The most popular way of explaining this, is to bring the number to just one blue eyed islander.
With just one, the answer is obvious. The guru says someone has blue eyes, the blue eyed person doesn't see anyone else with blue eyes, so they deduce their eye color.

Now bump the number up to two. Each blue eyed islander sees the other, and reasoning by the one blue eyed scenario assume the other will leave that night. When the other islander doesn't leave, they both deduce that their own eye color must be blue, and they leave on the 2nd night.

Each succesive blue eyed islander you add adds one more day of deduction.

Alrighty, this ones a bit harder to explain, but it is entirely my work. Which was kinda the whole point of these two entries.

The answer lies in the idea of belief.


To simplify the explanation, I'm gonna reduce the number of blue eyed islanders to 10, but this holds for any number.

So, imagine the ten islanders, who we'll call A, B, C, D, E, F, G, H, I, and J.

Consider A. A can only see 9 blue eyed islanders, and assumes his own eye color is brown. (if he assumed it was blue, he'd be assuming the qualifying statement and have some sort of backwards tautology thing go on.)

Now because A believes there are only 9 blue eyed islanders, he believes B can only see 8.
Being logically perfect, A knows that B will assume B's own eye color to be brown. A also knows that B knows that C will assume C's eye color to be brown.
So A believes B believes that C sees 7 blue eyed islanders.

You can follow this progression all the way to J:
A believes B believes C believes that D sees 6.
A believes B believes C believes D believes E sees 5.


Then, all the way at the bottom:
A believes B believes C believes D believes E believes F believes G believes H believes I believes that J sees 0 blue eyed islanders.

It's important to note, that A does not believe J sees 0 blue eyed, it's very obvious that A believes J sees 8.
Only through the progression of perception do we get this result.


This is what changes when the Guru speaks.
Once the Guru speaks, every one knows that every one knows that every one know..... that there is at least one blue eyed person.
Which means that A believes B believes C believes D believes E believes F believes G believes H believes I believes that J sees 1 blue eyed islanders, because A knows that everyone knows there is a blue eyed person.

This triggers the events above.