The smallest triangle in an annulus
Place n points inside an annulus with outer radius 1 and inner radius 0.5 so that the smallest triangle formed by any three points has the largest possible area.
Formal definition
- ContainerAn annulus centered at (1, 1) with outer radius 1 and inner radius 0.5 (i.e. 0.5 ≤ ‖p − (1, 1)‖ ≤ 1, with the central disk of radius 0.5 removed)
- 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
- 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. In an annulus, points can sit along the outer rim or stagger against the inner boundary, but any chord across the ring that lines up three points collapses that triangle's area to zero.
Current status
Heilbronn's triangle problem inside an annulus with outer radius 1 and inner radius 0.5 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.
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) ↓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-an-annulus.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-an-annulus.json
BibTeX (click to copy)
@misc{minmaxarena-heilbronn-in-an-annulus-2026-10,
title = {{The smallest triangle in an annulus} (P28)},
author = {{MinMax Arena}},
year = {2026},
note = {Machine-verified records, 2026-10 edition},
url = {https://minmaxarena.com/data/editions/2026-10/heilbronn-in-an-annulus.json},
license = {CC BY 4.0}
}Discussion
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