2025 AMC 12B Problem 13
Attempt Problem 13 of the 2025 AMC 12B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2025 AMC 12B solutions, or check the answer key.
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13.
A circle has been divided into sectors of different sizes. Then of the sectors are painted red, painted green, and 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?
Answer: D
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 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 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 -cycle, the triangular prism, which has perfect matchings. Assigning the three colors in ways gives colorings.
Thus, the correct answer is D.
Problem 13 in Other Years
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