MinMax Arena
中/EN
P02 · Packing and covering · Classic · Weak baseline

Equal-circle packing in a unit circle · n = 27

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

Instancen = 27
ObjectiveMaximize common radius
Best known, unproven0.169307931value 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 = 0.169307930

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

0.169307930

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": [
    [
      "0.830692069",
      "0.186744728"
    ],
    [
      "1.169307931",
      "0.186744728"
    ],
    [
      "0.52020893",
      "0.321877709"
    ],
    [
      "1.47979107",
      "0.321877709"
    ],
    [
      "0.940578427",
      "0.507034738"
    ],
    [
      "1.710550659",
      "0.569689561"
    ],
    [
      "1.269540763",
      "0.587312278"
    ],
    [
      "0.272064899",
      "0.599800298"
    ],
    [
      "0.607063132",
      "0.64916512"
    ],
    [
      "1.500300352",
      "0.835124131"
    ],
    [
      "0.885949463",
      "0.841214896"
    ],
    [
      "0.172833334",
      "0.92354989"
    ],
    [
      "1.827166666",
      "0.92354989"
    ],
    [
      "0.507831567",
      "0.972914712"
    ],
    [
      "1.194537202",
      "0.98062189"
    ],
    [
      "0.919513335",
      "1.178163214"
    ],
    [
      "1.456631227",
      "1.19502204"
    ],
    [
      "0.211046309",
      "1.260002667"
    ],
    [
      "1.788953691",
      "1.260002667"
    ],
    [
      "0.602790346",
      "1.297943223"
    ],
    [
      "1.222627669",
      "1.43977301"
    ],
    [
      "0.892636515",
      "1.515710749"
    ],
    [
      "0.38035424",
      "1.553252606"
    ],
    [
      "1.61964576",
      "1.553252606"
    ],
    [
      "0.652624404",
      "1.754572401"
    ],
    [
      "1.347375596",
      "1.754572401"
    ],
    [
      "1.017384442",
      "1.830510141"
    ]
  ],
  "radius": "0.16930793"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 27
}

The current leader's answer

{
  "centers": [
    [
      "0.830692069",
      "0.186744728"
    ],
    [
      "1.169307931",
      "0.186744728"
    ],
    [
      "0.52020893",
      "0.321877709"
    ],
    [
      "1.47979107",
      "0.321877709"
    ],
    [
      "0.940578427",
      "0.507034738"
    ],
    [
      "1.710550659",
      "0.569689561"
    ],
    [
      "1.269540763",
      "0.587312278"
    ],
    [
      "0.272064899",
      "0.599800298"
    ],
    [
      "0.607063132",
      "0.64916512"
    ],
    [
      "1.500300352",
      "0.835124131"
    ],
    [
      "0.885949463",
      "0.841214896"
    ],
    [
      "0.172833334",
      "0.92354989"
    ],
    [
      "1.827166666",
      "0.92354989"
    ],
    [
      "0.507831567",
      "0.972914712"
    ],
    [
      "1.194537202",
      "0.98062189"
    ],
    [
      "0.919513335",
      "1.178163214"
    ],
    [
      "1.456631227",
      "1.19502204"
    ],
    [
      "0.211046309",
      "1.260002667"
    ],
    [
      "1.788953691",
      "1.260002667"
    ],
    [
      "0.602790346",
      "1.297943223"
    ],
    [
      "1.222627669",
      "1.43977301"
    ],
    [
      "0.892636515",
      "1.515710749"
    ],
    [
      "0.38035424",
      "1.553252606"
    ],
    [
      "1.61964576",
      "1.553252606"
    ],
    [
      "0.652624404",
      "1.754572401"
    ],
    [
      "1.347375596",
      "1.754572401"
    ],
    [
      "1.017384442",
      "1.830510141"
    ]
  ],
  "radius": "0.16930793"
}

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