Recently, the German science journal Spektrum put online a riddle about squares being tiled to a rectangle:
The task was to determine the area of the rectangle tiled with $8$ squares, of which the middle square has area $1$.
Note that two pairs of squares each have identical side length.
Questions.
For what integers $n > 1$ is there a rectangle with real-valued size lengths such that the rectangle can be tiled by $n$ squares with pairwise distinct side lengths?
For what integers $n > 2$ is there a rectangle with real-valued size lengths such that the rectangle can be tiled by $n$ squares so that the squares have at least $n - 1$ different side lengths? (That is at most $2$ squares involved in the tiling have identical side length.)
Only one questions needs to be answered for acceptance.
