ChrisVer said:
As I said in a previous post, no you don't need a prophet for that case, because it is an extreme... watch out; not needing him does not mean you get something wrong if you include him. We can just say that in that particular case his existence is unecessary...
But in all other cases his presence is needed to reach a reliable answer... (as I mentioned the RRB state, in which without the prophet everybody'd die)...I guess the prophet makes the problem solvable in all cases then, and so he's necessary for the solution, and for some special cases his information is just repeating.
But I believe that the bbb case where all 3 islanders are b allows an extension to a very large number of cases.
To be precise, as I see it, we only need 3 or more islanders with b for the prophet's pronouncement to be useless. In that case, there is no new relevant common knowledge and they would have committed suicide on a count starting from finding out about the suicide law.
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Why? Consider one of the other islanders, X, besides the 3 (say A, B, C) who must have blue eyes - these other islanders may have blue or brown (let's go with red, r) eyes. Of course all the islanders know [someone has blue eyes] already (X sees A, B, C with blue eyes; A sees B, C with blue eyes; B sees A, C with blue eyes; etc.). Do all the islanders know that {all the islanders know that [someone has blue eyes]}?
Well, X knows A can see B, C with blue eyes, and X knows B can see A, C with blue eyes, etc. so X knows {A knows [someone has blue eyes]}, X knows {B knows [someone has blue eyes]}, etc. X knows all the other islanders besides the 3 (A, B, C) will make the same inference about A, B and C, so X knows that {all the islanders know that [someone has blue eyes]}. (So all islanders besides the 3 (A, B, C) know that {all the islanders know that [someone has blue eyes]}.)
A knows B can see C with blue eyes, so A knows that {B knows that [someone has blue eyes]}. By symmetry A knows that {C knows that [someone has blue eyes]} (here the someone is B), and B knows that {A knows that [someone has blue eyes]}, B knows that {C knows that [someone has blue eyes]}, so forth. Also, A knows X can see B, C with blue eyes, so A knows that {X knows that [someone has blue eyes]}. By symmetry, B knows that {X knows that [someone has blue eyes]} (here someone is A, C), and C knows that {X knows that [someone has blue eyes]} (here someone is A, B). Since the same as applies to X also applies to all islanders besides these 3, thus A, B, and C know that {all the islanders know that [someone has blue eyes]}.
So all islanders know that {all the islanders know that [someone has blue eyes]}.
So if there are at least 3 islanders with blue eyes, then all islanders know that {all the islanders know that [someone has blue eyes]} - no need for a prophetic announcement.
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It is true that the prophetic announcement cannot hurt, but the problem is that in many cases (those above) it does not suffice as a time-marker for starting the count. Rather, the suicide law itself (along with the villagers first seeing each other) would be the marker for the time to start in such cases.