MinMax Arena
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P58 · Extremal configurations · Classic

The smallest triangle in an equilateral triangle · n = 5

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 = 5
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
Open the full editor ↗
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

0.0722506975

the smallest triangle's area

Record holderNUE_13
Solution methodHuman
Challenge this record ↗
Record history (1 changes)
  1. NUE_13Human
    0.035048740589117021→0.0722506975
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": [
    [
      "1.000000000",
      "0.000000000"
    ],
    [
      "0.671832551",
      "0.567644526"
    ],
    [
      "0.329961566",
      "0.567832418"
    ],
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "0.500000000",
      "0.144501395"
    ]
  ]
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 5
}

The current leader's answer

{
  "points": [
    [
      "1.000000000",
      "0.000000000"
    ],
    [
      "0.671832551",
      "0.567644526"
    ],
    [
      "0.329961566",
      "0.567832418"
    ],
    [
      "0.000000000",
      "0.000000000"
    ],
    [
      "0.500000000",
      "0.144501395"
    ]
  ]
}

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