Covering an equilateral triangle with n equilateral triangles
Continuously cover the unit equilateral triangle with n congruent equilateral triangles. Overlap and protrusion are allowed; minimise the common side.
Formal definition
- ContainerThe closed equilateral triangle K = conv{(0,0),(1,0),(0.5,√3/2)}
- SubmissionA common side s and exactly n {x,y,turn,down} placements, each independently rotatable and flippable
- CoverEvery point of K belongs to at least one closed covering triangle
- Exact continuous verificationThe verifier enumerates container edges, piece edges and every intersection in Q(√3), then performs an exact vertical sweep rather than pixel sampling
- ScoreThe side s shared by every covering triangle; smaller is better
- ObjectiveMinimise s over all legal covers
Getting a feel for it
A similar container is still not a scaling exercise
Only square values of n can fill the similar container without area waste on a regular triangular grid. Other sizes force overlap or protrusion, and one extra piece may reorganise the whole layered arrangement.
The published nontrivial range starts at n=6
Friedman's table lists frontier constructions for n=6–9 and n=11–15. At n=9 the 3×3 triangular grid meets the area bound and is proved optimal. The n=10 row starts here from that grid plus one redundant triangle and is not presented as an external record. The site reconstructs the other published figures and separates their source values from the nine-decimal certificates.
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-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/equilateral-triangles-covering-an-equilateral-triangle.json
BibTeX (click to copy)
@misc{minmaxarena-equilateral-triangles-covering-an-equilateral-triangle-2026-08,
title = {{Covering an equilateral triangle with n equilateral triangles} (P77)},
author = {{MinMax Arena}},
year = {2026},
note = {Machine-verified records, 2026-08 edition},
url = {https://minmaxarena.com/data/editions/2026-08/equilateral-triangles-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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