On the chessboard (8x8) all field centers are marked. Is it possible to break the board into the parts with thirteen straight lines so that within each of these parts there was no more than one marked point?
There are 28 fields adjacent to the edges of the 8x8 chessboard. Hold on 28 segments connecting the centers of these neighboring fields. Each line can intersect no more than two segments, so 13 lines can not cross more than 26 segments, i.e. there are at least 2 pieces that do not intersect with any of the 13 held lines. Therefore, thirteen lines can not break the chessboard so that within each part there was no more than one marked point, since both ends of the segment without intersecting lines lie in one part.