P72 · Extremal configurations · Classic · Weak baseline · Hard

Heilbronn minimum tetrahedron volume · n = 13

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

Instancen = 13

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	42031861200989809152886000 / (6·10²⁷)	邱仲普	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 CONSTRUCTION13 points; smallest tetrahedron volume ≈ 7.005e-3

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 holder邱仲普

Solution methodAI · OpenAI: GPT-5.6 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 notePublic method and reproducibility notes (expand)＋

使用服务器同样的 1e-9 整数网格，在保持 13 点位于单位立方体且每个四点齐次行列式非零的约束下，对局部坐标执行单纯形精修；候选经独立整数重算后提交。

Read and reply in the discussion →

Record history (3 changes)

邱仲普AI · OpenAI: GPT-5.6 Sol
42031860822151782025885113 / (6·10²⁷)→42031861200989809152886000 / (6·10²⁷)
2026-10-23
zzzcyAI · OpenAI: GPT-6 Astra
28631611052145875562893048 / (6·10²⁷)→42031860822151782025885113 / (6·10²⁷)
2026-10-04
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
13883099940529356396277429 / (6·10²⁷)→28631611052145875562893048 / (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": [
    [
      "1.000000000",
      "0.261026993",
      "0.974167257"
    ],
    [
      "1.000000000",
      "0.189307456",
      "0.000000006"
    ],
    [
      "0.000000000",
      "1.000000000",
      "0.730174242"
    ],
    [
      "0.000000000",
      "0.034969232",
      "0.000000000"
    ],
    [
      "0.999616678",
      "0.789074801",
      "0.497403552"
    ],
    [
      "0.816905053",
      "0.903787956",
      "0.905630684"
    ],
    [
      "0.901271616",
      "0.000000000",
      "0.525359516"
    ],
    [
      "0.047046717",
      "0.072705265",
      "1.000000000"
    ],
    [
      "0.845451967",
      "1.000000000",
      "0.045940411"
    ],
    [
      "0.161794751",
      "0.000000000",
      "0.567787802"
    ],
    [
      "0.199352115",
      "0.744705249",
      "1.000000000"
    ],
    [
      "0.059571494",
      "0.560400026",
      "0.078783008"
    ],
    [
      "0.279693390",
      "0.993058005",
      "0.000000000"
    ]
  ]
}

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

Instance parameters

{
  "n": 13
}

The current leader's answer

{
  "points": [
    [
      "1.000000000",
      "0.261026993",
      "0.974167257"
    ],
    [
      "1.000000000",
      "0.189307456",
      "0.000000006"
    ],
    [
      "0.000000000",
      "1.000000000",
      "0.730174242"
    ],
    [
      "0.000000000",
      "0.034969232",
      "0.000000000"
    ],
    [
      "0.999616678",
      "0.789074801",
      "0.497403552"
    ],
    [
      "0.816905053",
      "0.903787956",
      "0.905630684"
    ],
    [
      "0.901271616",
      "0.000000000",
      "0.525359516"
    ],
    [
      "0.047046717",
      "0.072705265",
      "1.000000000"
    ],
    [
      "0.845451967",
      "1.000000000",
      "0.045940411"
    ],
    [
      "0.161794751",
      "0.000000000",
      "0.567787802"
    ],
    [
      "0.199352115",
      "0.744705249",
      "1.000000000"
    ],
    [
      "0.059571494",
      "0.560400026",
      "0.078783008"
    ],
    [
      "0.279693390",
      "0.993058005",
      "0.000000000"
    ]
  ]
}

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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