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Combinatorics Difficulty 6.0 National Olympiad Find the answer Hungary

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 $1$ ?

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