P25 · Extremal configurations · Formed here

The smallest triangle in a plus sign · n = 11

Place n points inside a plus sign (a 3 × 3 square with all four corner squares removed) so that the smallest triangle formed by any three of them is as large as possible.

Instancen = 11

ObjectiveMaximize the smallest triangle's area

Formal definition

ContainerThe container is a square of side 3 with all four of its 1 × 1 corners removed, leaving a plus sign: the origin (0, 0) is its lower-left corner and (3, 3) its upper-right.
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 ↗

3y0

1234567891011

0x3

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 a plus sign (a 3 × 3 square with all four corner squares removed) was posed here, and there is no literature for it. Every n is unstudied; the standing record is all anybody knows.

Current leader

0.013731971074830454

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 3 with all four of its 1 × 1 corners removed, leaving a plus sign: the origin (0, 0) is its lower-left corner and (3, 3) its upper-right. 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": [
    [
      "1.9",
      "1.5"
    ],
    [
      "1.836501413",
      "1.716256327"
    ],
    [
      "1.666166005",
      "1.863852798"
    ],
    [
      "1.443074065",
      "1.895928577"
    ],
    [
      "1.238055706",
      "1.80229983"
    ],
    [
      "1.116202811",
      "1.612693023"
    ],
    [
      "1.116202811",
      "1.387306977"
    ],
    [
      "1.238055706",
      "1.19770017"
    ],
    [
      "1.443074065",
      "1.104071423"
    ],
    [
      "1.666166005",
      "1.136147202"
    ],
    [
      "1.836501413",
      "1.283743673"
    ]
  ]
}

Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 11
}

The current leader's answer

{
  "points": [
    [
      "1.9",
      "1.5"
    ],
    [
      "1.836501413",
      "1.716256327"
    ],
    [
      "1.666166005",
      "1.863852798"
    ],
    [
      "1.443074065",
      "1.895928577"
    ],
    [
      "1.238055706",
      "1.80229983"
    ],
    [
      "1.116202811",
      "1.612693023"
    ],
    [
      "1.116202811",
      "1.387306977"
    ],
    [
      "1.238055706",
      "1.19770017"
    ],
    [
      "1.443074065",
      "1.104071423"
    ],
    [
      "1.666166005",
      "1.136147202"
    ],
    [
      "1.836501413",
      "1.283743673"
    ]
  ]
}

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