Challenge problem 1:
On an island there are 5 blue-eyed people and 5 brown-eyed people.
On the island the following things are common knowledge:
- Their are no reflective surfaces.
- No-one discusses eye color.
- If an islander deduces that his or her eyes are blue, then he or she will jump to their death into the sea the following midnight.
- The islanders are smart -- if, from the facts known to an islander, it follows logically from those facts that the islander has blue eyes, then the islander will indeed figure this out within a matter of minutes.
- Each day at noon the islanders meet in the town square.
Days go by and no-one jumps, because, although each person can see everyone else's eye color,
nobody knows their own. Then one day, a trusted visitor comes to the noon meeting and announces
- "There are people on this island with blue eyes."
- What happens?
- What precisely is wrong with the following line of reasoning:
- "What the visitor announced, everyone already knew. That is, everyone already knew that there were blue-eyed people on the island. Therefore, no new deductions are possible. Therefore, nothing different happens."
It suffices to answer (1-3) above, but if you get 1-3, I encourage you to try 4 also!
Submissions for this problem are due by Friday, April 9 at 5pm.