Just before I fell asleep last night, I was thinking of attempting a chain of reasoning that goes from top down instead of from bottom-up, which is how I see my previous thoughts on this topic:

If you can see N blue-eyed people, then it is possible that if you don’t have blue eyes then any one of those N blue-eyed people can see N-1 blue-eyed people. Therefore, you are sure everybody can see at least N-1 blues. Thus, it is common knowledge that there are N-1 blues or more people.

So, if there are 50 people, you are reasonably sure that everyone can see at least 49. “There are 49 or more blues” is common knowledge

However, if you think that someone sees 49 people, then that person is inclined to think that there exists someone who sees only 48 by the above argument.

The general principle is if you can personally see N people then you can be sure that:

*Everybody knows there are at least N-1 blue-eyed people.*

*Everybody knows that everybody knows there are at least N-2 blue-eyed.*

A pattern starts up.

*Everybody knows that everybody knows that everybody knows there are at least N-3 blue-eyed people.*

This can go on forever.

*Everybody knows that everybody knows that … there are at least 0 blue-eyed people.*

The stranger adds the information that:

*There is at least 1 blue-eyed person.*

*Everybody knows there is at least 1 blue-eyed person.*

*Everybody knows that everybody knows there is at least 1 blue-eyed person.*

*Everybody knows that everybody knows … there is at least 1 blue-eyed person.*

Thus, one of the many pieces of information given by the stranger must eventually contradict the statement concerning 0 blue-eyed people.

What I am really trying to emphasize here is that each individual has a series of beliefs. One of the dangers here is to think of each person’s beliefs in isolation without looking at the totality of statements that they believe, not just about what they think, but also about what everybody thinks, and about what everybody thinks that everybody thinks.

(This article was originally a comment which can be found here.)

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I think I’ve got it !

Thanks very much for the working-backwards explanation. I’ve been struggling with this problem for a few days, with my understanding, and indeed my belief in the answer was fading in and out, but your working backwards logic makes it clear. The Monty Hall problem is easy compared to this, and even that fooled me initially.

Wow !

Hywel

[…] further explanations of this puzzle here and here, where the synchronization can also be interpreted as between all “levels” of […]

Sorry, but I don’t think you can extend that pattern forever. Imagine N people . X of which have blue eyes, and Y have brown (such that X+Y=N). Imagine that the statement given was that there is at least one blue eyed person. I have brown eyes, but am also unaware of that. I can see X blues and can deduce that there are either X or X+1 blues (and, therefore, either N-X or N-(X+1) browns), depending on which I am. Imagine another person, “you”. You have blue eyes. You can see Y brown eyed people, and can therefore deduce that there are either N-Y or N-(Y+1) blue eyed people (and, consequently, Y or Y+1 brown eyed people), depending on what you are. Now, you deduce for this third person. He has blue eyes, and has been making deductions in parallel with you. You both imagine what the other sees. You deduce that he can either see N-Y or N-(Y+1) blue eyed people, and that he is deducing that there are either “N-(Y+1) or N-(Y+2)” or “N-Y or N-(Y+1)”, depending on what he is. But, you realize, he has been making deductions in tandem with you, and both of you can see that, in order for there to be N-(Y+2) blue eyed people, there must be Y+2 people of another eye color, and you both can see that there are Y brown eyes, so there can be at most N-Y blue eyeds, and you can see N-(Y+1) blues, so there must be at least N-(Y+1). So the pattern cannot extend past here, where the deductions start to contradict the observed facts.