# 2016 AIME II Problems/Problem 12

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. Insert figure of a smaller circle, a bigger circle, and 6 sections dividing the region between the concentric circles.

## Solution

Assume, WLOG, the left ring is color . Now, let be the number of ways to satisfy the conditions where there are sections ending on color . We make a table on . Note that because then adjacent sections are both color . We also have to multiply by by symmetry for other colors in the left ring, so the desired answer is .

Solution by Shaddoll