Lattice coverings by parallelotopes containing cube vertices

We investigate when a parallelotope containing vertices of the unit cube $Q_n=[0,1{]}^n$ covers ${\mathbb{R}}^n$ by integer translates. We prove the covering property for every parallelepiped containing five vertices of $Q_3$, and, in any dimension, for every parallelotope containing a facet of $Q_n$ and one vertex of the opposite facet. As a consequence, we obtain the sharp bound $1\mathrm{/}n$ for the volume of a pyramid contained in $Q_n$ with a parallelotope base. We also give an alternative proof of this bound.