Covering of a rectangle with squares from both sides

The paper provides an elementary proof of Kenyon's theorem that periodic tiling of a plane by squares with periods $(1,0)$ and $(0,\lambda)$ is possible only if $\lambda=p\pm\sqrt{q^2 - r^2}$ for some rational $p\geq q\geq r\geq 0$. A similar new result is proved about covering of a rectangle with squares from both sides in one layer. The paper also proves a necessary and sufficient condition for covering with equal squares.