Suppose we want to color the faces of a cube using either red or green paint. Use the "Counting Theorem" in Lecture 24 and the description of the rotational symmetry group of the cube in Lecture 25 to determine the total number of genuinely different painted cubes you can obtain. (Note: If you were to paint the top red and all other faces green, we consider that the same coloring as if we had painted the bottom red and all other faces green, since an obvious rotational symmetry takes one coloring to the other.)