Question

A circle has been divided into $6$ sectors of $6$ 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 six unequal sectors form a fixed cycle of $6$ distinguishable positions, so we want proper $3$ -colorings of a $6$ -cycle that use each color exactly twice. A $6$ -cycle has $2^6 + 2 = 66$ proper $3$ -colorings altogether. Of these, $6$ use only two colors (type $(3,3,0)$ ) and $36$ use one color three times (type $(3,2,1)$ ). That leaves $66 - 6 - 36 = 24.$ Therefore, the answer is D.

2025 AMC 10B Problem 16

Solution:

Problem 16 in Other Years