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Comment by jack9 | Hacker News Reader

> you know there must be three blue eyed people

Why? The visitor didn't synchronize anything. All of the villagers already knew there was at least 1 blue eyed person. Per the story, 99 or 100 depending on your perspective.

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Everyone commits suicide at noon in view of everyone else. So the lowest possible number of blue eyed people go up one per day.

Here's a longer explanation: Since people who know their eye color suicides at noon in plain sight of everyone else, we can set up induction.

If there's only one blue eyed person, they would suicide the first day at noon. Since everyone sees > 1 blue eyed people, they know this wouldn't happen. Since this doesn't happen, it's common knowledge that there is more than 1 blue eyed person. So if you only see one other blue eyed person, you must have blue eyes (but you don't know this yet because you see two blue eyed people).

On the second day, if there are only two blue eyed people, they would both suicide at noon. You see two blue eyed people. No one commits suicide because they either see two blue eyed people (if they have blue eyes) or three blue eyed people (if they don't). Now, everyone knows that there are more than two blue eyed people.

Since no one committed suicide the day before, you reason that there must be three blue eyed people, with the 2 people you see plus yourself. Thus, you know your eye color.

This is a bit long but I hope it's clear.

the argument i hear about induction is that it's only valid IF n=1 is a true situation, n=2 is a true situation, etc, up to n=k (where k is 100 here). But we are only given n=100, and we cannot say that n=1,2,3... is true. The best counter example is the surprise execution paradox: suppose you are a prisoner, about to be executed next week. However, a strange rule applies - if you are not surprised by the date of your execution, you get to go free. You reason that you cannot be executed on sunday, the last day, since by friday, you'd know, and thus not be surprised. by induction, friday is also not a possible day, since you'd know by thurs, etc. You conclude that you cannot be surprised, and this will not get executed. But come wed, you get executed, and is completely surprised!

This problem is different from the surprise paradox, which relies on probability changing each day, and a possibility that either (a) the executioner will be proven a liar by randomly guessing Sunday, or (b) you confidently guessing Wednesday and being right.

> Now, everyone knows that there are more than two blue eyed people.

That's incorrect, in a subtle way. The villagers (every single one) knew that before. It isn't discovered knowledge. Everyone (including the visitor) knows there are 99 or 100 because they can count them already, prior to the visitor.

So nobody is suiciding for 99 days. As an individual who sees 99 other blue eyes, why would you think or not think that you have blue eyes on the next day?

I meant strictly more, as in at least three. If you know that there are more than two blue eyed people and you can only see two, you know that you have blue eyes.

If you can see 99 blue eyes, you know that all the other people see either 98 or 99 (if there's a total of 99 blue eyes) or 99 or 100 (if there's 100 blue eyes). However, if some people see 98 and know that there's at least 99 people with blue eyes, then they would all suicide. Since they don't, you know everyone sees either 99 or 100 blue eyes. Since you see 99, you must have blue eyes.

As a villager, you already know everyone sees either 99 or 100 blue eyes. You see 99, but you don't know if everyone else sees 99 or 100 (if they have blue it's 99 if they dont it's 100). You knew it before the visitor, and after his statement.

This does not help in deducing if you have blue eyes.

> then they would all suicide

No they wouldn't. Nobody should suicide at all. The visitor saying someone has blue eyes, was already known by every member of the village and every member of the village saw more than 1 person with blue eyes prior. Nothing is different because he said something.

> Since you see 99, you must have blue eyes

I don't know there's 100, so I don't know I have blue eyes.

This is a solved problem. When a proof is written, you should challenge the proof, not just reassert your intuition.

You are ignoring islander A's view of B's view of C's view of ... Z's view of situation.

Think about 3 islander's, where A thinks "if I have green eyes, then B and C are in a situation equivalent to me not being here, and B sees exactly 1 blue eyed person C, and B knows that 'C sees 0 or 1 blue eyed people, but then C learned that 0 is impossible'"

Is the outsider not needed for this line of reasoning to work assuming that the total number of people doesn't change? All the 100 blue eyed would commit suicide after a hundred days after the creation of the island since they work out they have blue eyes. I think the main challenge in this puzzle is the lack of assumption on what kind of reasoning the islanders are capable of doing and what the initial conditions are.

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