Covering a disc with n equilateral triangles
Continuously cover the unit disc with n congruent equilateral triangles. Overlap and protrusion are allowed; minimise the common side.
Formal definition
- ContainerThe closed disc D = {(x,y) : (x−1)² + (y−1)² ≤ 1}
- SubmissionA common side s and exactly n {x,y,turn,down} placements, each independently rotatable and flippable
- CoverEvery point of D belongs to at least one closed covering triangle
- Exact continuous verificationThe 3n edges are extended to lines; on each face of that arrangement every triangle's membership is constant, so the disc is covered exactly when no face meeting the open disc is empty. Faces are sampled as trapezoids in vertical strips, all arithmetic in Q(√3); the circle enters only as whether a trapezoid meets the open disc, which is a rational comparison. No pixel sampling and no epsilon anywhere
- ScoreThe side s shared by every covering triangle; smaller is better
- ObjectiveMinimise s over all legal covers
Getting a feel for it
Six fit exactly; the problem starts at seven
Six equilateral triangles around the centre make a regular hexagon whose incircle is exactly the unit disc, side 2/√3, with nothing to spare. From the seventh piece on there is no ready answer to where the extra one goes and how the others make room.
Where the frontier is
Friedman's table records n = 1–18 in the reciprocal normalisation: the largest covered-disc radius r for n unit triangles, so the side here is s = 1/r. n = 8–13 have closed forms (Morandi, Cantrell); n = 7 and 14–18 are truncated decimals (Morandi, 2009), and nothing beyond n = 1, 2 and 6 is proved optimal. The source gives only approximately 200-pixel figures and no coordinates. This site reconstructed each pictured set of horizontal and ±60° edges and its contact topology, then used the exact continuous verifier to remove raster-rounding slivers. The literature target and the nine-decimal reconstruction remain separate, so reconstruction error is never presented as the authors' data.
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/equilateral-triangles-covering-a-disc.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/equilateral-triangles-covering-a-disc.json
BibTeX (click to copy)
@misc{minmaxarena-equilateral-triangles-covering-a-disc-2026-08,
title = {{Covering a disc with n equilateral triangles} (P78)},
author = {{MinMax Arena}},
year = {2026},
note = {Machine-verified records, 2026-08 edition},
url = {https://minmaxarena.com/data/editions/2026-08/equilateral-triangles-covering-a-disc.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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