The guru does provide very indirect meta-information. Everyone's inaction cannot be taken as information on the first day, but inaction over time can.

Consider the case with 100 brown and 1 blue person (+ guru). As soon as the guru speaks, the blue person knows their eye colour and leaves.

Consider the case with 100 brown and 2 blue people. After the guru speaks, both of the blue people still don't know their own eye colour - the guru could have been thinking of the other one. So both blue people stay on the island on the first night. The other one staying on the island is information to the each of the blue people. Because the other blue one didn't leave on the first night, they must be also unsure - and therefore they must have blue eyes themselves. Both leave on second night.

See also: http://img.spikedmath.com/comics/445-three-logicians-walk-in...

Exactly this; ProblemFactory has shown how the guru provides directly actionable information for the simplest case of the problem (1 blue eyed islander) and also highlighted the inductive step that leads to a proof for any number of blue eyed islanders.

Reducing to the simplest scenario and building (as ProblemFactory) has done, makes the result much more intuitive -- as it does in most proofs, really.

Also, the logicians comic is wonderful :)

I found this difficult to pinpoint, too, until I read about common knowledge. The example here is very relevant: http://en.wikipedia.org/wiki/Common_knowledge_(logic)

At first, the islanders do not know that the other islanders are making the same logical judgments.

The proof relies on the fact that each islander knows what the other islanders have deduced by each day.

By making an announcement to everyone at once, the new information that the guru provides is the common knowledge that (i) there is a blue eyed person, (ii) the islanders know there is a blue eyed person, (iii) the islanders know the other islanders know that there is a blue eyed person, and so on.

Note that if the guru went to each islander one by one and told them individually that there is a blue-eyed person, they do not learn anything new about the other islanders and nobody would leave.

the version of this I've seen before every one has blue eyes (except maybe the guru). day one some body with blue eyes looks around and knows there's at least one other person with blue eyes, so maybe the guru was talking about that person. day two no one has left so they know there's at least two people with blue eyes or else the blue eyed would have figured it out day one. of course they can still see at least two people with blue eyes, so no problem. Etc... day N hits and they come to the inescapable conclusion that they all have blue eyes and they all leave.

I'm not sure how that translates to two eye colors though. I'd guess basically the same, but I'd need to think some more.

CAUTION THIS IS PROBABLY A SPOILER:

Well, the simplest explanation is that the proof involves thinking about hypothetical worlds and each agent modelling everybody's behaviour in each hypothetical world, including what hypothetical-hypothetical worlds can arise, and hypothetical³ and so on. The Guru's statement exist in every world, no matter how hypothetical it is, while the subjective observation that there are blue-eyed people disappears in some of them.