  In this paper we study some cube packing problems. In particular we are
interested in compact subsets of $\mathbb{R}^n,n\geq 2$, which contain
boundaries of cubes with all side lengths in $(0,1)$. We show here that such
sets must have lower box dimension at least $n-0.5$ and we will also provide
sharp examples. We also show here that such sets must be large in general in a
precise sense which is also introduced in this paper.
