2019 AMC 10A Problem 8

Describe the pattern precisely. Above $\ell$, unit squares sit on the line with a short diagonal tail leaving the top-right corner toward the upper right. Below $\ell$, congruent squares hang from the line, shifted half a period to the right, with a tail leaving the bottom-left corner toward the lower left. The whole picture repeats every $4$ units along $\ell$.

• Translation parallel to $\ell$ by $4$ units: the pattern repeats, so this works.
• Rotation by $180^\circ$ about a point of $\ell$ midway between an upper square and the next lower square: a half-turn sends "square above, tail up-right from the top-right corner" to "square below, tail down-left from the bottom-left corner," which is exactly how the lower squares look, and the offset matches. This works.
• Reflection across $\ell$: an upper square would land directly below itself, but the lower squares are shifted, and the tail would point down-right instead of down-left. Fails.
• Reflection across a line perpendicular to $\ell$: every tail slanting up-right becomes one slanting up-left, and no tail in the figure does that. Fails.

Two of the four motions preserve the figure.

The answer is $\boxed{\textbf{(C)}\ 2}$.

Test each candidate motion on the smallest distinctive detail, here the little tails: a reflection reverses the slant of a tail while a half-turn reverses it twice and keeps it. Combined with the half-period offset of the lower row, this pins down which motions are symmetries. The figure is a frieze with a $180^\circ$ rotation and translations but no mirror lines.

• Assuming the reflection across the line works because there are squares on both sides, without checking that the lower squares are shifted and their tails reversed.
• Dismissing rotation because the figure "has a direction" along the line; a half-turn reverses that direction on both rows at once, which is consistent with the picture.