Covering an equilateral triangle with n equal discs
Place n centres in a unit equilateral triangle. Every centre has the same covering radius; every point of the triangle must lie in at least one disc, and that radius is minimised.
Formal definition
- ContainerThe closed convex polygon with vertices (0,0), (1,0), and (0.5,0.866025404), the site's unambiguous nine-decimal representative of an equilateral triangle
- SubmissionExactly n distinct points inside the container
- ScoreTake the distance to the nearest centre at every place in the container, then the largest of those distances; rational Voronoi polygons make the verifier exact
- ObjectiveMake the covering radius as small as possible
Getting a feel for it
Corners and interior compete for centres
The three sharp corners demand attention, but pushing every centre outward leaves a hole in the middle. The best configurations are usually not a simple equally spaced triangular grid.
Published records and verifiable constructions are included
Nurmela (2000) collected and extended high-precision constructions for n=2–36. The site reconstructs n=7–36 from the contact graphs in Figures 2–4, showing both the paper's continuous value and the verifier-exact nine-decimal certificate. The n=9 and n=10 rows are proved optimal; the other n≤36 rows remain challengeable known bests. The paper explicitly reports no searches beyond n=36, so n=37–40 ship only transparent site starts obtained by filling successive farthest holes of n=36, without invented external records.
SourceCurrent best solutions by n
Each n is an independent record with a page of its own. Open any of them to inspect the current construction, then challenge it.
Discussion (0) ↓Data and citation
Every sub-problem in this family — authoritative scores, proof status, coordinates and sources — lives at the stable address below, published under CC BY 4.0. Scores move as records fall, so cite the generatedAt timestamp the file carries.
GET https://minmaxarena.com/data/circles-covering-an-equilateral-triangle.json
Cite the frozen 2026-08 edition: records move, a frozen edition never does, so the citation is still checkable years later.
GET https://minmaxarena.com/data/editions/2026-08/circles-covering-an-equilateral-triangle.json
BibTeX (click to copy)
@misc{minmaxarena-circles-covering-an-equilateral-triangle-2026-08,
title = {{Covering an equilateral triangle with n equal discs} (P74)},
author = {{MinMax Arena}},
year = {2026},
note = {Machine-verified records, 2026-08 edition},
url = {https://minmaxarena.com/data/editions/2026-08/circles-covering-an-equilateral-triangle.json},
license = {CC BY 4.0}
}Discussion
Talk strategy, share methods, ask why you are stuck. Posts carry your public byline, the same name your records use; the #number after it is the account's signup ordinal, so a name cannot be worn by someone else. The floor is earned: break a record once, anywhere, and it is yours for good. New posts appear after an automated review.
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