[Math] 100 blue-eyed islanders puzzle: 3 questions

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I read the Blue Eyes puzzle here, and the solution which I find quite interesting. My questions:

  1. What is the quantified piece of information that the Guru provides that each person did not already have?

  2. Each person knows, from the beginning, no fewer than 99 blue-eyed people to be on the island. Then how is considering the 1 and 2-person cases relevant, if each person can dismiss these 2 cases immediately as possibilities?

  3. Why must they wait 99 nights if, on the first 98 or so of these nights, they're simply verifying something that they already know?

EDIT: Most answers seem to concentrate on question 1 which I understand partly: but I remain confused because of different answers.
Can someone answer questions 2 and 3?

Best Answer

Here's the story of one blue-eyed islander. The Guru said she saw someone with blue eyes. He looked around and thought "Hey, I don't see anyone with blue eyes. I guess she means me." And so he left right away.

Here's the story of two blue-eyed islanders. The Guru said she saw someone with blue eyes. They looked around and thought "OK, I see someone with blue eyes. I guess she means him," and they stayed. But the next day came, and they thought "Hey, that blue-eyed guy didn't figure it out. I guess he must have seen someone else with blue eyes, but I don't see anyone else with blue eyes. I guess that means me." And so they left together on the second day.

Here's the story of three blue-eyed islanders. The Guru said she saw someone with blue eyes. They looked around and thought "OK, I see two people with blue eyes. I guess she means one of them," and they stayed. A day passed, and nobody left, and they thought to themselves "OK, this is the day those two guys figure it out." But another day passed, and nobody left. The blue-eyed people thought "Wait; those two guys didn't figure it out yet. I guess they must have seen another person with blue eyes, but I don't see anyone else with blue eyes. I guess that means me." And so they left together on the third day.

Here's the story of four blue-eyed islanders. The Guru said she saw someone with blue eyes. They looked around and thought "OK, I see three people with blue eyes. I guess she means one of them," and they stayed. A day passed, and nobody left, but they were not worried; they knew it would take a couple of days. A second day passed, and nobody left, and they all thought to themselves "OK, this is the day those three guys figure it out." But another day passed, and nobody left. The blue-eyed people thought "Wait; those three guys didn't figure it out yet. I guess they all must have seen another person with blue eyes, but I don't see anyone else with blue eyes. I guess that means me." And so they left together on the fourth day.

...and this is why they have to wait the full 99 days. It's not important that the Guru can see someone with blue eyes, unless there's only one islander. What's really important is that, given that the Guru can see someone with blue eyes, "those blue-eyed guys" should be able to figure it out among themselves, and that takes a specific amount of time for a given number of blue-eyed islanders. It's only when they can't do this for a number of islanders that doesn't include you that it becomes clear you must have blue eyes too.

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