AMC12 2020 A

Answer (A): If two rotations that are not inverses of each other are applied in order, then there is a unique third rotation that will return $T$ to its original position. There are $3\cdot2=6$ of these. If two different reflections across the coordinate axes are applied in order, then the $180^\circ$ rotation is the unique transformation that will return $T$ to its original position. There are $2$ of these. If the first two transformations are a rotation of $180^\circ$ and a reflection, in either order, then the other reflection is the unique transformation that will return $T$ to its original position. There are $4$ such sequences. It remains to show that for all other choices of the first two transformations, none of the five choices for the third transformation will return $T$ to its original position. Two transformations that are inverses of each other already return $T$ to its original position, so applying a third transformation will move $T$ out of its original position. The other cases are a rotation of $90^\circ$ or $270^\circ$ followed by a horizontal or vertical reflection, or vice versa. In each case, a reflection would be needed to restore the correct orientation of the triangle, but neither a vertical reflection nor a horizontal reflection will return $T$ to its original position. In all, there are $6+2+4=12$ sequences of three of the given transformations that will return $T$ to its original position.