What is the maximum number of squares on an $8 \times 8$ chessboard on which pieces may be placed so that no two of these squares touch horizontally, vertically, or diagonally?
The answer is 16 .

Divide the chessboard into sixteen $2 \times 2$ regions (see figure). Notice that no two pieces can lie in the same region. So there are at most 16 pieces. By placing a piece in the lower left corner of each region, we see that 16 is achievable.