Question 27
Sixteen wooden $C$s are placed in a $4$-by-$4$ grid, all with the same orientation, and each is to be colored either red or blue. A quadrant operation on the grid consists of choosing one of the four $2$-by-$2$ subgrids of $C$s found at the corners of the grid and moving each $C$ in the subgrid to the adjacent square in the subgrid that is $90$ degrees away in the clockwise direction, without changing the orientation of the $C$. Given that two colorings are considered the same if and only if one can be obtained from the other by a series of quadrant operations, determine the number of distinct colorings of the $C$s.
$$
\begin{array}{cccc}C&C&C&C\\C&C&C&C\\C&C&C&C\\C&C&C&C\end{array}
$$
$$
\begin{array}{cccc}C&C&C&C\\C&C&C&C\\C&C&C&C\\C&C&C&C\end{array}
$$