Each 1x1 tile has a red side opposite a yellow side,and a blue side opposite a green side.An 8x8 chessboard is formed from 64 of these tiles,which may be turned around or turned over.When two tiles meet,the edges that come together must be of the same color.How many different chessboards can be formed? Turning the chessboard around or turning the chessboard over is not allowed.
Taken from CIMC 2015
Either all vertical edges are red or yellow and all horizontal edges blue or green, or vice versa. The colour of the horizontal edges must alternate in each column, and the colour of the vertical edges must alternate in each row. Because the tiles can be turned over, these alternations can all be chosen independently. Thus there are $8$ binary choices for the rows, $8$ binary choices for the columns and $1$ binary choice for the overall orientation, for a total of $2^{17}=131072$ possibilities.