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First I would like to comment that in this particular example not everyone leaves, just the ones with horns (doesn't make much difference but still)

It seems that the stranger provides additional information by letting the last horned guy in the chain of assumptions know that he has horns, but I can't quite pin down how exactly it is that this is new information.

I've spent roughly 7 hours thinking about this subject just in the last couple of days, and probably some more back in that original thread 8 months ago; and still can't quite get a "feeling" of understanding for the solution.

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Suppose there are 3 blue eyed monks. All 3 can see that there are 2 other monks with blue eyes. Everybody else can see there are 3 blue eyed monks. But what nobody knows is whether they themselves have blue eyes. Now a traveler comes and "Informs" them that there is at least 1 blue eyed Monk in their midst. But they all knew this already didn't they? Here's the thing. Yes, they all knew that Information. But what they DIDN'T KNOW was whether everybody else knew that everyone else knew that information.

That's what the Traveler now guarantees.

btw, Their reasoning is not really by "Induction". The proof by Induction shows that in any similiar situation where there are any number of blue eyed monks, the reasoning exists that produces the same conclusion for the number of days required with respect to the number of blue eyed monks. But for a specific case like 3 blue eyed monks, or 100 blue eyed monks, induction is not required. It is a straightforward chain of deductions on what the 100th monk thinks the 99 monks think the possible 98 monks think the possible 97 monks think the possible ... 1 monk thinks.


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