Three Cube Packing for All Dimensions

Let $V_n\left(d\right)$ denote the least number, such that every collection of n d-cubes with total volume 1 in d-dimensional (Euclidean) space can be packed parallelly into some d-box of volume $V_n\left(d\right)$. We show that $V_3\left(d\right)=\frac{r^{1-d}}{d}$ if $d\ge 11$ and $V_3\left(d\right)=\frac{\frac{1}{r}+1}{r^d+{\left(\frac{1}{r}-r\right)}^d+1}$ if $2\le d\le 10$, where r is the only solution of the equation $2(d-1)k^d+d{k}^{d-1}=1$ on $\left(\frac{\sqrt{2}}{2},1\right)$ and ${(k+1)}^d{(1-k)}^{d-1}\left(d{k}^2+d+k-1\right)=k^d\left(d{k}^{d+1}+d{k}^d+k^d+1\right)$ on $\left(\frac{\sqrt{2}}{2},1\right)$, respectively. The maximum volume is achieved by hypercubes with edges x, y, z, such that $x={\left(2r^d+1\right)}^{-1/d}$, $y=z=rx$ if $d\ge 11$, and $x={\left(r^d+{(\frac{1}{r}-r)}^d+1\right)}^{-1/d}$, $y=rx$, $z=(\frac{1}{r}-r)x$ if $2\le d\le 10$. We also proved that only for dimensions less than 11 are there two different maximum packings, and for all dimensions greater than 10, the maximum packing has the same two smallest cubes.