There is a classic problem of 'squaring the square', or constructing a perfect squared square, which is a unit square cut into a finite number of smaller squares whose sidelengths are all different. We know that these sidelengths must be rational so that we need only look for integral tilings; and it turns out solutions do exist to this problem (example shown below). A slightly more general (and much easier) problem which also has solutions is to look for rectangles that can be divided into distinct-sized squares.
A natural generalization is then to look for rectangles which can be tiled by different-sized similar rectangles. Obviously we should ask that our rectangle not be square, but indeed for any perfect squared square, we could simply scale it by an appropriate factor to create a dissection of any rectangle. Thus we could ask that at least one rectangle have a different orientation to the rest, and as far as I can see this makes the problem harder, and I have not found a solution or way of approaching it. It seems interesting to know though which rectangles permit such a tiling.
Thus I have the following question: Is it possible to divide any non-square rectangle into a finite number of rectangles all of different sizes such that all rectangles (including the tiled rectangle) are similar, and such there is a pair of rectangles oriented perpendicularly? If not, which rectangles permit such a dissection? In such a dissection (if one exists), must all of the rectangles be rational scalings of each other?
Lowest-order perfect squared square