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.)

Video solution:

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Written solution:

Each of $L,R,H,V$ moves every vertex to an adjacent corner of the square. Therefore after an odd number of transformations the labeling is in one of the four odd-parity states, and after an even number it is in one of the four even-parity states.

After any first $19$ transformations, the square is in an odd-parity state. From each odd-parity state, exactly one of $L,R,H,V$ sends the labeled vertices back to their original positions. Thus every sequence of the first $19$ transformations has exactly one valid final transformation.

There are $4^{19}=2^{38}$ choices for the first $19$ transformations, so there are $2^{38}$ valid sequences.

Thus, C is the correct answer.