Question 23
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.)