Combinatorial cube packings in the cube and the torus

We consider sequential random packing of cubes z+[0,1]nz+[0,1]n with z∈1NZn into the cube [0,2]n[0,2]n and the torus Rn/2ZnRn/2Zn as N→∞N→∞. In the cube case, [0,2]n[0,2]n as N→∞N→∞, the random cube packings thus obtained are reduced to a single cube with probability 1−O(1N). In the torus case, the situation is different: for n≤2n≤2, sequential random cube packing yields cube tilings, but for n≥3n≥3 with strictly positive probability, one obtains non-extensible cube packings.So, we introduce the notion of combinatorial cube packing, which instead of depending on NN depends on some parameters. We use them to derive an expansion of the packing density in powers of 1N. The explicit computation is done in the cube case. In the torus case, the situation is more complicated and we restrict ourselves to the case N→∞N→∞ of strictly positive probability. We prove the following results for torus combinatorial cube packings:
• We give a general Cartesian product construction.
• We prove that the number of parameters is at least n(n+1)2 and we conjecture it to be at most 2n−12n−1.
• We prove that cube packings with at least 2n−32n−3 cubes are extensible.
• We find the minimal number of cubes in non-extensible cube packings for nn odd and n≤6n≤6.