Wrap the whole group in an imaginary film. The film takes the shape of parts of cylinders, parts of planes, and parts of planets. The parts of the planets that contact the film are precisely those that are invisible to the other planets; a planet floating inside the group is entirely visible.

The invisible regions on the planets are bordered by spherical arcs, parallel two by two. If we translate all these arcs onto a single planet, they’ll partition it into several parts, each corresponding to the invisible region of some planet. Together these parts cover the planet completely.

A Soviet problem from the 22nd International Mathematical Olympiad in 1981. Here’s the same idea in two dimensions.