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P29 · Extremal configurations · Classic

Heilbronn's problem in a triangle · n = 9

Place n points inside the right triangle with vertices (0, 0), (1, 0) and (0, 1), maximizing the smallest triangle formed by any three of them.

Instancen = 9
ObjectiveMaximize the smallest triangle's area

Formal definition

  • ContainerA right triangle with vertices (0, 0), (1, 0) and (0, 1): the region x ≥ 0, y ≥ 0, x + y ≤ 1
  • SubmissionExactly n points, no three collinear
  • ConstraintsEvery point lies inside the triangle or on its boundary
  • ObjectiveMake the smallest triangle over all triples as large as possible; compared internally by twice the area, exactly
Open the full editor ↗
1y0
0x1
VERIFIED CONSTRUCTIONthe smallest triangle

Getting a feel for it

Where the room for improvement is

Even spreading is not enough: no three points may come close to collinear, and near-collinearity is exactly what tidy arrangements love to do. Optima are often asymmetric and hard even to guess.

Where the frontier is

n = 5 and 6 were proven by Yang, Zhang and Zeng, n = 7 by Sudermann-Merx — see arXiv:2607.15021; everything from n = 8 up is open.

Source
Current leader

0.013888888777777778

the smallest triangle's area

Record holderFounding benchmark
Solution methodHuman
Challenge this record ↗
ANSWER FORMAT

How to write your answer

The container is the right triangle with vertices (0, 0), (1, 0) and (0, 1): its interior is exactly x ≥ 0, y ≥ 0, x + y ≤ 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",
      "0.333333332"
    ],
    [
      "0",
      "0.83333333"
    ],
    [
      "0.166666666",
      "0.166666666"
    ],
    [
      "0.166666666",
      "0.499999998"
    ],
    [
      "0.333333332",
      "0.333333332"
    ],
    [
      "0.333333332",
      "0.499999998"
    ],
    [
      "0.499999998",
      "0"
    ],
    [
      "0.83333333",
      "0.166666666"
    ],
    [
      "0.999999996",
      "0"
    ]
  ]
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 9
}

The current leader's answer

{
  "points": [
    [
      "0",
      "0.333333332"
    ],
    [
      "0",
      "0.83333333"
    ],
    [
      "0.166666666",
      "0.166666666"
    ],
    [
      "0.166666666",
      "0.499999998"
    ],
    [
      "0.333333332",
      "0.333333332"
    ],
    [
      "0.333333332",
      "0.499999998"
    ],
    [
      "0.499999998",
      "0"
    ],
    [
      "0.83333333",
      "0.166666666"
    ],
    [
      "0.999999996",
      "0"
    ]
  ]
}

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