Here’s an old puzzle:
Everyone in the town of Erdosville has either blue eyes or brown eyes. However, Erdosville is cursed: anyone who can determine his or her own eye color to be blue will die the next midnight. Of course, Erdosville is a small and friendly community, and everyone sees everyone else every day. Naturally, there are no mirrors, and no one would ever discuss eye color. However, one day, a visitor stops by Erdosville. When he leaves, he announces to the entire community that he enjoyed his stay, but he noticed that at least one person in Erdosville has blue eyes.
Does this have any effect on the population?
Yes, it does. If there are n people in Erdosville with blue eyes, they will die n days later. (If you don’t see why, try to figure it out.)
It seems like a fairly weak puzzle to me though; the only purpose the visitor has is to start a counter. Exactly how strong is this counter-starting condition? Here are some variants (that I haven’t attempted to solve yet, but I don’t think they’re very difficult) that may perhaps shed some light on the question:
Suppose one person didn’t hear the announcement and that everyone knows that she misssed it. What is the effect in this case? Does it matter if she has blue eyes or brown eyes?
What if one person misses the announcement but gets told about it k days later (and everyone knows that)? Then what is the effect?
What happens if everyone is cursed so that as soon as a person can determine his or her own eye color to be either blue or brown then that person will die the next midnight? Do they need to be told that at least one person has blue eyes and at least one person has brown eyes?
What if everyone is only guaranteed to see everyone else once a week?
Does anyone have any other variants?