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.



Discrete disjoint covering of integer lattices
Let $S$ be the set of integer points $(x\_1,x\_2,\dots,x\_n)$ satisfying 


$$x\_1+2x\_2+3x\_3+\dots+nx\_n \equiv 0 \mod n+1,$$


and $T$ be the basis-and-origin simplex as described in Ben's comment. 


Then translates of $T$ by $S$ disjointly cover $\mathbb{Z}^n$ (since decreasing the $x\_i$ coordinate by $1$ changes the left hand side of the above relation by $i$, for any point not in $S$ there's exactly one direction we can move in to reach $S$). 


