Logic/Induction
Problem: 1000 perfect logicians are imprisoned. 500 have blue eyes, 499 people have brown eyes, and one has green eyes. They can see each other's eyes but not their own, and cannot communicate eye colors to each other. At the end of every day, those who have deduced their own eye color are freed. The guard states, "At least one of you has blue eyes." Who gets freed and when?
Answer: If one guy has blue eyes, he sees no other blues and realizes it's himself on the first day. If there are two blues, the guy who sees only one blue sees that that guy doesn't figure it out on the first day, so he must be blue as well. If there are 3 blues, each blue will see the other two blues not be freed after two days so they must be blue as well. By induction, the 500 blues will be freed on the 500th day.
Question: Since there are more than one blue present, each prisoner already knows that at least one prisoner has blue eyes, so seemingly, the guard's statement adds no new information, but seems to be a necessary requirement for induction. Why is that so? If it's not so, then the browns would be freed on the 499th day, right? Since they're perfect logicians and know that everyone can see at least one blue and one brown, can they not simply infer the base case without the guard stating it out loud?
Answer: If one guy has blue eyes, he sees no other blues and realizes it's himself on the first day. If there are two blues, the guy who sees only one blue sees that that guy doesn't figure it out on the first day, so he must be blue as well. If there are 3 blues, each blue will see the other two blues not be freed after two days so they must be blue as well. By induction, the 500 blues will be freed on the 500th day.
Question: Since there are more than one blue present, each prisoner already knows that at least one prisoner has blue eyes, so seemingly, the guard's statement adds no new information, but seems to be a necessary requirement for induction. Why is that so? If it's not so, then the browns would be freed on the 499th day, right? Since they're perfect logicians and know that everyone can see at least one blue and one brown, can they not simply infer the base case without the guard stating it out loud?