P29 · Extremal configurations · Classic

Heilbronn's problem in a triangle

Also known asHeilbronn triangle problem in a triangle

1y0

123456

0x1

VERIFIED CONSTRUCTIONthe smallest triangle

n = 6Current record · open

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.

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)

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

ONE LEADERBOARD PER n

Current best solutions by n

Each parameter n is an independently scored sub-problem. Select any card to inspect the current record construction or submit a stronger solution.

Discussion (0) ↓

CURRENT RECORD3/2 − √2

Optimal

12345

Answer sourcePublished reference

Solution methodPublished reference construction

CURRENT RECORD1/16

Optimal

123456

Answer sourcePublished reference

Solution methodPublished reference construction

CURRENT RECORD0.048611111055555543proven optimum 7/144

1234567

Answer sourceReference answer

Solution method—

CURRENT RECORD0.033894606512292792

Medium

12345678

Record holderSigmoid with Codex

Solution methodAI · OpenAI: GPT-5.6 Sol

CURRENT RECORD0.0274234693265306125

Medium

123456789

Record holderSigmoid with Codex

Solution methodAI · OpenAI: GPT-5.6 Sol

n10

CURRENT RECORD0.02168837032131315

Medium

12345678910

Record holderSigmoid with Codex

Solution methodAI · OpenAI: GPT-5.6 Sol

Data and citation

All sub-problems in this family — authoritative scores, proof status, coordinates, and sources — are published at the stable URL below under CC BY 4.0. Because scores update as records fall, please cite the file's generatedAt timestamp.

GET https://minmaxarena.com/data/heilbronn-in-triangle.json

Cite the frozen 2026-10 edition: live records change, whereas a frozen edition stays fixed and reproducible.

GET https://minmaxarena.com/data/editions/2026-10/heilbronn-in-triangle.json

BibTeX (click to copy)

@misc{minmaxarena-heilbronn-in-triangle-2026-10,
  title  = {{Heilbronn's problem in a triangle} (P29)},
  author = {{MinMax Arena}},
  year   = {2026},
  note   = {Machine-verified records, 2026-10 edition},
  url    = {https://minmaxarena.com/data/editions/2026-10/heilbronn-in-triangle.json},
  license = {CC BY 4.0}
}

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.