Question

Each of the $9$ squares in a $3 \times 3$ grid is to be colored red, blue, or yellow in such a way that each red square shares an edge with at least one blue square, each blue square shares an edge with at least one yellow square, and each yellow square shares an edge with at least one red square. Colorings that can be obtained from one another by rotations and/or reflections are to be considered the same. How many different colorings are possible?

$3$

$9$

$12$

$18$

$27$

Solution:

Each red square needs a blue neighbor, each blue a yellow, and each yellow a red, forcing all three colors to appear in an interlocking pattern. A systematic check gives $84$ valid colorings of the labeled grid. Under the $8$ symmetries of the square, only two diagonal reflections fix any colorings — $6$ each — so Burnside's lemma gives $\tfrac{1}{8}(84 + 6 + 6) = 12$ distinct colorings.

Thus, the correct answer is C.

2025 AMC 12B Problem 17

Solution:

Problem 17 in Other Years