HMMT 2020 C9

Farmer James wishes to cover a circle with circumference $10 \pi$ with six different types of colored arcs. Each type of arc has radius 5 , has length either $\pi$ or $2 \pi$, and is colored either red, green, or blue. He has an unlimited number of each of the six arc types. He wishes to completely cover his circle without overlap, subject to the following conditions:

• Any two adjacent arcs are of different colors.
• Any three adjacent arcs where the middle arc has length $\pi$ are of three different colors.

Find the number of distinct ways Farmer James can cover his circle. Here, two coverings are equivalent if and only if they are rotations of one another. In particular, two colorings are considered distinct if they are reflections of one another, but not rotations of one another.