Orient the board so that a white square is in the top left hand corner. Label each square with a piece on it with $1$, and $0$ otherwise. Let $R_1,R_2,\ldots, R_8$ be the rows, and $C_1,C_2,\ldots, C_8$ be the columns.
Add together $R_1+R_3+R_5+R_7+C_1+C_3+C_5+C_7$, which is even. Each white square in these squares gets accounted for $2$ times, so they have an even contribution to the sum. So this leaves the black squares, and an even minus an even is even, and all the black squares are accounted for exactly once, so there are an even number of pieces on the black squares.
The answer is $1$).