Comment by nhatcher

Oh, I don't think I understood you at all the first time around.

But you are absolutely right. The center of an hexagon may or may not coincide with the vetex of a marked hat.

In particular in the set of statements, for a given (m, n) all 12 these might be false:

There is Hat (or an anti-Hat) in vertex (m, n) with rotation 60×i (with i=0,..,5)

I'll update the text when I get a chance