Well, the problem is fundamentally that the people should've been playing the game before anyone said anything, No, that's not true. The proof-by-induction doe

Well, the problem is fundamentally that the people should've been playing the game before anyone said anything, No, that's not true. The proof-by-induction doesn't work without the guru, because the initial case, N=1, is a failure. If there is only one blue-eyed person (call him "Adam"), he looks around and see a lot of brown-eyed people but has no way to know whether his own eyes are blue, or not. Meanwhile, the other 999 people see a single blue-eyed person, Adam, who isn't leaving the island, but they think it's just because Adam doesn't realize that there are any blue-eyed people at all. Once the guru makes the observation then the chain of induction is set in motion. So, yes, that observation most certainly does convey information. The real question is: why do our minds persist in believing that the guru's statement conveys no information? IMHO, it's a matter of psychology and anthropology. Proof-by-induction is, literally speaking, an inhuman trick. No creature on Earth can do it except humans, and most humans can't do it without special training. (We call those trained humans "math students".) Because there weren't any math students around during most of evolutionary history, our mental models of other people's behavior do not include proof-by-induction. When we see a problem like this -- one which involves an island containing 1,000 math students who apply inductive reasoning flawlessly and continuously -- our mental model of behavior just goes haywire. This scenario is literally outside our mind's design parameters, so it's completely nonintuitive. Just solving the problem at all requires a heroic mindhack -- you have to plod along through a set recipe, starting with the N=1 case and proceeding by a tedious chain of cases. It's fun to imagine a creature who could just see the answer to this problem, in the same way that many, many humans can spot a fleeting facial expression on the face of a Survivor contestant and just know, immediately, who will be voted off the island. (That is a task that our brain was designed for.) Perhaps some savants can do it. Perhaps Ramanujan could have done it. Perhaps Garry Kasparov can do it. Perhaps you can do it (but, if you can, it may be hard to explain how...)