[LeadbellyDan]

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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.



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I guess the basis for why this works must be something like this but what you say doesnt explain why this works no matter how many have blue eyes.

If there were, say, 50 blue eyed horned monks (you started the exampled mixing so I thought I'd go all the way) within the group then the blue eyed horned monks would all know that everybody else knows that there are at least 48 blue eyed horned monks.

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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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I think what this stems from is that there is no monk which everyone knows has blue eyes. The knowledge of the group as a whole - what everyone agrees upon - does not include "Billy has blue eyes," "Tom has blue eyes," "Sally has blue eyes," etc and so it also cannot include "there is at least one blue eyes (possibly horned) monk."

*Im nearly starting to understand this but I'm too tired and can't think any more. I'll just post this for now - sorry if its incoherent. I find this fascinating btw - I'm suprised its not more well known.*