Suppose you have marked all the 64 centers of unitsquares of a chessboard.
At least how many lines do you need, such that they divide the plane in a way, such that no two marked points lie in the same region (part of the plane)?
My guess is 14, and I was told that that is the correct answer, How can I prove it? Please help.
Big hint
The centers of the chessboard form an $8$ times $8$ grid of dots. There are $28$ line segments on the border of this grid which connect two adjacent border dots. Namely, there are $7$ horizontal segments on the top border, $7$ vertical segments on the right border, etc.
There needs to be a line passing through each of these $28$ border segments; if there was a border segment without a line through it, then the two points at the ends of the segment would not be separated.
How many border segments can a single line pass though?