I know how to pack $5$ unit squares in a square of side length $2+\frac{\sqrt{2}}{2}$. Is there an $\varepsilon>0$ such that there exists a packing of $9$ unit cubes in a cube of side length $3-\varepsilon$?


(Inspired by [this question](https://mathoverflow.net/questions/131047/how-to-determine-the-number-of-a-cube-within-a-bigger-cube).)



Is this cube packing possible?
Yes, since one can pack 10 unit cubes in a cube of side length $\:\:2+\left(\hspace{-0.023 in}\frac12\hspace{-0.05 in}\cdot\hspace{-0.03 in}\sqrt2\right) \;\;$.


See <https://erich-friedman.github.io/packing/cubincub> .


