P30 · Packing and covering · Classic

Equal circles in a quadrant · n = 14

Place n circles of one common radius inside a quadrant (quarter-disc) of radius 1, making that radius as large as possible.

Instancen = 14

ObjectiveMaximize common radius

Best known, unproven0.115456141value from Eckard Specht's table of best known packings, which does not mark this one as proved

Formal definition

ContainerThe container is a quarter-disc of radius 1: centred at the origin (0, 0), with its straight edges along the axes from 0 to 1 and the arc in the first quadrant.
SubmissionExactly n circles: one shared radius and n centres; every circle uses the same radius
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 ↗

1y0

1234567891011121314

0x1

VERIFIED CONSTRUCTIONr = 0.115456141

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

The best known values come from Specht's ccq survey table, none proven; all fourteen n are open.

Source

Current leader

0.115456141

common radius

Matches the best known

Record holderFounding benchmark

Solution methodHuman

Challenge this record ↗

ANSWER FORMAT

How to write your answer

The container is a quarter-disc of radius 1: centred at the origin (0, 0), with its straight edges along the axes from 0 to 1 and the arc in the first quadrant. Coordinates and radii share one unit and are written as plain decimals such as "0.25", to at most nine decimal places.

Submit radius and centers. Write every number as a decimal string such as "0.25". Every circle shares one radius.

The current leader's answer

{
  "centers": [
    [
      "0.876976463",
      "0.115456142"
    ],
    [
      "0.408474176",
      "0.116853829"
    ],
    [
      "0.642166151",
      "0.117891909"
    ],
    [
      "0.116855661",
      "0.164627948"
    ],
    [
      "0.817211966",
      "0.33850028"
    ],
    [
      "0.340395803",
      "0.340398032"
    ],
    [
      "0.574110875",
      "0.341462243"
    ],
    [
      "0.116850398",
      "0.408476805"
    ],
    [
      "0.701755825",
      "0.538476183"
    ],
    [
      "0.341444765",
      "0.574093223"
    ],
    [
      "0.11787891",
      "0.64216962"
    ],
    [
      "0.538476183",
      "0.701755825"
    ],
    [
      "0.33850028",
      "0.817211966"
    ],
    [
      "0.115456142",
      "0.876976463"
    ]
  ],
  "radius": "0.115456141"
}

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": [
    [
      "0.876976463",
      "0.115456142"
    ],
    [
      "0.408474176",
      "0.116853829"
    ],
    [
      "0.642166151",
      "0.117891909"
    ],
    [
      "0.116855661",
      "0.164627948"
    ],
    [
      "0.817211966",
      "0.33850028"
    ],
    [
      "0.340395803",
      "0.340398032"
    ],
    [
      "0.574110875",
      "0.341462243"
    ],
    [
      "0.116850398",
      "0.408476805"
    ],
    [
      "0.701755825",
      "0.538476183"
    ],
    [
      "0.341444765",
      "0.574093223"
    ],
    [
      "0.11787891",
      "0.64216962"
    ],
    [
      "0.538476183",
      "0.701755825"
    ],
    [
      "0.33850028",
      "0.817211966"
    ],
    [
      "0.115456142",
      "0.876976463"
    ]
  ],
  "radius": "0.115456141"
}

Submit radius and centers. Write every number as a decimal string such as "0.25". Every circle shares one radius. · Verifier v1.0.0