$[$ Extremal properties (other) $]$

What is the maximum number of cells on an $8 \times 8$ chessboard that can be cut by a single straight line? #
A line can intersect 15 cells (see the figure). Now let's prove that a line cannot intersect more than 15 cells. The number of cells a line intersects is one less than the number of intersection points 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 intersection points 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 intersection points 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.

![](https://cdn.mathpix.com/cropped/2024_05_06_55a07599adfd5314dc75g-03.jpg?height=669&width=675&top_left_y=269&top_left_x=697)

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