Many thanks to Mary Jane Sterling for suggesting this problem.  Will Shortz, puzzle editor for the New York Times, presented it on National Public Radio's Weekend Edition.  He gave no solution to the puzzle.

Each square of a four by four checkerboard is to be labeled with a zero or a one in such a way that the sum of the entries in the neighborhood of each square is always odd -- the neighborhood being the squares immediately adjacent (up, down, right, left, and diagonally) together with the square itself. Come up with as many ways of doing this as possible, or show that it can never be done.  For example, the checkerboard on the left fails to satisfy the condition since the square in the top left corner has a neighborhood sum of 2.