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P09 · Packing and covering · Classic

Equal circles in a half-disc · n = 5

Place n circles of one common radius inside a half-disc of radius 1, making that radius as large as possible.

Instancen = 5
ObjectiveMaximize common radius
Best known, unproven0.261203874value from Eckard Specht's table of best known packings, which does not mark this one as proved

Formal definition

  • ContainerThe container is a half-disc of radius 1: its diameter lies along y = 0 from (0, 0) to (2, 0), centred at (1, 0), with the arc above.
  • 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
0x2
VERIFIED CONSTRUCTIONr = 0.261203874

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

Every best known value comes from Specht's csc survey table, which marks none of them proven; all fourteen n are open.

Source
Current leader

0.261203874

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 half-disc of radius 1: its diameter lies along y = 0 from (0, 0) to (2, 0), centred at (1, 0), with the arc above. 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": [
    [
      "0.308919505",
      "0.261203875"
    ],
    [
      "1",
      "0.261203875"
    ],
    [
      "1.691080495",
      "0.261203875"
    ],
    [
      "0.654459753",
      "0.653009687"
    ],
    [
      "1.345540247",
      "0.653009687"
    ]
  ],
  "radius": "0.261203874"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 5
}

The current leader's answer

{
  "centers": [
    [
      "0.308919505",
      "0.261203875"
    ],
    [
      "1",
      "0.261203875"
    ],
    [
      "1.691080495",
      "0.261203875"
    ],
    [
      "0.654459753",
      "0.653009687"
    ],
    [
      "1.345540247",
      "0.653009687"
    ]
  ],
  "radius": "0.261203874"
}

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