Question

How many different patterns can be made by shading exactly two of the nine unit squares in a $3 \times 3$ grid? Patterns that can be matched by flips and/or turns are not considered different.

$3$

$6$

$8$

$12$

$18$

Solution:

Classify the two shaded squares up to rotations and reflections. Patterns that include a corner: two adjacent corners (same edge), two diagonally opposite corners, corner with the center, corner with an adjacent edge-middle, and corner with a far edge-middle. That is $5$ patterns.

Patterns with no corner: two adjacent edge-middles, two opposite edge-middles, and an edge-middle with the center. That is $3$ more.

In total there are $5 + 3 = 8$ distinct patterns.

Thus, the correct answer is C .

1990 AMC 8 Problem 25

Solution:

Problem 25 in Other Years