P29 · Extremal configurations · Classic

Heilbronn's problem in a triangle · n = 7

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 = 7

ObjectiveMaximize the smallest triangle's area

Record comparison

the smallest triangle's area · Higher is better

Record	Value / interval	Author / holder	Source
Proven optimum	7/144	See source attribution	arxiv.org ↗
Site record	0.048611111055555543	Reference answer	Record details ↓

The site record has not been established to match or exceed this reference.

FinishedA construction reaching the theoretical ceiling is already published, leaving nothing further to improve

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)

Inspect in editor ↗

Layout shown: site record

1y0

1234567

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.

Answer sourceReference answer

Solution method—

The optimum is proven; no further improvement is possibleSubmit 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 (1 changes)

Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.0486111109722222225→0.048611111055555543
2026-09-02

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": [
    [
      "0.166666668",
      "0.000000000"
    ],
    [
      "0.750000003",
      "0.000000000"
    ],
    [
      "0.000000000",
      "0.250000000"
    ],
    [
      "0.833333332",
      "0.166666668"
    ],
    [
      "0.000000000",
      "0.833333328"
    ],
    [
      "0.249999997",
      "0.750000003"
    ],
    [
      "0.333333332",
      "0.333333334"
    ]
  ]
}

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

Instance parameters

{
  "n": 7
}

The current leader's answer

{
  "points": [
    [
      "0.166666668",
      "0.000000000"
    ],
    [
      "0.750000003",
      "0.000000000"
    ],
    [
      "0.000000000",
      "0.250000000"
    ],
    [
      "0.833333332",
      "0.166666668"
    ],
    [
      "0.000000000",
      "0.833333328"
    ],
    [
      "0.249999997",
      "0.750000003"
    ],
    [
      "0.333333332",
      "0.333333334"
    ]
  ]
}

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

DISCUSSION

Discussion

Discuss strategies, share construction methods, or ask questions. Any signed-in user can post under their public byline and unique #UID. New posts appear after moderation.

No threads yet. Start the first discussion.