6.1. On a plane, 55 points are marked - the vertices of a certain regular 54-gon and its center. Petya wants to paint a triplet of the marked points in red so that the painted points are the vertices of some equilateral triangle. In how many ways can Petya do this?
Answer: 72

Solution. Assume that our 54-gon is inscribed in a circle, and its vertices divide this circle into 54 arcs of length 1.

If one of the painted points is the center, then the other two painted points must be the ends of an arc of size $54 / 6=9$. There are 54 such possibilities.

Otherwise, all three painted vertices must form an equilateral triangle, the vertices of which divide the circle into 3 arcs of size $54 / 3=18$. There are $-54 / 3=18$ such possibilities.

In total: $54+18=72$ options.