P25 · Extremal configurations · Formed here

The smallest triangle in a plus sign

VERIFIED CONSTRUCTIONthe smallest triangle

Place n points inside a cross shape (a 3 × 3 square missing all four 1 × 1 corners) so that the smallest triangle formed by any three points has the largest possible area.

Formal definition

ContainerA cross-shaped region formed by removing all four 1 × 1 corners from a 3 × 3 square: the bounding box runs from (0, 0) at the lower-left to (3, 3) at the upper-right
SubmissionExactly n points, with no three collinear
ConstraintsEvery point must lie inside the container or on its boundary
ScoreA(P) is the minimum triangle area over all triples of points; internally, the verifier compares 2A(P) using 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

How to push the minimum area up

Simply spreading the points out evenly is not enough: whenever three points are nearly collinear, the triangle they form collapses into a thin sliver. Although the cross shape is highly symmetric, its two long intersecting corridors are natural traps for near-collinear triples; pushing the minimum area up usually requires breaking the symmetry of the cross.

Current status

Heilbronn's triangle problem inside a cross shape (a 3 × 3 square missing all four 1 × 1 corners) is an original variant introduced on this site, with no prior published literature. Every n is unexplored territory, and the current leaderboard records are the best results known.

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.

Record holderbowen2147483647

Solution methodHuman

CURRENT RECORD0.551127991

Medium

1234567

Record holder邱仲普

Solution methodAI · DeepSeek: DeepSeek V4 Pro 0423

CURRENT RECORD0.386736246100135666

Medium

123456789

Record holder邱仲普

Solution methodAI · OpenAI: GPT-5.6 Sol

n10

CURRENT RECORD0.346153846

Medium

12345678910

Record holderSigmoid with Codex

Solution methodAI · OpenAI: GPT-5.6 Sol

n11

CURRENT RECORD0.226341024449149533

Medium

1234567891011

Record holder邱仲普

Solution methodAI · OpenAI: GPT-5.6 Sol

n12

CURRENT RECORD0.194331691151725144

Medium

123456789101112

Record holder邱仲普

Solution methodAI · OpenAI: GPT-5.6 Sol

n13

CURRENT RECORD0.162665613549321623

Medium

12345678910111213

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Sol

n14

CURRENT RECORD0.1337600709190406705

Medium

1234567891011121314

Record holder邱仲普

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-a-cross.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-cross.json

BibTeX (click to copy)

@misc{minmaxarena-heilbronn-in-a-cross-2026-10,
  title  = {{The smallest triangle in a plus sign} (P25)},
  author = {{MinMax Arena}},
  year   = {2026},
  note   = {Machine-verified records, 2026-10 edition},
  url    = {https://minmaxarena.com/data/editions/2026-10/heilbronn-in-a-cross.json},
  license = {CC BY 4.0}
}

DISCUSSION

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