The following problem is an exercise from a handout on recursion:
The figure below shows a ring made of six small sections which you are to paint on a wall. You have four paint colors available and you will paint each of the six sections a solid color. Find the number of ways you can choose to paint the sections if no two adjacent sections can be painted with the same color.
I thought that this can be solved with simple counting. The idea is to select a section which can be painted in $4$ ways. Then the four sections taken counterclockwise can each be painted in $3$ ways. Finally, the last section can be painted in $2$ ways. So the answer would be $4\cdot3^4\cdot2=648$. But the given answer is $732$. What is wrong in my solution and what is the correct solution?