100 blue eyes island

Question: There is an island with exactly 201 residents, 100 with blue eyes, 100 with brown eyes, and the island leader (who has green eyes). To leave the island, one must know their own eye color. There are no reflective surfaces on the island and no on can communicate with each other other than the leader to the residents. No one on the island knows how many of each eye color there is. Everyone on the island is a perfect logician, meaning that if there us a solution they'll find it. Every morning the leader gives anyone a chance to leave the island by guessing their eye color. One morning, the leader gathers all 200 residents to make an announcement, he says "at least 1 person on this island has blue eyes" How many people leave the island and in how many days after the announcement? Notes: this is known as one of the hardest riddles ever

All 100 blue eyed people in 100 days.

EXPLANATION

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.

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.

Hope this answers your question... ...HACKGH FOR LIFE...

You are most definitely right. You are awesome. I didn't expect anyone get it this easily.

Yeah... Thanks man.... You are awesome too.... The presence of great minds like you guys always makes things easy.... HACKGH IS AWESOME..