The smallest triangle in an L
Place n points inside an L-shape (a 2 × 2 square missing its top-right 1 × 1 corner) so that the smallest triangle formed by any three points has the largest possible area.
Formal definition
- ContainerAn L-shaped region formed by removing the top-right 1 × 1 square from a 2 × 2 square: the lower-left corner is at (0, 0), and the removed region is where x > 1 and y > 1
- 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 L-shaped region where two corridors meet at a corner, lining points up neatly along the straight edges or diagonals almost inevitably creates near-collinear triples, so high-scoring configurations are typically completely asymmetric.
Current status
Heilbronn's triangle problem inside an L-shape (a 2 × 2 square missing its top-right 1 × 1 corner) 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-l.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-l.json
BibTeX (click to copy)
@misc{minmaxarena-heilbronn-in-an-l-2026-10,
title = {{The smallest triangle in an L} (P24)},
author = {{MinMax Arena}},
year = {2026},
note = {Machine-verified records, 2026-10 edition},
url = {https://minmaxarena.com/data/editions/2026-10/heilbronn-in-an-l.json},
license = {CC BY 4.0}
}Discussion
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