In the case of a message designed to last hundreds of
millions of years all bets are off. Over such a time scale,
any given point is likely to be hit by a meteorite large
enough to destroy artifacts on the surface made out of any
known material, so to survive for 4108 years the message
would need to be securely buried at depth. Therefore,
it would also need a beacon that is detectable
through at least several meters of regolith. This could be
long-wavelength radio, a strong magnetic field, or (in
principle) something more exotic such as a neutrino
generator. It could also, of course, involve some form of
more advanced technology as yet unknown to us. In any
case, there would probably be no visual evidence left on
the moon’s surface to attract our attention that anything
of interest was in the area. However, there might still be a
way to narrow the search. The north and south poles are
obvious choices for placing messages, but thanks to the
moon’s tidally-locked orbit, there are other points that
can be uniquely pin-pointed over a long period of time,
such as the sub-Terran point (the point directly below the
Earth) and its antipode, and the centers of the leading and
trailing hemispheres in the moon’s orbit. One of these
might be deliberately chosen as the site of a message in
the knowledge that these points would have significance
to a scientifically literate terrestrial community. But without
any surface features to indicate the presence of a
subsurface artifact, detection would have to depend on
ground-penetrating radar or deliberate excavations by
future manned expeditions.