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**atreyyu** We have an equilateral triangle of side $\displaystyle n$. It's been cut into $\displaystyle m$ equilateral triangles of side 1 and some rhombi whose acute angles is $\displaystyle 60^\circ$. I have to show that $\displaystyle m\geq n$. So, it's straightforward when I draw it, that there are less than $\displaystyle (n^2-n)/2$ pairs of different smaller triangles with one common side. I could describe the situation as I see it, which is trying to stick in another rhombus step-by-step to show when it eventually becomes impossible. But is there a more formal way to do that? If not, what's the most formal way of presenting what I have described?