An equilateral triangle can be divided into smaller equilateral triangles by drawing $n$ lines parallel to each side with equal spacing, as this image shows:
Moreover, the vertices can be labeled with $1,2,3$ so that each smaller triangle contains exactly one of $1,2,3$. To do this, we start with $1$ on the first row, $2,3$ on the second row, $3,1,2$ on the third row, and so on.
Is there an analogous division and labeling for higher dimensions? For instance, if we start with a regular tetrahedron?