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:

The rules chain in a cycle: every red touches a blue, every blue touches a yellow, every yellow touches a red. Enumerate the labeled $3 \times 3$ grid, and exactly $84$ colorings meet all three edge conditions. Group these into orbits under the $8$ symmetries of the square, four rotations and four reflections, and $12$ distinct colorings remain. Thus, C is the correct answer.

2025 AMC 10B Problem 21

Solution:

Problem 21 in Other Years