I think this is a property spheres. It seems to me that any two spheres that are touching have a straight line from one center to the other center exactly through the point of contact. Try thinking of just two spheres and adding more in step-by-step.
Then the result follows because all the spheres are defined as centered on the cube/sub-cubes respectively.
Then the result follows because all the spheres are defined as centered on the cube/sub-cubes respectively.