11.44*. What is the maximum number of cells on an $8 \times 8$ chessboard that can be cut by a single straight line?
11.44. A line can intersect 15 cells (Fig. 11.9). We will now prove that a line cannot intersect more than 15 cells. The number of cells intersected by a line is one less than the number of points of intersection with the segments that form the sides of the cells. Inside the square, there are 14 such segments. Therefore, inside the square, there are no more than 14 points of intersection of the line with the sides of the cells. No line can intersect the boundary of the board in more than two points, so the number of points of intersection with the segments does not exceed 16. Therefore, the maximum number of cells on an $8 \times 8$ chessboard that can be intersected by a single line is 15.