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P02 · Packing and covering · Classic · Weak baseline

Equal-circle packing in a unit circle · n = 20

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

Instancen = 20
ObjectiveMaximize common radius
Best known, unproven0.195224011value 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 = 0.195224010

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

0.195224010

common radius

Record holderFounding benchmark
Solution methodHuman
Challenge this record ↗
Record history (6 changes)
  1. AnonymousHuman
    0.185048285→0.195224010
  2. AnonymousHuman
    0.185000000→0.185048285
  3. 随便填的,加油我相信你可以Human
    0.156462547→0.185000000
  4. AnonymousHuman
    0.142180975→0.156462547
  5. Jev LiHuman
    0.132584492→0.142180975
  6. Jev LiHuman
    0.062573786→0.132584492
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": [
    [
      "0.804775989",
      "0.219261902"
    ],
    [
      "1.195224011",
      "0.219261902"
    ],
    [
      "0.460280522",
      "0.403034944"
    ],
    [
      "1.539719478",
      "0.403034944"
    ],
    [
      "0.804775989",
      "0.609709924"
    ],
    [
      "1.195224011",
      "0.609709924"
    ],
    [
      "0.242826237",
      "0.727323846"
    ],
    [
      "1.757173763",
      "0.727323846"
    ],
    [
      "0.57801851",
      "0.927563198"
    ],
    [
      "1.42198149",
      "0.927563198"
    ],
    [
      "1",
      "1"
    ],
    [
      "0.203598333",
      "1.115796277"
    ],
    [
      "1.796401667",
      "1.115796277"
    ],
    [
      "0.699199341",
      "1.298730093"
    ],
    [
      "1.300800659",
      "1.298730093"
    ],
    [
      "0.351830421",
      "1.477012149"
    ],
    [
      "1.648169579",
      "1.477012149"
    ],
    [
      "1",
      "1.547665068"
    ],
    [
      "0.65263108",
      "1.725947123"
    ],
    [
      "1.34736892",
      "1.725947123"
    ]
  ],
  "radius": "0.19522401"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 20
}

The current leader's answer

{
  "centers": [
    [
      "0.804775989",
      "0.219261902"
    ],
    [
      "1.195224011",
      "0.219261902"
    ],
    [
      "0.460280522",
      "0.403034944"
    ],
    [
      "1.539719478",
      "0.403034944"
    ],
    [
      "0.804775989",
      "0.609709924"
    ],
    [
      "1.195224011",
      "0.609709924"
    ],
    [
      "0.242826237",
      "0.727323846"
    ],
    [
      "1.757173763",
      "0.727323846"
    ],
    [
      "0.57801851",
      "0.927563198"
    ],
    [
      "1.42198149",
      "0.927563198"
    ],
    [
      "1",
      "1"
    ],
    [
      "0.203598333",
      "1.115796277"
    ],
    [
      "1.796401667",
      "1.115796277"
    ],
    [
      "0.699199341",
      "1.298730093"
    ],
    [
      "1.300800659",
      "1.298730093"
    ],
    [
      "0.351830421",
      "1.477012149"
    ],
    [
      "1.648169579",
      "1.477012149"
    ],
    [
      "1",
      "1.547665068"
    ],
    [
      "0.65263108",
      "1.725947123"
    ],
    [
      "1.34736892",
      "1.725947123"
    ]
  ],
  "radius": "0.19522401"
}

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