From the upper bound $\sqrt{2}n$ on the edge length of the smallest regular simplex containing the unit cube in $\mathbb{R}^n$, shortly outlined in the question http://mathoverflow.net/questions/139161/smallest-regular-simplex-containing-the-unit-cube-in-rn, it follows that roughly $n^n2^{n/2}$ unit simplices are sufficient to cover the unit cube by translation.