P08 · Packing and covering · Formed here

Equal circles in an L · n = 10

Place n non-overlapping circles of one common radius inside the L-shaped region, making that radius as large as possible.

Instancen = 10

ObjectiveMaximize common radius

Formal definition

ContainerA 2 × 2 square with its top-right 1 × 1 removed: the origin (0, 0) at the lower-left; the notch is where x and y both exceed 1
SubmissionExactly n circles: one shared radius and n centres
ConstraintsEvery circle lies wholly inside the L and clear of the notch; no two overlap in their interiors, tangency allowed
ObjectiveMake the common radius as large as possible

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2y0

12345678910

0x2

VERIFIED CONSTRUCTIONr = 0.100000000

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

Our own variant: equal-circle packing in an L was posed here, and there is no literature for it. Every n is unstudied; the standing record is all anybody knows.

Current leader

0.1

common radius

Record holderFounding benchmark

Solution methodHuman

Challenge this record ↗

ANSWER FORMAT

How to write your answer

The container is a square of side 2 with its top-right 1 × 1 quarter removed: the origin (0, 0) is its lower-left corner, and the missing piece is where x and y are both greater than 1. 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": [
    [
      "0.1",
      "0.1"
    ],
    [
      "0.3",
      "0.1"
    ],
    [
      "0.5",
      "0.1"
    ],
    [
      "0.7",
      "0.1"
    ],
    [
      "0.9",
      "0.1"
    ],
    [
      "1.1",
      "0.1"
    ],
    [
      "1.3",
      "0.1"
    ],
    [
      "1.5",
      "0.1"
    ],
    [
      "1.7",
      "0.1"
    ],
    [
      "1.9",
      "0.1"
    ]
  ],
  "radius": "0.1"
}

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

Instance parameters

{
  "n": 10
}

The current leader's answer

{
  "centers": [
    [
      "0.1",
      "0.1"
    ],
    [
      "0.3",
      "0.1"
    ],
    [
      "0.5",
      "0.1"
    ],
    [
      "0.7",
      "0.1"
    ],
    [
      "0.9",
      "0.1"
    ],
    [
      "1.1",
      "0.1"
    ],
    [
      "1.3",
      "0.1"
    ],
    [
      "1.5",
      "0.1"
    ],
    [
      "1.7",
      "0.1"
    ],
    [
      "1.9",
      "0.1"
    ]
  ],
  "radius": "0.1"
}

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