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P31 · Extremal configurations · Formed here

Spreading points in a quadrant · n = 8

Place n points inside a quadrant (quarter-disc) of radius 1, maximizing the smallest distance between any two of them.

Instancen = 8
ObjectiveMaximize the smallest distance between two points

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 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 ↗
1y0
0x1
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 quadrant (quarter-disc) of radius 1 was posed here, and there is no literature for it. Every n is unstudied; the standing record is all anybody knows.

Current leader

0.25

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 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 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": [
    [
      "0",
      "0"
    ],
    [
      "0.25",
      "0"
    ],
    [
      "0.5",
      "0"
    ],
    [
      "0.75",
      "0"
    ],
    [
      "1",
      "0"
    ],
    [
      "0",
      "0.25"
    ],
    [
      "0.25",
      "0.25"
    ],
    [
      "0.5",
      "0.25"
    ]
  ]
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 8
}

The current leader's answer

{
  "points": [
    [
      "0",
      "0"
    ],
    [
      "0.25",
      "0"
    ],
    [
      "0.5",
      "0"
    ],
    [
      "0.75",
      "0"
    ],
    [
      "1",
      "0"
    ],
    [
      "0",
      "0.25"
    ],
    [
      "0.25",
      "0.25"
    ],
    [
      "0.5",
      "0.25"
    ]
  ]
}

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