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?

$12$

$16$

$18$

$24$

$28$

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.

2025 AMC 12B Problem 13

Solution:

Problem 13 in Other Years