Not only is there an odd number of tricolor triangles, but they come in pairs according to their orientation (RGB clockwise/anticlockwise). Proof: define a continuously differentiable vector field on the plane, by letting the field at each vertex be 0, and the field in the center of each edge be a vector of magnitude 1 pointing in the direction R->G->B->R (or 0 if the two adjacent vertices are the same color). Extend the field to the complete edges, then the interiors of the triangles by some interpolat
Proof of #4, but with unnecessary calculus:
Not only is there an odd number of tricolor triangles, but they come in pairs according to their orientation (RGB clockwise/anticlockwise). Proof: define a continuously differentiable vector field on the plane, by letting the field at each vertex be 0, and the field in the center of each edge be a vector of magnitude 1 pointing in the direction R->G->B->R (or 0 if the two adjacent vertices are the same color). Extend the field to the complete edges, then the interiors of the triangles by some interpolat