P02 · Packing and covering · Classic · Weak baseline

Equal-circle packing in a unit circle · n = 14

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

Instancen = 14

ObjectiveMaximize common radius

Proven optimum0.231030727optimality proved by Ekanayake and LaFountain (2024)

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

1234567891011121314

0x2

VERIFIED CONSTRUCTIONr = 0.231030727

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

0.231030727

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": [
    [
      "0.426517068",
      "1.512280263"
    ],
    [
      "1.768651589",
      "0.977898507"
    ],
    [
      "1.242932803",
      "0.270412860"
    ],
    [
      "1.617219354",
      "0.541356336"
    ],
    [
      "0.658542316",
      "0.903138033"
    ],
    [
      "0.398029103",
      "0.521518255"
    ],
    [
      "1.073743269",
      "0.700384592"
    ],
    [
      "1.320634857",
      "1.090955115"
    ],
    [
      "1.284453276",
      "1.714422896"
    ],
    [
      "0.780932503",
      "0.262895412"
    ],
    [
      "1.642553117",
      "1.422420683"
    ],
    [
      "0.905433904",
      "1.293708555"
    ],
    [
      "0.236448670",
      "1.091121393"
    ],
    [
      "0.823648230",
      "1.748474311"
    ]
  ],
  "radius": "0.231030727"
}

Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 14
}

The current leader's answer

{
  "centers": [
    [
      "0.426517068",
      "1.512280263"
    ],
    [
      "1.768651589",
      "0.977898507"
    ],
    [
      "1.242932803",
      "0.270412860"
    ],
    [
      "1.617219354",
      "0.541356336"
    ],
    [
      "0.658542316",
      "0.903138033"
    ],
    [
      "0.398029103",
      "0.521518255"
    ],
    [
      "1.073743269",
      "0.700384592"
    ],
    [
      "1.320634857",
      "1.090955115"
    ],
    [
      "1.284453276",
      "1.714422896"
    ],
    [
      "0.780932503",
      "0.262895412"
    ],
    [
      "1.642553117",
      "1.422420683"
    ],
    [
      "0.905433904",
      "1.293708555"
    ],
    [
      "0.236448670",
      "1.091121393"
    ],
    [
      "0.823648230",
      "1.748474311"
    ]
  ],
  "radius": "0.231030727"
}

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