### Pigeonhole in a Chessboard

Let's mark the centers of all squares of an 8×8 chess board. Is it possible to cut the board with 13 straight lines (none passing through a single mispoint) so that every piece had at most 1 marked point?

Solution

Let's mark the centers of all squares of an 8×8 chess board. Is it possible to cut the board with 13 straight lines (none passing through a single mispoint) so that every piece had at most 1 marked point?

No, it's not possible. Indeed, 28 small squares line up the boundary of the 8×8 chessboard. It takes 28 segments to connect the midpoints of consecutive squares. This way we obtain a 28-gon. Any line through this 28-gon intersects it in at most 2 points. 13 straight lines may intersect at most 26 sides of the polygon. Therefore, by the Pigeonhole Principle, at least two sides of the polygon have the property that each falls entirely (endpoints included) inside one of the pieces.

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