P32 · Extremal configurations · Formed here

The smallest triangle in a quadrant

1y0

1234567

0x1

VERIFIED CONSTRUCTIONthe smallest triangle

n = 7Current record · open

Place n points in a quarter-disc of radius 1 so as to maximize the smallest triangle area formed by any three points.

Formal definition

ContainerA quarter-disc of radius 1 centered at the origin (0, 0), with straight edges along the axes from 0 to 1 and the arc in the first quadrant
SubmissionExactly n points, with no three collinear
ConstraintsEvery point lies inside the container 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

Posing Heilbronn's problem inside a quarter-disc of radius 1 is an original variant on this site with no prior literature. Every instance of n is open.

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 RECORD0.1501415529977804115

Medium

12345

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Sol

CURRENT RECORD0.103553390593154502

Medium

123456

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Sol

CURRENT RECORD0.074864898783171395

Medium

1234567

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Terra

CURRENT RECORD0.0531009072806833215

Medium

12345678

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Sol

CURRENT RECORD0.042266484924465307

Medium

123456789

Record holderzzzcy

Solution methodAI · OpenAI: GPT-6 Astra

n10

CURRENT RECORD0.03575195604793944

Medium

12345678910

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Sol

n11

CURRENT RECORD0.029657024229360187

Medium

1234567891011

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Sol

n12

CURRENT RECORD0.024920114270038166

CURRENT RECORD0.021931103524233824

Medium

12345678910111213

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Sol

n14

CURRENT RECORD0.019480944690694145

Medium

1234567891011121314

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 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-a-quadrant.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-a-quadrant.json

BibTeX (click to copy)

@misc{minmaxarena-heilbronn-in-a-quadrant-2026-10,
  title  = {{The smallest triangle in a quadrant} (P32)},
  author = {{MinMax Arena}},
  year   = {2026},
  note   = {Machine-verified records, 2026-10 edition},
  url    = {https://minmaxarena.com/data/editions/2026-10/heilbronn-in-a-quadrant.json},
  license = {CC BY 4.0}
}

DISCUSSION

Discussion

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