# Size of minimal covering, overlapping and disjoint

There are two ways to cover a geometric shape with primitive units: you can allow the units to overlap, or require that they be disjoint. Of course the number of units in the case of disjoint covering may be larger. But how much larger?

If the units are squares, the number of units in a disjoint covering might be arbitrarily larger. For example, consider an $n\times (n+1)$ rectangle. For every $n$, it can be covered by two overlapping squares. But if the squares have to be disjoint, at least $O(\log n)$ squares are required.

What if the units are rectangles? Is there a sequence of shapes that can be covered by two rectangles, but whose disjoint-rectangles-cover-number can be arbitrarily large?

EDIT: This question is related: Tiling an L-shape with "almost square"s

I suspect that if you take two $1 \times 4$ strips of paper, call them $A$ and $B$, and lay out one horizontally, and the other rotated by, say, 30 degrees, but crossing over the first, so that they form a kind of squashed "X" shape, you get something that can be covered by two rectangles (the two strips $A$ and $B$), but may not be coverable by any finite number of disjoint rectangles -- the sharp corners of $B - A$ seem likely to make this impossible.