IR-TST 2013 Problem 18

A special kind of parallelogram tile is made up by attaching the legs of two right isosceles triangles of side length $1$ . We want to put a number of these tiles on the floor of an $n\times n$ room such that the distance from each vertex of each tile to the sides of the room is an integer and also no two tiles overlap. Prove that at least an area $n$ of the room will not be covered by the tiles.

Proposed by Ali Khezeli