Covering an equilateral triangle with n equal discs
Place n centres in a unit equilateral triangle so that closed discs of equal radius centred at those points cover the entire triangle, minimizing their shared radius.
Formal definition
- ContainerThe closed convex polygon with vertices (0,0), (1,0), and (0.5,0.866025404), defining the unit equilateral triangle on the nine-decimal coordinate grid
- SubmissionExactly n distinct points inside the container
- ScoreThe maximum over the container of the distance to the nearest centre, computed exactly via rational Voronoi polygons
- ObjectiveMinimize the covering radius r(P)
Getting a feel for it
Balancing corners and interior
The three 60° corners pull centres outward, whereas pushing too many centres toward the boundary opens uncovered pockets in the interior. Outside special triangular numbers, optimal configurations depart from a uniform triangular grid.
Literature constructions and open cases
Nurmela (2000) compiled and extended high-precision constructions for n = 2–36. This benchmark reconstructs nine-decimal coordinate certificates for n = 7–36 from the contact graphs in Figures 2–4, displaying both the paper's continuous value and the certificate's exact score. Optimality is proved for n = 9 and n = 10, while the remaining n ≤ 36 cases represent best-known constructions; n = 37–40 extend beyond the published table with baseline initializations.
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 (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/circles-covering-an-equilateral-triangle.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/circles-covering-an-equilateral-triangle.json
BibTeX (click to copy)
@misc{minmaxarena-circles-covering-an-equilateral-triangle-2026-10,
title = {{Covering an equilateral triangle with n equal discs} (P74)},
author = {{MinMax Arena}},
year = {2026},
note = {Machine-verified records, 2026-10 edition},
url = {https://minmaxarena.com/data/editions/2026-10/circles-covering-an-equilateral-triangle.json},
license = {CC BY 4.0}
}Discussion
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