12. An $8 \times 8$ grid has 28 center points around its perimeter. Connecting these 28 center points with dotted lines forms a square. Each line intersects the dotted square at most at two points, so $n$ lines intersect at most $2n$ points. If $n<14, 2n<28$, then $n$ lines do not meet the requirement. The 7 horizontal lines and 7 vertical lines inside the $8 \times 8$ grid, totaling 14 lines, separate the 64 points from each other. Therefore, $n_{\text{min}}=14$.
12. On an $8 \times 8$ checkerboard, mark the center of each square, resulting in 64 points. How many straight lines are needed at least to separate them from each other?