Covering tori with squares

We find that an n × n toroidal checkerboard can be covered with $\text{?}\text{n}\text{k}\text{?}\text{n}\text{k}\text{??}$k × k squares, and no fewer. To prove this result we also need to prove that the unit Euclidean torus can be covered with ?α^?1?α^?1?? squares of side α, and no fewer.