For a given integer $k\ge3$, cover the unit square with $k$ rectangles so that the longest of the rectangles' diagonals be as short as possible. You may assume that the rectangles tile the square, though this assumption may be redundant. Call such a partition optimal. The solutions is obvious in the easy cases when $k$ is the square of an integer and for a few small values of $k$ only (unpublished). In each of the solved cases, the diagonals of all rectangles turn out to be rational and their diagonals are equal.

Question. In an optimal partition, must the sides of all rectangles be rational and their diagonals be equal?

The analogous question for covering the $n$-dimensional cube with rectangular boxes can be asked for every $n\ge3$ as well.