Covering a disc with n equilateral triangles

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$$D\subseteq T_1\cup \cdots \cup T_n$$
• Exact continuous verificationExtending the 3n triangle edges into a line arrangement partitions the plane into faces on which triangle membership is constant; the disc is covered if and only if no uncovered face intersects the open disc. The verifier decomposes faces into vertical trapezoids over Q(√3) and tests open-disc intersection via exact rational comparisons
• ScoreThe common side s shared by all covering triangles; smaller is better$$s\left(T\right)=s$$
• ObjectiveMinimise s over all valid covers$$\underset{T}{\min }s\left(T\right)$$