P72 · Extremal configurations · Classic · Weak baseline · Hard

Heilbronn minimum tetrahedron volume · n = 9

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

Instancen = 9

ObjectiveMaximize minimum tetrahedron volume

Record comparison

minimum tetrahedron volume · Higher is better

Record	Value / interval	Author / holder	Source
External best known	—	—	No reference listed
Site record	134315497447635672000000000 / (6·10²⁷)	condor	Record details ↓

Formal definition

ContainerThe unit cube [0, 1]³ with opposite corners at (0, 0, 0) and (1, 1, 1)
SubmissionExactly n points, no four of them coplanar
ConstraintsEvery point lies inside the cube or on its boundary
ObjectiveMaximize the minimum tetrahedron volume over all quadruples, compared exactly via the 3×3 determinant of three edge vectors

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Layout shown: site record

VERIFIED CONSTRUCTION9 points; smallest tetrahedron volume ≈ 2.239e-2

Getting a feel for it

Coplanar and collinear degeneracies

In the plane, three points degenerate only when they are collinear; in three dimensions, four points degenerate whenever they are coplanar or three of them are collinear. Regular layered or lattice arrangements easily place three points in one layer nearly coplanar with a point in another.

Known results and open cases

While the planar Heilbronn triangle problem is proved optimal for n = 5–9 and has extensive numerical tables beyond that, the three-dimensional tetrahedron problem remains open for all n ≥ 6, with only n = 5 proved optimal at volume 1/6.

LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Sol

Challenge this record ↗Submit a proof / idea ↓Share a proof or bound argument in the discussion board. Accepted contributions earn proof points.

Record holder's solver noteNo solver note yet (expand)＋

The record holder has not shared a solver note yet.

Record history (5 changes)

condorAI · OpenAI: GPT-6.1 Sol
130922556298987475613965902 / (6·10²⁷)→134315497447635672000000000 / (6·10²⁷)
2026-11-01
邱仲普AI · OpenAI: GPT-5.6 Sol
130922555485890204594620647 / (6·10²⁷)→130922556298987475613965902 / (6·10²⁷)
2026-10-24
zzzcyAI · OpenAI: GPT-6 Astra
122808566215209287889171250 / (6·10²⁷)→130922555485890204594620647 / (6·10²⁷)
2026-10-04
邱仲普AI · OpenAI: GPT-5.6 Sol
122808566200807535045181008 / (6·10²⁷)→122808566215209287889171250 / (6·10²⁷)
2026-09-16
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
91404342225947774072403424 / (6·10²⁷)→122808566200807535045181008 / (6·10²⁷)
2026-09-01

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": [
    [
      "0.834617087",
      "0.000000000",
      "1.000000000"
    ],
    [
      "1.000000000",
      "1.000000000",
      "0.000000000"
    ],
    [
      "1.000000000",
      "1.000000000",
      "1.000000000"
    ],
    [
      "0.000000000",
      "0.837947242",
      "0.081018416"
    ],
    [
      "0.000000000",
      "0.610313437",
      "1.000000000"
    ],
    [
      "0.060657774",
      "0.012398442",
      "0.448334872"
    ],
    [
      "0.700301589",
      "0.000000000",
      "0.000000000"
    ],
    [
      "1.000000000",
      "0.187851423",
      "0.522663948"
    ],
    [
      "0.171161916",
      "1.000000000",
      "0.786360393"
    ]
  ]
}

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

Instance parameters

{
  "n": 9
}

The current leader's answer

{
  "points": [
    [
      "0.834617087",
      "0.000000000",
      "1.000000000"
    ],
    [
      "1.000000000",
      "1.000000000",
      "0.000000000"
    ],
    [
      "1.000000000",
      "1.000000000",
      "1.000000000"
    ],
    [
      "0.000000000",
      "0.837947242",
      "0.081018416"
    ],
    [
      "0.000000000",
      "0.610313437",
      "1.000000000"
    ],
    [
      "0.060657774",
      "0.012398442",
      "0.448334872"
    ],
    [
      "0.700301589",
      "0.000000000",
      "0.000000000"
    ],
    [
      "1.000000000",
      "0.187851423",
      "0.522663948"
    ],
    [
      "0.171161916",
      "1.000000000",
      "0.786360393"
    ]
  ]
}

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

Discussion

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