The answer is that on the 100th day, all 100 blue-eyed people will leave. It's pretty convoluted logic and it took me a while to believe the solution, but here's a rough guide to how to get there. Note -- while the text of the puzzle is very carefully worded to be as clear and unambiguous as possible (thanks to countless discussions with confused readers), this solution is pretty thrown-together. It's correct, but the explanation/wording might not be the best. If you're really confused by something, let me know.
If you consider the case of just one blue-eyed person on the island, you can show that he obviously leaves the first night, because he knows he's the only one the Guru could be talking about. He looks around and sees no one else, and knows he should leave. So: [THEOREM 1] If there is one blue-eyed person, he leaves the first night.
If there are two blue-eyed people, they will each look at the other. They will each realize that "if I don't have blue eyes [HYPOTHESIS 1], then that guy is the only blue-eyed person. And if he's the only person, by THEOREM 1 he will leave tonight." They each wait and see, and when neither of them leave the first night, each realizes "My HYPOTHESIS 1 was incorrect. I must have blue eyes." And each leaves the second night.
So: [THEOREM 2]: If there are two blue-eyed people on the island, they will each leave the 2nd night.
If there are three blue-eyed people, each one will look at the other two and go through a process similar to the one above. Each considers the two possibilities -- "I have blue eyes" or "I don't have blue eyes." He will know that if he doesn't have blue eyes, there are only two blue-eyed people on the island -- the two he sees. So he can wait two nights, and if no one leaves, he knows he must have blue eyes -- THEOREM 2 says that if he didn't, the other guys would have left. When he sees that they didn't, he knows his eyes are blue. All three of them are doing this same process, so they all figure it out on day 3 and leave.
This induction can continue all the way up to THEOREM 99, which each person on the island in the problem will of course know immediately. Then they'll each wait 99 days, see that the rest of the group hasn't gone anywhere, and on the 100th night, they all leave.
Before you email me to argue or question: This solution is correct. My explanation may not be the clearest, and it's very difficult to wrap your head around (at least, it was for me), but the facts of it are accurate. I've talked the problem over with many logic/math professors, worked through it with students, and analyzed from a number of different angles. The answer is correct and proven, even if my explanations aren't as clear as they could be. If you're not convinced, you can contact me (info available on my website) for further discussion. If you're satisfied with this answer, here are a couple questions that may force you to further explore the structure of the puzzle:
a potato
His eyes are blue
Blue moon is a board game which was published by Kosmos/Fantasy Flight games in the year 2004
Michael mcintyre's eyes are brown.
Taking out the word riddles makes this What is life if life isn't life but a life is a life yet life doesn't make sense like life should? This riddle makes no sense- a riddle is a riddle. a riddle is not, not a riddle because a riddle is a riddle. This riddle acts as if riddles are different than riddle (no s). To answer the riddle I would answer: Riddles are riddle.
A Potato
a potato
What is the hardest riddle and the answer? is the riddle. The next answer is the answer.
What flies by, has two arms, but no beak, wings, or eyes? A Clock!
clock
potato
The buss drivers eyes are what ever color your eyes are you are the driver If you still don't get it think about it for a while re read the riddle and think a little harder!
The question, "What has 1000 eyes, 2000 shoes, and eats grass?" is a riddle. The answer to this riddle is 500 horses.
green eyes
His eyes are blue.
If you mean a song, than that is Mika - Blue Eyes :)
Most do not have blue eyes and the ones with blue eyes are usually blind.