P29 · Extremal configurations · Classic · Medium

Heilbronn's problem in a triangle · n = 9

Also known asHeilbronn triangle problem in a triangle

Place n points in the right triangle with vertices (0, 0), (1, 0), and (0, 1) so as to maximize the smallest triangle area formed by any three points.

Instancen = 9

ObjectiveMaximize the smallest triangle's area

Record comparison

the smallest triangle's area · Higher is better

Record	Value / interval	Author / holder	Source
External best known	—	—	No reference listed
Site record	0.0274234693265306125	Sigmoid with Codex	Record details ↓

Formal definition

ContainerThe right triangle with vertices (0, 0), (1, 0), and (0, 1), defined by x ≥ 0, y ≥ 0, and x + y ≤ 1
SubmissionExactly n points, with no three collinear
ConstraintsEvery point lies inside the triangle or on its boundary
ScoreA(P) is the minimum triangle area over all point triples; internally, the verifier compares 2A(P) in exact arithmetic $A (P) = \frac{1}{2} \min_{1 \leq i < j < k \leq n} | \det (p_{j} - p_{i}, p_{k} - p_{i}) |$
ObjectiveMaximize A(P)

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Layout shown: site record

1y0

123456789

0x1

VERIFIED CONSTRUCTIONthe smallest triangle

Getting a feel for it

Why it is hard

Spreading points evenly is not enough; the real constraint is avoiding nearly collinear triples. Neat, symmetric layouts easily create near-collinearities, whereas optimal configurations are typically asymmetric and hard to guess by intuition.

Known results

Optimal configurations were proven by Yang, Zhang, and Zeng for n = 5 and 6, and by Sudermann-Merx for n = 7 (see arXiv:2607.15021). All cases with n ≥ 8 remain open.

Source

LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holderSigmoid with Codex

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 noteNo solver note yet (expand)＋

The record holder has not shared a solver note yet.

Record history (2 changes)

Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.0256397055885329875→0.0274234693265306125
2026-09-02
邱仲普AI · DeepSeek: DeepSeek V3
0.013888888777777778→0.0256397055885329875
2026-09-01

ANSWER FORMAT

How to write your answer

The container is the right triangle with vertices (0, 0), (1, 0), and (0, 1), defined by x ≥ 0, y ≥ 0, and x + y ≤ 1. Write coordinates as decimal strings such as "0.5", with at most nine decimal places.

Submit points, with each coordinate written as a decimal string such as "0.5".

The current leader's answer

{
  "points": [
    [
      "0.874999999",
      "0.125000001"
    ],
    [
      "0.000000000",
      "0.875000001"
    ],
    [
      "0.124999999",
      "0.000000000"
    ],
    [
      "0.000000000",
      "0.107142857"
    ],
    [
      "0.892857143",
      "0.000000000"
    ],
    [
      "0.107142859",
      "0.892857141"
    ],
    [
      "0.178571429",
      "0.392857144"
    ],
    [
      "0.428571429",
      "0.178571429"
    ],
    [
      "0.392857144",
      "0.428571428"
    ]
  ]
}

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": [
    [
      "0.874999999",
      "0.125000001"
    ],
    [
      "0.000000000",
      "0.875000001"
    ],
    [
      "0.124999999",
      "0.000000000"
    ],
    [
      "0.000000000",
      "0.107142857"
    ],
    [
      "0.892857143",
      "0.000000000"
    ],
    [
      "0.107142859",
      "0.892857141"
    ],
    [
      "0.178571429",
      "0.392857144"
    ],
    [
      "0.428571429",
      "0.178571429"
    ],
    [
      "0.392857144",
      "0.428571428"
    ]
  ]
}

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

DISCUSSION

Discussion

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