In this above picture, there are 15 cups, to make a perfect pyramid, you start the bottom row with 5 cups.
What is the formulae to start the bottom row, if I have $n$ cups.
[Math] Display of cups in triangle/pyramid
triangles
Related Solutions
You have to cut out four isosceles triangles with base $x$ and altitude $x/\sqrt2$. See picture below for a possible cutting pattern.
EDIT. Here's a way to get the triangles with only four cuts. Notice however that they are smaller than before.
Well, strict mathematical fractals don't exists in Nature or in reality, because their infiniteness would yield various paradox while the physical world is finite (e.g. at some point you get atoms). And basic mathematical fractals are too regular for Nature, where fractal-like patterns have more irregular variations.
Still, Pascal triangle with modulo looks quite like Sierpinski triangle, and some cell phone ultra-compact antenna are not without similarities.
Also, systems to amortise energy at all frequencies (sound, water waves) have more or less fractal shape.
NB: I won't say "cellular automata" are a "technology".
Best Answer
Let's start with the other way round. If the base has $k$ cups, the total number of cups $n$ will be given by:
$$n=k+(k-1)+(k-2)+\ldots+1=\frac{k^2+k}{2}$$ Or: $$k^2+k-2n=0$$ Now, just solve the quadratic equation to get: $$k=\frac{-1\pm\sqrt{1+8n}}{2}$$ Only one of these solution will be positive, and of course, only certain $n$ will have integer $k$.