If I have $n$ unit squares and want to build a bigger square out of the ones I already have, it is obvious that $n$ itself has to be a perfect square.
But after doing some elementary math it turned out for me that in order to construct a bigger equilateral triangle out of $n$ unit equilateral triangles, $n$ has to be a perfect square as well! I was completely shocked.
I also know that the square, triangle and hexagon are the only regular polygons you can tile a plane with (and the whole thing I've described doesn't work if you consider hexagons, as was pointed out to me by @Travis), so that was pretty much the end of the journey there.
Is there some kind of explanation to this? It seems very odd to me that you always need a perfect square.
If you build a larger triangle out of copies of a triangle (it need not be equilateral), say, $n$ to a side, then any linear dimension of the large triangle is $n$ times as long as that as the small triangle, and so the area of the large triangle is $n^2$ times that of the small, and therefore it takes $n^2$ small triangles to make a large one. The same argument shows that the same applies to any shape for which this works, not just triangles and squares.