[mkl]

ε > 0. Yes, overlapping triangles. Add an arbitrary + 1 to what? You need to arrange the small triangles so that they cover the big triangle of side length n+ε. n² unit equilateral triangles cover a big equilateral triangle of side length n without overlap, so at least n²+1 are needed for side length n+ε, and the paper shows (not especially clearly, IMHO) that at most n²+2 are needed.

[crote]

Figure 1 on its own is a pretty decent demonstration, once you zoom in quite a bit.

The annoying part about the paper is figure 2: it shows a different method of doing so, without mentioning that it is unrelated to figure 1. It is also drawn in a less obvious style, which really hurts its readability.

[mkl]

Yes, Figure 1 is okay, but Figure 2 is not very clear; most of the triangles are missing and have to be imagined. I also think the examples for a single n are not the best argument for the general case - the reader can extrapolate other diagrams and a proof but I think a slightly longer paper could be clearer.