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

Equal-circle packing in a unit circle · n = 4

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

Instancen = 4
ObjectiveMaximize common radius
Proven optimum√2 − 1the best a certificate can hold 0.414213562a square, r = √2 − 1

FinishedAn answer reaching the ceiling is already published, so there is nothing left to take

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 editor and look ↗
2y0
0x2
VERIFIED CONSTRUCTIONr = 0.414213562

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

0.414213562

common radius

OptimalThe optimum is proven; there is nothing left to gain
Record holderReference answer
Solution method—
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": [
    [
      "1.585786438",
      "1"
    ],
    [
      "1",
      "1.585786438"
    ],
    [
      "0.414213562",
      "1"
    ],
    [
      "1",
      "0.414213562"
    ]
  ],
  "radius": "0.414213562"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 4
}

The current leader's answer

{
  "centers": [
    [
      "1.585786438",
      "1"
    ],
    [
      "1",
      "1.585786438"
    ],
    [
      "0.414213562",
      "1"
    ],
    [
      "1",
      "0.414213562"
    ]
  ],
  "radius": "0.414213562"
}

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