Squaring and not squaring one or more planes

A set of natural numbers tiles the plane if a square-tiling of the plane exists using exactly one square of side length n for every n in the set. In [9] it is shown that N, the set of all natural numbers, tiles the plane. We answer here a number of questions from that paper. We show that there is a simple tiling of the plane (no nontrivial subset of squares forms a rectangle). We show that neither the odd numbers nor the prime numbers tile the plane. We show that N can tile many, even infinitely many planes.