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P02 · Packing and covering · Classic · Weak baseline

Equal-circle packing in a unit circle · n = 18

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

Instancen = 18
ObjectiveMaximize common radius
Best known, unproven1/(1 + √2 + √6)the best a certificate can hold 0.205604646r = 1/(1 + √2 + √6), due to Pirl (1969); believed optimal but not provedSubmissions are decimal, so a record chases the writable value. If you believe you have the exact closed form, send it in. minmaxarena.com@gmail.com

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 = 0.205604646

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

0.205604646

common radius

Matches the best known
Record holderFounding benchmark
Solution methodHuman
Challenge this record ↗
Record history (6 changes)
  1. AnonymousHuman
    0.197748799→0.205604646
  2. lirdHuman
    0.195293898→0.197748799
  3. AnonymousHuman
    0.193242090→0.195293898
  4. AnonymousHuman
    0.188658982→0.193242090
  5. AnonymousHuman
    0.178838479→0.188658982
  6. AnonymousHuman
    0.069459271→0.178838479
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": [
    [
      "0.794395353",
      "0.232673012"
    ],
    [
      "1.205604647",
      "0.232673012"
    ],
    [
      "0.438277659",
      "0.438277659"
    ],
    [
      "1.561722341",
      "0.438277659"
    ],
    [
      "0.794395353",
      "0.643882306"
    ],
    [
      "1.205604647",
      "0.643882306"
    ],
    [
      "0.232673012",
      "0.794395353"
    ],
    [
      "1.767326988",
      "0.794395353"
    ],
    [
      "0.588790706",
      "1"
    ],
    [
      "1.411209294",
      "1"
    ],
    [
      "0.232673012",
      "1.205604647"
    ],
    [
      "1.767326988",
      "1.205604647"
    ],
    [
      "0.794395353",
      "1.356117694"
    ],
    [
      "1.205604647",
      "1.356117694"
    ],
    [
      "0.438277659",
      "1.561722341"
    ],
    [
      "1.561722341",
      "1.561722341"
    ],
    [
      "0.794395353",
      "1.767326988"
    ],
    [
      "1.205604647",
      "1.767326988"
    ]
  ],
  "radius": "0.205604646"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 18
}

The current leader's answer

{
  "centers": [
    [
      "0.794395353",
      "0.232673012"
    ],
    [
      "1.205604647",
      "0.232673012"
    ],
    [
      "0.438277659",
      "0.438277659"
    ],
    [
      "1.561722341",
      "0.438277659"
    ],
    [
      "0.794395353",
      "0.643882306"
    ],
    [
      "1.205604647",
      "0.643882306"
    ],
    [
      "0.232673012",
      "0.794395353"
    ],
    [
      "1.767326988",
      "0.794395353"
    ],
    [
      "0.588790706",
      "1"
    ],
    [
      "1.411209294",
      "1"
    ],
    [
      "0.232673012",
      "1.205604647"
    ],
    [
      "1.767326988",
      "1.205604647"
    ],
    [
      "0.794395353",
      "1.356117694"
    ],
    [
      "1.205604647",
      "1.356117694"
    ],
    [
      "0.438277659",
      "1.561722341"
    ],
    [
      "1.561722341",
      "1.561722341"
    ],
    [
      "0.794395353",
      "1.767326988"
    ],
    [
      "1.205604647",
      "1.767326988"
    ]
  ],
  "radius": "0.205604646"
}

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