Pechkovskiy A.n.

On a chessboard of size $8 \times 8$, 64 points are marked - the centers of all cells. Is it possible to separate all the points from each other by drawing 13 lines that do not pass through these points?
Answer: No. Indeed, consider a square passing through the centers of all 28 boundary cells. It is clear that 13 lines intersect it in no more than 26 points, and therefore divide it into no more than 26 parts, meaning that two "boundary" centers will end up in the same part. Therefore, to separate the 28 boundary, and thus all centers, at least 14 lines are needed.