MinMax Arena
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P58 · Extremal configurations · Classic

The smallest triangle in an equilateral triangle · n = 6

Place n points inside an equilateral triangle of side 1 so that the smallest triangle formed by any three of them is as large as possible.

Instancen = 6
ObjectiveMaximize the smallest triangle's area

Formal definition

  • ContainerThe container is an equilateral triangle of side 1: its base runs from (0, 0) to (1, 0) and its apex is at (1/2, √3/2).
  • 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
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0.866025404y0
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

The equilateral version was pushed by AlphaEvolve's large-scale mathematical discovery runs: its n = 11 configuration has been reproduced but never beaten by the agents on EinsteinArena, and no per-n optimum is proven.

Source
Current leader

0.023094010484529947

the smallest triangle's area

Record holderFounding benchmark
Solution methodHuman
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ANSWER FORMAT

How to write your answer

The container is an equilateral triangle of side 1: its base runs from (0, 0) to (1, 0) and its apex is at (1/2, √3/2). 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": [
    [
      "0.730940107",
      "0.288675134"
    ],
    [
      "0.615470054",
      "0.488675133"
    ],
    [
      "0.384529947",
      "0.488675133"
    ],
    [
      "0.269059893",
      "0.288675134"
    ],
    [
      "0.384529946",
      "0.088675135"
    ],
    [
      "0.615470054",
      "0.088675135"
    ]
  ]
}
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": [
    [
      "0.730940107",
      "0.288675134"
    ],
    [
      "0.615470054",
      "0.488675133"
    ],
    [
      "0.384529947",
      "0.488675133"
    ],
    [
      "0.269059893",
      "0.288675134"
    ],
    [
      "0.384529946",
      "0.088675135"
    ],
    [
      "0.615470054",
      "0.088675135"
    ]
  ]
}

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