On 2D packings of cubes in the torus

The 2D packings of cubes (i.e. squares) in the torus ${\mathcal {T}}^{2}={[0,1)}^{2}$ are considered. We obtain the exact expression $N_{2} (\lambda ) = \left \lfloor \lambda \lfloor \lambda \rfloor \right \rfloor$ for the quantity $N_{2} (\lambda )$, the maximal number of 2D cubes in a packing. (Here $1/\lambda$ is the length of sides of cubes, $\lambda \in {\bold {R}}, \lambda >2$.) Corresponding best packings are constructed. Both rank 1 best lattice packings and rank 2 best lattice packings are given.