Recently I posted the "traveler problem," which was also stated here a few months ago with different but equivalent vocabulary. The real question amounts to "why the traveler (stranger)?"
link to original
The answer given in the above thread was not quite correctly justified, although it contains the essence of the idea. Basically the proof of the statement "everyone dies (leaves)" is by induction, and saying that the traveler is necessary to start the induction is essentially saying that any proof that the traveler is unnecessary probably wouldn't use induction.
So, why the traveler? In other words, what is precisely the new knowledge that the islanders (villagers) have?
(Some clarifications: "Rational agents" are defined to have certain knowledge and are not allowed to act on anything but knowledge that follows from their original knowledge. In this problem, the islanders know that there are two types of eye color (i.e. horns or no horns), and they know the customs, and they know that everyone else is also a rational agent with exactly this knowledge.)
Hint in white:
<font color="white">Consider what the islanders know about the other islanders' knowledge. How does this change when the traveler makes the announcement? </font>