Covering a square with n equilateral triangles

Formal definition

• ContainerThe closed unit square K = [0,1] × [0,1]
• Piecesn equilateral triangles of common side s; each piece may be translated and rotated independently, with the third vertex on either side of its reference edge
• SubmissionGive the common side s together with the reference-edge midpoint, rotation parameter, and orientation for each of the n triangles
• CoverEvery point of the square lies in at least one closed triangle; triangles may overlap or protrude$$K\subseteq T_1\cup \cdots \cup T_n$$
• Exact rotationThe certificate represents rotations by rational turn = tan(θ/2). By the rational half-angle parameterisation, cos θ and sin θ are rational, so all vertices and intersections are computed exactly in Q(√3)
• Continuous verificationThe verifier builds the planar arrangement of triangle and square edges and performs an exact algebraic vertical sweep over Q(√3) to check for uncovered faces
• ScoreThe common side s shared by all submitted triangles; smaller is better$$s\left(T\right)=s$$
• ObjectiveMinimise s over all valid covers$$\underset{T}{\min }s\left(T\right)$$