2025 AMC 12B Problem 13

Question

A circle has been divided into

6

sectors of different sizes. Then

2

of the sectors are painted red,

2

painted green, and

2

painted blue so that no two neighboring sectors are painted the same color. One such coloring is shown below.

How many different colorings are possible?

12

16

18

24

28

Answer: D

Concepts:graph theory multiplication principle

Difficulty rating: 1660

Small hint:

Because same-colored sectors cannot touch, the two sectors of each color form a non-adjacent pair

Big hint:

Count the ways to split the

6

sectors into three non-adjacent pairs, then assign the three colors to the pairs

Solution:

The two sectors of each color must be a non-adjacent pair, so a coloring is a way to split the

6

cyclic sectors into three non-adjacent pairs together with an assignment of the three colors. The non-adjacent pairs are the edges of the complement of the

6

-cycle, the triangular prism, which has

4

perfect matchings. Assigning the three colors in

3!

ways gives

4 \times 6 = 24

colorings.

Thus, the correct answer is D.