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P02 · Packing and covering · Classic · Weak baseline

Equal-circle packing in a unit circle · n = 26

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

Instancen = 26
ObjectiveMaximize common radius
Best known, unproven0.171580252value 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 = 0.171580251

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

0.171580251

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": [
    [
      "0.961757873",
      "0.172463404"
    ],
    [
      "1.300400405",
      "0.22796446"
    ],
    [
      "0.629677334",
      "0.258959919"
    ],
    [
      "1.587497026",
      "0.415939624"
    ],
    [
      "0.361140715",
      "0.47261201"
    ],
    [
      "0.948934049",
      "0.515384213"
    ],
    [
      "1.285992274",
      "0.579811574"
    ],
    [
      "0.642274296",
      "0.669393719"
    ],
    [
      "1.773784629",
      "0.704134108"
    ],
    [
      "0.202226398",
      "0.776758967"
    ],
    [
      "1.164371857",
      "0.900697233"
    ],
    [
      "1.506991337",
      "0.919959202"
    ],
    [
      "0.82621613",
      "0.959091033"
    ],
    [
      "0.48335998",
      "0.973540676"
    ],
    [
      "1.827297998",
      "1.043096423"
    ],
    [
      "0.183593989",
      "1.140572058"
    ],
    [
      "1.045864507",
      "1.222745583"
    ],
    [
      "1.412851223",
      "1.255815259"
    ],
    [
      "0.610371036",
      "1.292331083"
    ],
    [
      "1.738854736",
      "1.374663793"
    ],
    [
      "0.310605044",
      "1.459362465"
    ],
    [
      "1.199358394",
      "1.529663746"
    ],
    [
      "0.856204279",
      "1.531757791"
    ],
    [
      "1.523630883",
      "1.641942347"
    ],
    [
      "0.556685433",
      "1.699779357"
    ],
    [
      "1.029594832",
      "1.82789095"
    ]
  ],
  "radius": "0.171580251"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 26
}

The current leader's answer

{
  "centers": [
    [
      "0.961757873",
      "0.172463404"
    ],
    [
      "1.300400405",
      "0.22796446"
    ],
    [
      "0.629677334",
      "0.258959919"
    ],
    [
      "1.587497026",
      "0.415939624"
    ],
    [
      "0.361140715",
      "0.47261201"
    ],
    [
      "0.948934049",
      "0.515384213"
    ],
    [
      "1.285992274",
      "0.579811574"
    ],
    [
      "0.642274296",
      "0.669393719"
    ],
    [
      "1.773784629",
      "0.704134108"
    ],
    [
      "0.202226398",
      "0.776758967"
    ],
    [
      "1.164371857",
      "0.900697233"
    ],
    [
      "1.506991337",
      "0.919959202"
    ],
    [
      "0.82621613",
      "0.959091033"
    ],
    [
      "0.48335998",
      "0.973540676"
    ],
    [
      "1.827297998",
      "1.043096423"
    ],
    [
      "0.183593989",
      "1.140572058"
    ],
    [
      "1.045864507",
      "1.222745583"
    ],
    [
      "1.412851223",
      "1.255815259"
    ],
    [
      "0.610371036",
      "1.292331083"
    ],
    [
      "1.738854736",
      "1.374663793"
    ],
    [
      "0.310605044",
      "1.459362465"
    ],
    [
      "1.199358394",
      "1.529663746"
    ],
    [
      "0.856204279",
      "1.531757791"
    ],
    [
      "1.523630883",
      "1.641942347"
    ],
    [
      "0.556685433",
      "1.699779357"
    ],
    [
      "1.029594832",
      "1.82789095"
    ]
  ],
  "radius": "0.171580251"
}

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