MathDB

For

n>1

consider an

n\times n

chessboard and place identical pieces at the centers of different squares.
[*] Show that no matter how

2n

identical pieces are placed on the board, that one can always find

4

pieces among them that are the vertices of a parallelogram. [*] Show that there is a way to place

(2n-1)

identical chess pieces so that no

4

of them are the vertices of a parallelogram.

Problem Statement