P24 · Extremal configurations · Formed here

The smallest triangle in an L · n = 14

Place n points inside an L (a 2 × 2 square with its top-right 1 × 1 removed) so that the smallest triangle formed by any three of them is as large as possible.

Instancen = 14

ObjectiveMaximize the smallest triangle's area

Formal definition

ContainerThe container is a square of side 2 with its top-right 1 × 1 quarter removed: the origin (0, 0) is its lower-left corner and the missing piece is where x and y are both greater than 1.
SubmissionExactly n points, no three collinear
ConstraintsEvery point lies inside the container 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 ↗

2y0

1234567891011121314

0x2

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

Our own variant: Heilbronn's problem in an L (a 2 × 2 square with its top-right 1 × 1 removed) was posed here, and there is no literature for it. Every n is unstudied; the standing record is all anybody knows.

Current leader

0.006874879632653493

the smallest triangle's area

Record holderFounding benchmark

Solution methodHuman

Challenge this record ↗

ANSWER FORMAT

How to write your answer

The container is a square of side 2 with its top-right 1 × 1 quarter removed: the origin (0, 0) is its lower-left corner and the missing piece is where x and y are both greater than 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": [
    [
      "0.9",
      "0.5"
    ],
    [
      "0.860387547",
      "0.673553496"
    ],
    [
      "0.749395921",
      "0.812732593"
    ],
    [
      "0.589008374",
      "0.889971165"
    ],
    [
      "0.410991626",
      "0.889971165"
    ],
    [
      "0.250604079",
      "0.812732593"
    ],
    [
      "0.139612453",
      "0.673553496"
    ],
    [
      "0.1",
      "0.5"
    ],
    [
      "0.139612453",
      "0.326446504"
    ],
    [
      "0.250604079",
      "0.187267407"
    ],
    [
      "0.410991626",
      "0.110028835"
    ],
    [
      "0.589008374",
      "0.110028835"
    ],
    [
      "0.749395921",
      "0.187267407"
    ],
    [
      "0.860387547",
      "0.326446504"
    ]
  ]
}

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": [
    [
      "0.9",
      "0.5"
    ],
    [
      "0.860387547",
      "0.673553496"
    ],
    [
      "0.749395921",
      "0.812732593"
    ],
    [
      "0.589008374",
      "0.889971165"
    ],
    [
      "0.410991626",
      "0.889971165"
    ],
    [
      "0.250604079",
      "0.812732593"
    ],
    [
      "0.139612453",
      "0.673553496"
    ],
    [
      "0.1",
      "0.5"
    ],
    [
      "0.139612453",
      "0.326446504"
    ],
    [
      "0.250604079",
      "0.187267407"
    ],
    [
      "0.410991626",
      "0.110028835"
    ],
    [
      "0.589008374",
      "0.110028835"
    ],
    [
      "0.749395921",
      "0.187267407"
    ],
    [
      "0.860387547",
      "0.326446504"
    ]
  ]
}

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