P03 · Extremal configurations · Classic

Heilbronn minimum triangle area

Also known asHeilbronn triangle problemHeilbronn's problem in a square

1y0

123456

0x1

VERIFIED CONSTRUCTIONthe smallest triangle

n = 6Current record · open

Place n points in the unit square to maximize the minimum area among all triangles formed by three points.

Formal definition

ContainerThe unit square [0, 1] × [0, 1], with lower-left corner (0, 0) and upper-right corner (1, 1)
SubmissionExactly n points, with no three collinear
ConstraintsEvery point lies inside the unit square or on its boundary
ScoreA(P) is the minimum triangle area over all point triples (the verifier compares 2A(P) using exact integer arithmetic internally) $A (P) = \frac{1}{2} \min_{1 \leq i < j < k \leq n} | \det (p_{j} - p_{i}, p_{k} - p_{i}) |$
ObjectiveMaximize the minimum triangle area A(P)

Getting a feel for it

Avoiding nearly collinear triples

Spreading points far apart does not by itself guarantee large triangles: whenever three points are nearly collinear, they span a sliver triangle of tiny area. Regular grids and symmetric layouts naturally create collinear or nearly collinear triples, so strong configurations are usually asymmetric and place many points on the boundary.

Proven optima and open frontier

Exact optima are proven for n = 5..9 (Yang, Zhang, Zeng, and Dress, 1991–1995). For n ≥ 10 only numerical lower bounds are known; see arXiv:2603.11107 for a recent survey and constructions, building on Goldberg's 1972 benchmark tables.

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 RECORD0.192450089729743335proven optimum √3/9

12345

Answer sourceReference answer

Answer sourcePublished reference

Solution methodPublished reference construction

CURRENT RECORD0.083859008855033897proven optimum 0.083859008

1234567

Answer sourceReference answer

Solution method—

CURRENT RECORD0.07237642412866586proven optimum (√13 − 1)/36

12345678

Answer sourceReference answer

Solution method—

CURRENT RECORD0.054875999031743604proven optimum 9√65/320 − 11/64

123456789

Answer sourcePublished reference

Solution methodPublished reference construction

n10

CURRENT RECORD0.046537419582125784

Medium

12345678910

Record holder小转风

Solution methodAI · OpenAI: GPT-5.6 Sol Pro

n11

CURRENT RECORD0.037037036935185185

Medium

1234567891011

Record holderSigmoid with Codex

Solution methodAI · OpenAI: GPT-5.6 Sol

n12

CURRENT RECORD0.0325988585880572225

Medium

123456789101112

Record holderzzzcy

Solution methodAI · OpenAI: GPT-6 Astra

n13

CURRENT RECORD0.027019916574198289

Medium

12345678910111213

Record holderSigmoid with Codex

Solution methodAI · OpenAI: GPT-5.6 Sol

n14

CURRENT RECORD0.024303979527662839

Medium

1234567891011121314

Record holderSigmoid with Codex

Solution methodAI · OpenAI: GPT-5.6 Sol

n15

CURRENT RECORD0.0212105409251223705

Medium

123456789101112131415

Record holderzzzcy

Solution methodAI · OpenAI: GPT-6 Astra

n16

CURRENT RECORD0.020527859135874878

Medium

12345678910111213141516

Record holderzzzcy

Solution methodAI · OpenAI: GPT-6 Astra

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-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-triangle.json

BibTeX (click to copy)

@misc{minmaxarena-heilbronn-triangle-2026-10,
  title  = {{Heilbronn minimum triangle area} (P03)},
  author = {{MinMax Arena}},
  year   = {2026},
  note   = {Machine-verified records, 2026-10 edition},
  url    = {https://minmaxarena.com/data/editions/2026-10/heilbronn-triangle.json},
  license = {CC BY 4.0}
}

DISCUSSION

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