The cubic sphere packing has centers of spheres at all points (i,j,k) where i, j, and k are integers. The radius of each sphere is 1/2.
The face centered cubic packing has centers at each point on the lattice generated by the three vectors (0,1,1), (1,0,1), and (1,1,0). The fcc lattice is just the set of points in the integer lattice whose sum is an even integer. Points in the fcc lattice are a distance of square root of 2 from the neighboring point, so we take spheres of radius 21/2/2.
The cubic lattice sphere packing can easily be tiled by cubes containing one sphere. The fcc lattice sphere packing can be tiled by cubes as in the following figure.
Now we can compute the densities of these two packings.