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I don't know where else to ask this, but I don't really understand one part in the famous eye color island question/puzzle. Here is an example of the question/puzzle on mathematics stack exchange:

This puzzle is obviously not mine and is famous, including being on Wikipedia.

Eye color probability puzzle

In the correct answer, I have a problem with this statement about the case where there are two blue-eyed islanders:

After no one leaves on the first day, each blue eyed person can correctly deduce that the other blue eyed person must have seen someone with blue eyes, and it must be themselves.

How can the blue eyed person deduce that the other blue eyed person saw themselves as blue eyed? What if a brown eyed person deduced that?

One way I can see how the blue eyed person could deduce they must have blue eyes if no one left on the first day was if they knew the number of brown eyed people on the island. Then they could deduce there must have been someone else with blue eyes. But if they knew the number of brown eyed people, and they knew the number of people, then they would know the number of blue eyed people, which would be extra information (knowing the number of blue eyed people on the island) inconsistent with the original puzzle.

Can anyone explain it to me in simple English terms?

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    $\begingroup$ If you're just talking about the setup with 2 blue and 1 brown eye then a blue eyed person doesn't know his own color - it could be blue or brown. But if it were brown then the other blue eyed person he sees can deduce that they have blue eyes because at least one has blue eyes. $\endgroup$ – 伽罗瓦 Sep 11 at 3:25
  • $\begingroup$ Thanks, but I'm talking about the setup with 3 blue (including 1 oracle) and 1 brown eye. I quoted the part about no one leaving the first day, which I think means there isn't only 1 blue in addition to the oracle. $\endgroup$ – Yukang Jiang Sep 11 at 3:28
  • $\begingroup$ What is an oracle here? $\endgroup$ – 伽罗瓦 Sep 11 at 3:30
  • $\begingroup$ The blue eyed person that visits the island and tells the islanders there is at least one blue eyed among them. $\endgroup$ – Yukang Jiang Sep 11 at 3:31
  • $\begingroup$ I never included any oracle in what I said. There are 2 blue and 1 brown on the island. I make mention of "the other blue eyed person". $\endgroup$ – 伽罗瓦 Sep 11 at 3:37
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Let's say there are two blue-eyed people Alice and Bob, and one brown-eyed person Charlie. Let's consider what happens from Alice's persective, and then from Charlie's perspective.

Alice knows that Bob has blue eyes, and Charlie has brown eyes. Alice doesn't know her own eye color, but she can go through the following reasoning. If Alice had brown eyes, then Bob would be the only blue-eyed person on the island. So, it would be news to Bob that there is a blue-eyed person, since he's only seen people with brown eyes before. Therefore Bob would deduce he must have blue eyes, and would leave on the first day.

So, after Alice sees Bob not leave the first day, Alice deduces that she must have blue eyes, since if she didn't then Bob would have left.

You then ask, why can't Charlie go through the same reasoning? Let's look at Charlie's perspective. He knows that both Alice and Bob have blue eyes. Moreover, he knows that Bob knows Alice has blue eyes. So he knows Bob won't be surprised to learn there's someone with blue eyes--he's already seen Alice's eyes! The fact that Bob doesn't leave the first day therefore doesn't tell Charlie anything new, since he already knew Bob wouldn't leave.

In other words, the difference between Alice and Charlie's perspective is that Alice thinks Bob could be the only blue-eyed person, while Charlie knows that he isn't. When Bob doesn't leave, this tells them that Bob isn't the only blue-eyed person (so that Bob already knew someone has blue eyes), which is new information to Alice but not to Charlie.

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  • $\begingroup$ Ok I upvoted, so if Charlie can't leave the first or second day, that means Alice and Bob have to leave the second day. $\endgroup$ – Yukang Jiang Sep 11 at 3:48
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How can the blue eyed person deduce that the other blue eyed person saw themselves as blue eyed? What if a brown eyed person deduced that?

Note that a brown eyed person (A) can't possibly deduce that some other blue eyed pewrson (B) saw themselves (A) as blue eyed ... for the very simple reason that A has brown eyes, not blue. In short, it is impossible for me to conclude I have blue eyes if I have brown eyes.

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