P10 · Packing and covering · Formed here

Equal circles in a plus sign · n = 6

Place n circles of one common radius inside a plus sign (a 3 × 3 square with all four corner squares removed), making that radius as large as possible.

Instancen = 6

ObjectiveMaximize common radius

Formal definition

ContainerThe container is a square of side 3 with all four of its 1 × 1 corners removed, leaving a plus sign: the origin (0, 0) is its lower-left corner and (3, 3) its upper-right.
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 ↗

3y0

123456

0x3

VERIFIED CONSTRUCTIONr = 0.386972588

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 a plus sign (a 3 × 3 square with all four corner squares removed) was posed here, and there is no literature for it. Every n is unstudied; the standing record is all anybody knows.

Current leader

0.386972588

common radius

Record holderNUE_13

Solution methodHuman

Challenge this record ↗

Record history (1 changes)

NUE_13Human
0.24→0.386972588
2026-08-26

ANSWER FORMAT

How to write your answer

The container is a square of side 3 with all four of its 1 × 1 corners removed, leaving a plus sign: the origin (0, 0) is its lower-left corner and (3, 3) its upper-right. 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": [
    [
      "1.510434624",
      "0.493043519"
    ],
    [
      "1.866347348",
      "1.375478261"
    ],
    [
      "0.394217471",
      "1.398782489"
    ],
    [
      "1.134174233",
      "1.626304228"
    ],
    [
      "1.500000000",
      "2.501739171"
    ],
    [
      "2.605782529",
      "1.606130355"
    ]
  ],
  "radius": "0.386972588"
}

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

Instance parameters

{
  "n": 6
}

The current leader's answer

{
  "centers": [
    [
      "1.510434624",
      "0.493043519"
    ],
    [
      "1.866347348",
      "1.375478261"
    ],
    [
      "0.394217471",
      "1.398782489"
    ],
    [
      "1.134174233",
      "1.626304228"
    ],
    [
      "1.500000000",
      "2.501739171"
    ],
    [
      "2.605782529",
      "1.606130355"
    ]
  ],
  "radius": "0.386972588"
}

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