  We consider tilings and packings of $\RR^d$ by integral translates of cubes
$[0,2[^d$, which are $4\ZZ^d$-periodic. Such cube packings can be described by
cliques of an associated graph, which allow us to classify them in dimension
$d\leq 4$. For higher dimension, we use random methods for generating some
examples.
  Such a cube packing is called {\em non-extendible} if we cannot insert a cube
in the complement of the packing.
  In dimension 3, there is a unique non-extendible cube packing with 4 cubes.
We prove that $d$-dimensional cube packings with more than $2^d-3$ cubes can be
extended to cube tilings. We also give a lower bound on the number $N$ of cubes
of non-extendible cube packings.
  Given such a cube packing and $z\in \ZZ^d$, we denote by $N_z$ the number of
cubes inside the $\4t$-cube $z+[0,4[^d$ and call {\em second moment} the
average of $N_z^2$. We prove that the regular tiling by cubes has maximal
second moment and give a lower bound on the second moment of a cube packing in
terms of its density and dimension.

Combinatorics