A problem from cut-the-knot:
Imagine a rectangular grid (lattice in the math parlance) with the distance between the nodes equal to 1. Also, let there be given a shape with the area less than 1 but otherwise arbitrary. Show that it's possible to place this shape onto the plane in such a manner that no grid nodes will fall inside the shape.
The proof just uses translation. If rotation is considered, can the area be less than π/2 instead of 1?
I'm thinking a circle with area π/2 just touching 4 grid nodes. Intuitively, any shape slightly less than this circle can be rotated to avoid these 4 grid nodes. But I can neither prove it nor think out a counter example.