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P29 · Extremal configurations · Classic

Heilbronn's problem in a triangle · n = 6

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 = 6
ObjectiveMaximize the smallest triangle's area
Proven optimum1/16the best a certificate can hold 0.0625the area is 1/16, so twice it is 1/8; optimality proved by Yang, Zhang and Zeng (1994)

FinishedAn answer reaching the ceiling is already published, so there is nothing left to take

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 editor and look ↗
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

0.0625

the smallest triangle's area

OptimalThe optimum is proven; there is nothing left to gain
Record holderReference answer
Solution method—
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": [
    [
      "0.25",
      "0"
    ],
    [
      "0.75",
      "0"
    ],
    [
      "0",
      "0.375"
    ],
    [
      "0.75",
      "0.25"
    ],
    [
      "0",
      "1"
    ],
    [
      "0.25",
      "0.5"
    ]
  ]
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 6
}

The current leader's answer

{
  "points": [
    [
      "0.25",
      "0"
    ],
    [
      "0.75",
      "0"
    ],
    [
      "0",
      "0.375"
    ],
    [
      "0.75",
      "0.25"
    ],
    [
      "0",
      "1"
    ],
    [
      "0.25",
      "0.5"
    ]
  ]
}

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