Covering a square with n equilateral triangles
Cover the unit square with n congruent equilateral triangles. Every piece may be translated, rotated and flipped independently; arbitrary overlap and protrusion are allowed. Minimise the common side.
Formal definition
- ContainerThe closed unit square K = [0,1] × [0,1]
- Piecesn equilateral triangles of common side s; every piece may be translated and rotated independently, with the third vertex on either side of its reference edge
- SubmissionGive the common side s plus one reference-edge midpoint, rotation and orientation for each triangle; submit exactly n pieces
- CoverEvery point of the square lies in at least one closed triangle; triangles may overlap and protrude
- Exact rotationThe certificate writes rotations as turn = tan(θ/2). Half-angle identities make cos θ and sin θ rational, so every vertex and intersection is computed exactly in Q(√3)
- Continuous verificationThe verifier builds the planar arrangement of triangle and square edges, then performs an exact algebraic vertical sweep for uncovered faces; it uses neither pixels nor random sampling
- ScoreThe side s shared by every submitted triangle; smaller is better
- ObjectiveMinimise s over all legal covers
Getting a feel for it
Why this is not tiling
A square has 90° corners and an equilateral triangle has 60° corners. The boundary angles do not match, so good covers must overlap and protrude. Adding one piece can reorganise the whole contact pattern rather than append one more tile.
The published frontier
Friedman's public table gives the best known constructions for n=1–12, with nontrivial layouts already at n=2 and independent rotations from n ≥ 4. This site reconstructs the pictured arrangements and rechecks continuous coverage instead of copying numbers alone.
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-square.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-square.json
BibTeX (click to copy)
@misc{minmaxarena-equilateral-triangles-covering-a-square-2026-08,
title = {{Covering a square with n equilateral triangles} (P76)},
author = {{MinMax Arena}},
year = {2026},
note = {Machine-verified records, 2026-08 edition},
url = {https://minmaxarena.com/data/editions/2026-08/equilateral-triangles-covering-a-square.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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