Each edge and diagonal of a regular hexagon (15 in total) is colored red or black such that the hexagon appears unchanged when rotated by 60 degrees clockwise about its center. How many such colorings are possible?
There are 15 line segments to be colored:
(1) 6 edges
(2) 3 diagonals that connect opposite vertices
(3) 6 other diagonals
For the appearance of the figure to be unchanaged when it is rotated by 60 degrees, all segments of the same type must be the same one of the two colors.
So the number of different colorings is 2*2*2 = 8.