MinMax Arena
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P77 · Extremal configurations · Classic

Covering an equilateral triangle with n equilateral triangles

VERIFIED CONSTRUCTIONcommon side 0.307692309
n = 11Current record · open

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 triangleK⊆T1∪⋯∪Tn
  • Exact continuous verificationThe verifier computes all container edges, piece edges, and intersections exactly in Q(√3), then performs a vertical sweep to verify full coverage
  • ScoreThe common side s shared by all covering triangles; smaller is betters(T)=s
  • ObjectiveMinimise s over all valid coversminTs(T)

Getting a feel for it

Non-square piece counts

Only when n is a perfect square can a regular triangular grid tile the container without overlap or protrusion. Other values of n require overlapping, rotated, or protruding pieces, and adding one piece often alters the layered structure.

Known constructions and literature data

Friedman's table lists constructions for n = 6–9 and n = 11–15; at n = 9, the 3 × 3 triangular grid meets the area lower bound and is provably optimal. For n = 10, the table uses the n = 9 grid plus one redundant piece as an initial baseline. All other instances are reconstructed from the published diagrams, with literature values and nine-decimal certificate scores listed separately.

Source
ONE LEADERBOARD PER n

Current 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 (1) ↓
n6
CURRENT RECORD0.428571429best known 3/7
Hard
Answer sourceDavid Cantrell
Solution methodPublished reference construction
n7
CURRENT RECORD0.400000001best known 2/5
Matches the best known
Answer sourceErich Friedman
Solution methodPublished reference construction
n8
CURRENT RECORD0.375000001best known 3/8
Matches the best known
Answer sourceDavid Cantrell
Solution methodPublished reference construction
n9
CURRENT RECORD1/3
Optimal
Answer sourcearea-tight triangular tiling
Solution methodPublished reference construction
n10
CURRENT RECORD0.333333334
Hard
Answer sourceMinMax Arena
Solution methodMinMax Arena reference construction
n11
CURRENT RECORD0.307692309best known 4/13
Hard
Answer sourceErich Friedman
Solution methodPublished reference construction
n12
CURRENT RECORD0.300000001best known 3/10
Matches the best known
Answer sourceErich Friedman
Solution methodPublished reference construction
n13
CURRENT RECORD0.285714287best known 2/7
Hard
Answer sourceErich Friedman
Solution methodPublished reference construction
n14
CURRENT RECORD0.272727274best known 3/11
Hard
Answer sourceDavid Cantrell
Solution methodPublished reference construction
n15
CURRENT RECORD0.266666667best known 4/15
Hard
Record holderzzzcy
Solution methodAI · OpenAI: GPT-6 Astra

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/equilateral-triangles-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/equilateral-triangles-covering-an-equilateral-triangle.json
BibTeX (click to copy)
@misc{minmaxarena-equilateral-triangles-covering-an-equilateral-triangle-2026-10,
  title  = {{Covering an equilateral triangle with n equilateral triangles} (P77)},
  author = {{MinMax Arena}},
  year   = {2026},
  note   = {Machine-verified records, 2026-10 edition},
  url    = {https://minmaxarena.com/data/editions/2026-10/equilateral-triangles-covering-an-equilateral-triangle.json},
  license = {CC BY 4.0}
}
DISCUSSION

Discussion

Discuss strategies, share construction methods, or ask questions. Any signed-in user can post under their public byline and unique #UID. New posts appear after moderation.

论n=10的最优解hadsajnc#1463 replies