yy2bggggs wrote:There's a different way to get at this, too. I'll present that some time this weekend.
That's fine. However, understand that I presented a logical argument, and still don't see anything wrong with it (other than my initial misstatement of the induction direction).
What I'm saying is wrong is that there's only one bit of relevant information. The particular piece
of information you cited is indeed relevant, and is one bit of information, but there's information communicated at each tribal meeting.
The inductive argument of what would happen at day 1, day 2, etc is perfectly valid, and provides a solution to this puzzle, but it's not the whole picture. The tribe is in a completely distinct state after the first tribal meeting; after the second, it's in another completely distinct state.
Even the initial announcement by the guru gives more information that the content of the announcement. The guru, by announcing this to the whole tribe, starts the chain--but if the guru gave the same exact announcement
, individually, person to person, to every member of the tribe, nothing would happen at all.
The tribe goes into a different state when the guru makes the announcement to the whole tribe, than it would be in if the guru just made the announcement individually. This is because in the former case, information is gained
, but in the latter case, it is not.
This all points to the same thing--information gained depends upon other tribal members being present. So obviously, it involves other members; however, it's also obviously not just what other members know, because that's just a symmetric situation to what a particular individual knows.
That's as far as I'll dare go at the moment explaining this, but here's the different situation. Let's consider the original puzzle--this time, we're going to make variations on the state of the tribe.
Guru announces to the tribe that the guru sees a blue eyed person. Given n blue eyed people, everyone knows their eye color by day n.
Guru announces to each individual, one on one, that she sees a blue eyed person. Nothing ever happens.
Guru announces to the tribe that she sees n blue eyed people. Everyone leaves the first day.
Guru announces to each individual, one on one, that she sees n blue eyed people. Everyone leaves the first day.
Guru announces to the tribe that she sees (at least) k blue eyed people, 1>k>n. What happens?
Guru announces to a group of j blue eyed people (subgroup of the tribe), that she sees (at least) k blue eyed people (in the whole tribe). 1>j>k>n. What happens?
Same as above, 1>k>j>n. What happens?