11.50 Place chess pieces on a square grid chessboard so that on every line passing through the center of any square and parallel to the sides or diagonals of the chessboard squares, there is at least one chess piece (the pieces are placed at the centers of the squares). How many chess pieces are needed at a minimum on (1) an $8 \times 8$ grid; (2) an $n \times n$ grid?
[Solution] From the example in the figure below, we can see that 16 pieces are enough on an $8 \times 8$ chessboard. For a general $n$, when $n$ is even, place $2n$ pieces, and when $n$ is odd, place $2n+1$ pieces.

- On the other hand, when $n$ is even, the $2n$ dotted lines in the figure do not intersect, so at least $2n$ pieces are needed to meet the requirements. When $n$ is odd, in addition to the $2n-2$ dotted lines in the figure, each of which must have at least one point, there are 6 lines between points $A, B, C, D$, and these 6 lines must have at least 3 different points. Therefore, the chessboard must have at least $2n+1$ points.

In summary, when $n$ is even and odd, the chessboard must have at least $2n$ and $2n+1$ pieces, respectively, to meet the requirements.