Painting chess board The task is to paint each of the $64$ squares on a chess board either blue or red. I need to find the number of distinct ways this can be done given that any $2\times 2$ square on the board has two red and two blue squares. I've tried solving it for a $4\times 4$ board, but I am getting no where. Would appreciate any help

For an $m\times n$ chessboard there are $2^m+2^n-2$ ways.

Case I. There are two horizontally adjacent squares of the same color: $2^m-2$ ways.

Case II. There are two vertically adjacent squares of the same color: $2^n-2$ ways.

Case III. None of the above: $2$ ways.

Hint for Case I: There are $2^m-2$ ways to color one row so that two adjacent squares have the same color. The rest of the coloring is determined from that; colors must alternate in each column. (Note, therefore, that Cases I and II do not overlap.)