3 ms·

If n=3, then each of the blue-eyed people know that there are blue-eyed people and know that the other blue-eyed people know. However, they don't know that all

If n=3, then each of the blue-eyed people know that there are blue-eyed people and know that the other blue-eyed people know. However, they don't know that all the blue-eyed people know that the blue-eyed people know. This is the piece of information that is learned by the statement given. In general, if there are N blue-eyed people, then it is the Nth abstraction of "he knows that I know that he knows that I know that..." that is learned by the statement.

voyou 13y ago
"they don't know that all the blue-eyed people know that the blue-eyed people know" Yes, they do. In the three-person case, a blue-eyed person can see two other blue-eyed people, A and B, and they know that A can see B, and vice-versa, so they know that A and B both know that there are blue-eyed people, and they know that both A and B would be able to us the same logic they used, so they also know that A and B know that A and B know that there are blue-eyed people.
vytasgd 13y ago
Can you go into further detail about the nth abstraction of "he knows that I know..." and what effect that has on the population? The way I see it is that as soon as everybody knows that there are blue-eyed ppl and everybody knows that everybody else knows there are blue-eyed ppl the countdown begins... additionally, at that point you can already get to an infinite number of abstractions of the "he knows that I know that he.." statement because you both know that the other knows blue-eyed ppl exist, so you both know that the other knows that you know that blue-eyed ppl exist, so you both know....
Mindless2112 13y ago
At n = 3, every blue-eyed person knows everything every other blue-eyed person knows and that every blue-eyed person knows it, so you can already construct an infinitely-long chain of "I know that he knows that I know that he knows..." at that point. (Of course, I am assuming that all the people think/reason exactly the same and that they all know that they think/reason exactly the same, which are assumptions of the original problem.)

voyou 13y ago

"they don't know that all the blue-eyed people know that the blue-eyed people know" Yes, they do. In the three-person case, a blue-eyed person can see two other blue-eyed people, A and B, and they know that A can see B, and vice-versa, so they know that A and B both know that there are blue-eyed people, and they know that both A and B would be able to us the same logic they used, so they also know that A and B know that A and B know that there are blue-eyed people.

vytasgd 13y ago

Can you go into further detail about the nth abstraction of "he knows that I know..." and what effect that has on the population? The way I see it is that as soon as everybody knows that there are blue-eyed ppl and everybody knows that everybody else knows there are blue-eyed ppl the countdown begins... additionally, at that point you can already get to an infinite number of abstractions of the "he knows that I know that he.." statement because you both know that the other knows blue-eyed ppl exist, so you both know that the other knows that you know that blue-eyed ppl exist, so you both know....

Mindless2112 13y ago

At n = 3, every blue-eyed person knows everything every other blue-eyed person knows and that every blue-eyed person knows it, so you can already construct an infinitely-long chain of "I know that he knows that I know that he knows..." at that point. (Of course, I am assuming that all the people think/reason exactly the same and that they all know that they think/reason exactly the same, which are assumptions of the original problem.)