Is there a method to make a rep-n rep-tile for any number n, using only triangles? And if there is no such method, what's the smallest number for which there's no example? I'm only considering rep-tiles which are all of the same size, as they make up the larger congruent rep-tile.
I first asked the more general question, can you do it for any shape, not just triangles? But then I felt confident that you can do it with rectangles, for any n.
You can indeed do it for rectangles, by taking a ratio of sides equal to $\sqrt{n}$.
For triangles, M. Beeson (2012), Triangle Tiling I: the tile is similar to ABC or has a right angle (pre-print) gives the following results:
When the tile is similar to $ABC$, we always have "quadratic tilings" when $N$ is a square. If the tile is similar to $ABC$ and is not a right triangle, then $N$ is a square. If $N$ is a sum of two squares, $N = e^2 + f^2$, then a right triangle with legs $e$ and $f$ can be $N$-tiled by a tile similar to $ABC$; these tilings are called "biquadratic". If the tile and $ABC$ are $30-60-90$ triangles, then $N$ can also be three times a square. If $T$ is similar to ABC, these are all the possible triples $(ABC, T, N)$.
$n = 6$ is not a square, a sum of two squares, or three times a square, so this is the first $n$ for which there is no example.
