Is there a covering of $\mathbb{Z}^n$ by disjoint translates of the basis-and-origin minimal integer $n$-simplex? By haphazard I have such coverings for $\mathbb{Z}$, $\mathbb{Z}^2$ and $\mathbb{Z}^3$, where the wanted translations are lattices spanned by $\{2\}$, $\{(2,-1),(-1,2)\}$, and $\{(1,1,-1),(1,-1,1),(-1,1,1)\}$, but rhyme nor reason can I see in this sequence of families to extend.