A six-pointed star is drawn inside a hexagon, with its points at the hexagon's corners. How many triangles are there?

Smallest (6 star points and 6 notches between the star and the hexagon's sides): 12. Next size, a point with its neighbouring piece: 12. Corner triangles cut off by one side of the star: 6. The 2 big triangles of the star: 2. Total $12+12+6+2=32$.