P21 · Extremal configurations · Formed here

Spreading points in a plus sign · n = 14

Place n points inside a plus sign (a 3 × 3 square with all four corner squares removed), maximizing the smallest distance between any two of them.

Instancen = 14

ObjectiveMaximize the smallest distance between two points

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 points, no two coinciding
ConstraintsEvery point lies inside the container or on its boundary
ObjectiveMake the smallest pairwise distance as large as possible; compared internally by its square, exactly

Open the full editor ↗

3y0

1234567891011121314

0x3

VERIFIED CONSTRUCTIONthe closest pair

Getting a feel for it

Where the room for improvement is

Spreading points IS packing equal circles: discs of half the minimum distance around each point must not overlap. Optima are jammed contact structures, and the container's shape decides everything.

Where the frontier is

Our own variant: point spreading 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

0.5

the smallest distance between two points

Record holderFounding benchmark

Solution methodHuman

Challenge this record ↗

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 are written as plain decimals such as "0.5", to at most nine decimal places.

Submit points, each coordinate written as a decimal string such as "0.5".

The current leader's answer

{
  "points": [
    [
      "1",
      "0"
    ],
    [
      "1.5",
      "0"
    ],
    [
      "2",
      "0"
    ],
    [
      "1",
      "0.5"
    ],
    [
      "1.5",
      "0.5"
    ],
    [
      "2",
      "0.5"
    ],
    [
      "0",
      "1"
    ],
    [
      "0.5",
      "1"
    ],
    [
      "1",
      "1"
    ],
    [
      "1.5",
      "1"
    ],
    [
      "2",
      "1"
    ],
    [
      "2.5",
      "1"
    ],
    [
      "3",
      "1"
    ],
    [
      "0",
      "1.5"
    ]
  ]
}

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

Instance parameters

{
  "n": 14
}

The current leader's answer

{
  "points": [
    [
      "1",
      "0"
    ],
    [
      "1.5",
      "0"
    ],
    [
      "2",
      "0"
    ],
    [
      "1",
      "0.5"
    ],
    [
      "1.5",
      "0.5"
    ],
    [
      "2",
      "0.5"
    ],
    [
      "0",
      "1"
    ],
    [
      "0.5",
      "1"
    ],
    [
      "1",
      "1"
    ],
    [
      "1.5",
      "1"
    ],
    [
      "2",
      "1"
    ],
    [
      "2.5",
      "1"
    ],
    [
      "3",
      "1"
    ],
    [
      "0",
      "1.5"
    ]
  ]
}

Submit points, each coordinate written as a decimal string such as "0.5". · Verifier v1.0.0