It is known that a square can be tiled with $n$ rectangles whose length is double their width for any $n > 4$. In particular, no two rectangles can overlap and no part of any rectangle is outside the square. The rectangles can be of different size.
I am now investigating the same problem for rectangles whose length is triple their width. Clearly if we can tile a square with $m$ rectangles then we can also tile it with $m+4$ rectangles using the "building up" method from the video. Also by placing the square into a $2 \times 2$ square grid (adding 3 additional squares), we can also tile it with $m+9$ rectangles. Using these two facts I was able to show that we can tile a square for any $n > 26$. However I still don't know which $n \leq 26$ do not have a tiling (other than the trivial ones). How can I find them?
I am also interested in the general version of this problem, ie., when the length of the rectangles is $k$ times their width.