New results on torus cube packings and tilings

We consider the sequential random packing of integral translates of cubes $[0,N]^n$ into the torus $\mathbb Z^n/2N\mathbb Z^n$. Two particular cases are of special interest: (1) $N=2$, which corresponds to a discrete case of tilings, and (2) $N=\infty$, which corresponds to a case of continuous tilings. Both cases correspond to some special combinatorial structure, and we describe here new developments.