There is a special town, Spotsville, where it is wonderful to live. However, to get into this town everyone must willingly take an oath, and everyone that takes the oath always keeps their promise. There is a special fountain in Spotsville from which everyone drinks. The water cures all ills, and guarantees a long and happy life. However, it also causes some people to grow a red spot on the back of their neck.
The oath that everyone takes is as follows: if you ever discover that you are one of the people who has a spot on the back of his neck, you must, near midnight on the same day, ring the town bell, and then leave the town forever. Because everyone in the town is nice, no one ever tells anyone else that they have a spot on the back of their neck. In addition, mirrors and other reflecting surfaces are very carefully controlled so that no one ever sees a spot on the back of their own neck. No one asks. No one tells.
It happens that exactly three people in Spotsville have red spots on the back of their necks. Spotsville is small, and everyone sees everyone else every day. Thus there is little mystery about who has the spots, except that no one knows whether or not they themesleves have one. In other words, most of the people in town see three spots, but three of the people (the ones with the spots on their necks) see only two. Similarly, non-spotted people know there are either three or four spots in town, and spotted people know there are either two or three spots in town.
One day a woman comes to town. Everyone knows that she always tells the truth. She calls a town meeting, and everyone comes. She speaks one sentence, as follows, and then leaves the town: "There is at least one person in town with a red spot on the back of their neck."
What happens? Does the system remain stable? If so, in what state? Does the system change? If so, how? Does anyone leave?
Interesting variation answer: what would be the only reason for your answer above to change?
Hints: (a) There are no tricks to this, the answer follows as an exercise in logic. (b) Everyone already knows that there is at least one spot in town because they can all see either two, or three, already. (c) This is a very HARD but very FUN problem, and the answer is interesting, but does not require any special knowledge.
Original author unknown.