Six looks like a magic number related to triangle. Unfortunately, I am awful with calculations. Most likely I won't be of any help with this problem.

I am now preoccupied with trying to answer the Why? question at the end of

http://www.cut-the-knot.org/Curriculum/Arithmetic/FlipThem.shtml

The case of odd numbers of triangles and the ones flipped on a move is simple. But when the total is even I am stuck with particular cases.