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P16 · Extremal configurations · Formed here

Spreading points in a right triangle · n = 15

Place n points inside a right isosceles triangle with legs 1, maximizing the smallest distance between any two of them.

Instancen = 15
ObjectiveMaximize the smallest distance between two points

Formal definition

  • ContainerThe container is a right isosceles triangle with legs of length 1: the right angle sits at the origin (0, 0) and the other vertices are (1, 0) and (0, 1).
  • 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 right isosceles triangle with legs 1 was posed here, and there is no literature for it. Every n is unstudied; the standing record is all anybody knows.

Current leader

0.2

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 right isosceles triangle with legs of length 1: the right angle sits at the origin (0, 0) and the other vertices are (1, 0) and (0, 1). 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"
    ],
    [
      "0.2",
      "0"
    ],
    [
      "0.4",
      "0"
    ],
    [
      "0.6",
      "0"
    ],
    [
      "0.8",
      "0"
    ],
    [
      "1",
      "0"
    ],
    [
      "0",
      "0.2"
    ],
    [
      "0.2",
      "0.2"
    ],
    [
      "0.4",
      "0.2"
    ],
    [
      "0.6",
      "0.2"
    ],
    [
      "0.8",
      "0.2"
    ],
    [
      "0",
      "0.4"
    ],
    [
      "0.2",
      "0.4"
    ],
    [
      "0.4",
      "0.4"
    ],
    [
      "0.6",
      "0.4"
    ]
  ]
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 15
}

The current leader's answer

{
  "points": [
    [
      "0",
      "0"
    ],
    [
      "0.2",
      "0"
    ],
    [
      "0.4",
      "0"
    ],
    [
      "0.6",
      "0"
    ],
    [
      "0.8",
      "0"
    ],
    [
      "1",
      "0"
    ],
    [
      "0",
      "0.2"
    ],
    [
      "0.2",
      "0.2"
    ],
    [
      "0.4",
      "0.2"
    ],
    [
      "0.6",
      "0.2"
    ],
    [
      "0.8",
      "0.2"
    ],
    [
      "0",
      "0.4"
    ],
    [
      "0.2",
      "0.4"
    ],
    [
      "0.4",
      "0.4"
    ],
    [
      "0.6",
      "0.4"
    ]
  ]
}

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