The smallest triangle in an equilateral triangle
Also known asHeilbronn triangle problem in an equilateral triangle
Place n points inside an equilateral triangle of side 1 so that the smallest triangle formed by any three of them is as large as possible.
Formal definition
- ContainerThe container is an equilateral triangle of side 1: its base runs from (0, 0) to (1, 0) and its apex is at (1/2, √3/2).
- SubmissionExactly n points, no three collinear
- ConstraintsEvery point lies inside the container or on its boundary
- ScoreA(P) is the smallest triangle area over all triples; the verifier compares 2A(P) exactly
- ObjectiveMake A(P) as large as possible
Getting a feel for it
Where the room for improvement is
Spreading points evenly is only part of the job, because any three nearly collinear points immediately collapse the area of the triangle they form. Regular grids naturally create collinear or near-collinear triples, so strong configurations usually break symmetry.
Where the frontier is
Affine maps between triangles preserve area ratios, so the classical Heilbronn optima for n = 5 and 6 carry over directly; those continuous cases are solved and archived, with nine-decimal grid notes retained. Case n = 11 includes a site-verifiable version of AlphaEvolve's published construction, while the remaining sizes are still open.
SourceCurrent 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 (1) ↓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-equilateral.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-equilateral.json
BibTeX (click to copy)
@misc{minmaxarena-heilbronn-in-equilateral-2026-10,
title = {{The smallest triangle in an equilateral triangle} (P58)},
author = {{MinMax Arena}},
year = {2026},
note = {Machine-verified records, 2026-10 edition},
url = {https://minmaxarena.com/data/editions/2026-10/heilbronn-in-equilateral.json},
license = {CC BY 4.0}
}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.
Grouped by player and sub-problem, showing the latest note first. Earlier versions and replies are preserved; click any note to read or reply. Notes can be published alongside record-breaking submissions.