KöMaL 2017 A704

A regular triangle has side length $n$. We divided its sides into $n$ equal parts and drew a line segment parallel with each side through the dividing points. A lattice of $1+2+\cdots+(n+1)$ intersection points is thus formed. For which positive integers $n$ can this lattice be partitioned into triplets of points which are the vertices of a regular triangle of side length $1$?