P72 · Extremal configurations · Classic · Weak baseline · Hard

Heilbronn minimum tetrahedron volume · n = 5

Place n points in the unit cube so that the smallest tetrahedron formed by any four of them is as large as possible.

Instancen = 5

ObjectiveMaximize minimum tetrahedron volume

Formal definition

ContainerThe unit cube: the origin (0, 0, 0) at one corner, (1, 1, 1) at the opposite one
SubmissionExactly n points, no four of them coplanar
ConstraintsEvery point lies inside the cube or on its boundary
ObjectiveMake the smallest tetrahedron over all quadruples as large as possible; compared internally by six times the volume, which is the determinant of the three edge vectors, exactly

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VERIFIED CONSTRUCTION5 points; the smallest tetrahedron has volume ≈ 4.883e-4

Getting a feel for it

Why this is not the plane problem with a z

In the plane three points degenerate one way: they line up. In space four points degenerate two ways — they share a plane, or three of them already share a line. Tidy arrangements walk straight into the first: any neat layering puts three points of one layer nearly in a plane with a point of another.

Where the frontier is

The plane version is proved optimal for n = 5..9 and has long-maintained numerical tables beyond that. Space has no comparable table: even small n are open, published constructions are scattered, and this arena cites no external value for any row. Every score here is a site record.

Current leader

2929687500000000000000000 / (6·10²⁷)

minimum tetrahedron volume

Record holderFounding benchmark

Solution methodHuman

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ANSWER FORMAT

How to write your answer

The container is a cube of side 1, with one corner at the origin (0, 0, 0) and the opposite corner at (1, 1, 1). Coordinates are plain decimals such as "0.5", to at most nine places.

Submit points: exactly n triples [x,y,z], each coordinate a decimal string in [0,1] with at most nine places.

The current leader's answer

{
  "points": [
    [
      "0",
      "0",
      "0"
    ],
    [
      "0.25",
      "0.0625",
      "0.015625"
    ],
    [
      "0.5",
      "0.25",
      "0.125"
    ],
    [
      "0.75",
      "0.5625",
      "0.421875"
    ],
    [
      "1",
      "1",
      "1"
    ]
  ]
}

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": [
    [
      "0",
      "0",
      "0"
    ],
    [
      "0.25",
      "0.0625",
      "0.015625"
    ],
    [
      "0.5",
      "0.25",
      "0.125"
    ],
    [
      "0.75",
      "0.5625",
      "0.421875"
    ],
    [
      "1",
      "1",
      "1"
    ]
  ]
}

Submit points: exactly n triples [x,y,z], each coordinate a decimal string in [0,1] with at most nine places. · Verifier v1.0.0

DISCUSSION

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