Problem 1296

Solution:

Number both the rows and the columns from $1$ to $8$, and say that black squares are the ones where the rows and columns have the same parity. We will use, e.g., "even rows" to refer to rows $2, 4, 6, 8$. Choosing $8$ squares all in different rows and columns is equivalent to matching rows to columns.

For each of the $8$ rows, we first decide whether they will be matched with a column of the same parity as itself (resulting in a black square) or with one of a different parity (resulting in a white square). Since we want to choose $4$ squares of each color, the $4$ rows matched to same-parity columns must contain $2$ even rows and $2$ odd rows. There are $\binom{4}{2}^{2} = 6^{2}$ ways to choose $2$ odd rows and $2$ even rows to match with same-parity columns.

After choosing the above, we have fixed which $4$ rows should be matched with odd columns (while the other $4$ should be matched with even columns). Then there are $(4!)^{2} = 24^{2}$ ways to assign the columns to the rows, so the answer is $(6 \cdot 24)^{2} = 144^{2} = 20736$.