2020 AMC 12B Problem 19

Square ABCD in the coordinate plane has vertices at the points A(1,1), B(-1,1), C(-1,-1), and D(1,-1). Consider the following four transformations:

\quad\bullet\qquad L, a rotation of 90^{\circ} counterclockwise around the origin;

\quad\bullet\qquad R, a rotation of 90^{\circ} clockwise around the origin;

\quad\bullet\qquad H, a reflection across the x-axis; and

\quad\bullet\qquad V, a reflection across the y-axis.

Each of these transformations maps the squares onto itself, but the positions of the labeled vertices will change. For example, applying R and then V would send the vertex A at (1,1) to (-1,-1) and would send the vertex B at (-1,1) to itself. How many sequences of 20 transformations chosen from \{L, R, H, V\} will send all of the labeled vertices back to their original positions? (For example, R, R, V, H is one sequence of 4 transformations that will send the vertices back to their original positions.)

\textbf{(A)}\ 2^{37} \qquad\textbf{(B)}\ 3\cdot 2^{36} \qquad\textbf{(C)}\ 2^{38} \qquad\textbf{(D)}\ 3\cdot 2^{37} \qquad\textbf{(E)}\ 2^{39}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.