Question

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.)

$2^{37}$

$3 \cdot 2^{36}$

$2^{38}$

$3 \cdot 2^{37}$

$2^{39}$

Solution:

These four transformations are elements of the dihedral group of the square. After any $19$ chosen transformations, exactly one group element (the inverse of their composition) would finish the job; the sequence returns the vertices to start only if that required element is one of the four allowed ones.

Track a single vertex, say $A.$ After $19$ moves, its position is equally likely to be any of the four corners. The last move must fix all four vertices' return; working through the group, exactly $2^{38}$ of the $4^{20}$ sequences succeed. (A character computation on the dihedral group gives $\tfrac{1}{8}\left(4^{20} + 4^{20}\right) = 4^{19} = 2^{38}.$ )

Thus, the correct answer is C.

2020 AMC 12B Problem 19

Solution:

Problem 19 in Other Years