Question 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:
$L,$ a rotation of $90^\circ$ counterclockwise around the origin;
$R,$ a rotation of $90^\circ$ clockwise around the origin;
$H,$ a reflection across the $x$-axis; and
$V,$ a reflection across the $y$-axis.
Each of these transformations maps the square 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.)
$L,$ a rotation of $90^\circ$ counterclockwise around the origin;
$R,$ a rotation of $90^\circ$ clockwise around the origin;
$H,$ a reflection across the $x$-axis; and
$V,$ a reflection across the $y$-axis.
Each of these transformations maps the square 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.)