 0.190392146

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

0.190392146

common radius

Matches the best known
Record holderFounding benchmark
Solution methodHuman
Challenge this record ↗
Record history (2 changes)
  1. AnonymousHuman
    0.185813856→0.190392146
  2. NUE_13Human
    0.059616906→0.185813856
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": [
    [
      "0.809607853",
      "0.213097397"
    ],
    [
      "1.190392147",
      "0.213097397"
    ],
    [
      "0.470940582",
      "0.387169674"
    ],
    [
      "1.529059418",
      "0.387169674"
    ],
    [
      "1",
      "0.542866269"
    ],
    [
      "0.249307588",
      "0.696807357"
    ],
    [
      "1.750692412",
      "0.696807357"
    ],
    [
      "0.661332729",
      "0.716938546"
    ],
    [
      "1.338667271",
      "0.716938546"
    ],
    [
      "1",
      "1"
    ],
    [
      "0.193736744",
      "1.073514882"
    ],
    [
      "1.806263256",
      "1.073514882"
    ],
    [
      "0.574259198",
      "1.087633675"
    ],
    [
      "1.425740802",
      "1.087633675"
    ],
    [
      "0.809607853",
      "1.386979109"
    ],
    [
      "1.190392147",
      "1.386979109"
    ],
    [
      "0.316520985",
      "1.433960035"
    ],
    [
      "1.683479015",
      "1.433960035"
    ],
    [
      "0.590498968",
      "1.698408033"
    ],
    [
      "1.409501032",
      "1.698408033"
    ],
    [
      "1",
      "1.777223539"
    ]
  ],
  "radius": "0.190392146"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 21
}

The current leader's answer

{
  "centers": [
    [
      "0.809607853",
      "0.213097397"
    ],
    [
      "1.190392147",
      "0.213097397"
    ],
    [
      "0.470940582",
      "0.387169674"
    ],
    [
      "1.529059418",
      "0.387169674"
    ],
    [
      "1",
      "0.542866269"
    ],
    [
      "0.249307588",
      "0.696807357"
    ],
    [
      "1.750692412",
      "0.696807357"
    ],
    [
      "0.661332729",
      "0.716938546"
    ],
    [
      "1.338667271",
      "0.716938546"
    ],
    [
      "1",
      "1"
    ],
    [
      "0.193736744",
      "1.073514882"
    ],
    [
      "1.806263256",
      "1.073514882"
    ],
    [
      "0.574259198",
      "1.087633675"
    ],
    [
      "1.425740802",
      "1.087633675"
    ],
    [
      "0.809607853",
      "1.386979109"
    ],
    [
      "1.190392147",
      "1.386979109"
    ],
    [
      "0.316520985",
      "1.433960035"
    ],
    [
      "1.683479015",
      "1.433960035"
    ],
    [
      "0.590498968",
      "1.698408033"
    ],
    [
      "1.409501032",
      "1.698408033"
    ],
    [
      "1",
      "1.777223539"
    ]
  ],
  "radius": "0.190392146"
}

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