Packing of non-blocking cubes into the unit cube (Q6600956)

A sequence of three-dimensional cubes with edge lengths \(l_1,l_2,\ldots\) is called non-blocking if \(l_{i_1}+l_{i_2} \le 1\) whenever \(i_1 \ne i_2\). It is shown that every non-blocking sequence \((C_i)_{i=1}^\infty\) of cubes with edges parallel to the coordinate axes and of total volume at most \(1/3\) admits parallel packing into the unit cube \([0,1]^3\). That is, there are suitable translates \(\tau_1(C_1),\tau_2(C_2),\ldots\) that are placed in \([0,1]^3\) and have mutually disjoint interiors.