P22 · Extremal configurations · Classic

The smallest triangle in a disc

Also known asHeilbronn triangle problem in a discHeilbronn's problem in a circle

2y0

1234567

0x2

VERIFIED CONSTRUCTIONthe smallest triangle

n = 7Current record · open

Place n points inside a disk of radius 1 so that the smallest triangle formed by any three points has the largest possible area.

Formal definition

ContainerA circle of radius 1 centered at (1, 1), with both coordinates in [0, 2]
SubmissionExactly n points, with no three collinear
ConstraintsEvery point lies inside the circle 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

Boundary polygons and symmetry breaking

For very small n, placing all points at the vertices of a regular n-gon on the boundary avoids near-collinearities; as n grows, however, triangles formed by three consecutive boundary points shrink at rate 1/n³, forcing some points into the interior at carefully staggered angles.

Literature benchmarks and open frontier

Known benchmark values for the disk Heilbronn problem (including closed algebraic forms for several small n) are compiled on Wolfram MathWorld, though rigorous optimality proofs analogous to those for the unit square have not yet been established.

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 RECORD0.657163890086059118best known √(5(5 − √5)/2)/4

Hard

12345

Record holderSigmoid with Codex

Solution methodAI · OpenAI: GPT-5.6 Sol

CURRENT RECORD0.433012701891298228best known √3/4

Hard

123456

Record holderSigmoid with Codex

Solution methodAI · OpenAI: GPT-5.6 Sol

CURRENT RECORD0.2943675263010013505best known [0.2943675263771180045, 0.2943675263771180055]

Hard

1234567

Record holderSigmoid with Codex

Solution methodAI · OpenAI: GPT-5.6 Sol

CURRENT RECORD0.216941869489467186best known [0.216941869558779055, 0.216941869558779065]

Hard

12345678

Record holderSigmoid with Codex

Solution methodAI · OpenAI: GPT-5.6 Sol

CURRENT RECORD0.1737637524206077925best known [0.173763748337212, 0.173763752508585]

Matches the best known

123456789

Record holderSigmoid with Codex

Solution methodAI · OpenAI: GPT-5.6 Sol

n10

CURRENT RECORD0.1503837331060307best known [0.15038373318043529645, 0.15038373318043529655]

Hard

12345678910

Record holderSigmoid with Codex

Solution methodAI · OpenAI: GPT-5.6 Sol

n11

CURRENT RECORD0.113938117339945312best known [0.113, 0.114)

Matches the best known

1234567891011

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Sol

n12

CURRENT RECORD0.1049153827599845435best known [0.104915382696098, 0.104915382840488]

Matches the best known

123456789101112

Record holderSigmoid with Codex

Solution methodAI · OpenAI: GPT-5.6 Sol

n13

CURRENT RECORD0.085689772555098089best known [0.0856, 0.0857)

Matches the best known

12345678910111213

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Sol

n14

CURRENT RECORD0.0767158855873489425best known [0.0758572510609005, 0.0758572510609015]

Numerically ahead (under review)

1234567891011121314

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-in-a-circle.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-circle.json

BibTeX (click to copy)

@misc{minmaxarena-heilbronn-in-a-circle-2026-10,
  title  = {{The smallest triangle in a disc} (P22)},
  author = {{MinMax Arena}},
  year   = {2026},
  note   = {Machine-verified records, 2026-10 edition},
  url    = {https://minmaxarena.com/data/editions/2026-10/heilbronn-in-a-circle.json},
  license = {CC BY 4.0}
}

DISCUSSION

Discussion

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