BLUE EYES PUZZLE

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I saw this puzzle - and thought it was fanstastic - and the answer is completey logical once you figure it out!
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A group of people live on an remote island. They are all perfect logicians and if a conclusion can be logically deduced, they will do it instantly. No one knows the colour of their eyes and they are not allowed to discuss eye colour. Every night at midnight, a ferry stops at the island. If anyone has figured out the color of their own eyes, they must leave the island that midnight. They all follow the rules unquestionably (I added this bit on purpose - since it was the only sticking point I had).

On this island live 100 blue-eyed people, 100 brown-eyed people, and the Guru. The Guru has green eyes, and does not know her own eye colour. Everyone on the island knows the rules and is constantly aware of everyone else’s eye colour, and keeps a constant count of the total number of each (excluding themselves). So any given blue-eyed person can see 100 people with brown eyes and 99 people with blue eyes, but that does not tell them their own eye color; it could be 101 brown and 99 blue. Or 100 brown, 99 blue, and the one could have red eyes.

The Guru then says "One of you has blue eyes". This causes a ripple of shock amongst all of the people for although it was known by each member individually it was not COMMON knowledge.

Who leaves the island, and on what night?

- Not a trick question - don't cheat by looking for the answer - A clue will be given tomorrow if no one has solved it.

i reckon the answer is going to be something like the 100 people with blue eyes leave after 99 or 100 days or something like that, unfortuantely the problem is so badly worded that i can't even begin to explain why without getting frustrated

I'm dying to know the answer and rationalle.

DiVad is correct... the wording is that way intentionally. It is only to give you the minimum info.

Imagine if there were only 6 people -

1 has blue eyes
4 have brown eyes
1 guru

The guru tells them that one of them has blue eyes. All of the people look at each other.
The brown eyed people have no idea what colour their eyes are because they see 3 brown eyed people and 1 blue eyed person. They logically assume they may have blue eyes or may not. In this scenario - the blue eyed person looks around and sees no one with blue eyes. They therefore know that they are the blue eyed person and leave that night.

Expand this. Imagine if there were 7 people: -


2 have blue eyes
4 have brown eyes
1 guru

They all look at each other. The brown eyed people see two blue eyed people and cannot deduce their own eye colour. The blue eyed people only see one other blue eyed person. They deduce that if I don't have blue eyes, then that person with blue eyes knows that they are the only blue eyed person (see above) so therefore they will leave tonight. If they don't leave tonight then that means that they can see someone else with blue eyes. Since I can only see one person with blue eyes that that other person must therefore be me. Both blue eyed people think the same thoughts. On the first dawn they notice the blue eyed person has not left so they both know they have blue eyes and leave that night.


So expand this. Imagine if there were 8 people: -

3 have blue eyes
4 have brown eyes
1 guru

They all look at each other. The brown eyed people see three blue eyed people and cannot deduce their own eye colour. The blue eyed people see two other blue eyed people. The blue eyed people all deduce that if the two blue eyed people they see are the only blue eyed people around then they would leave on the second night. On the third dawn the blue eyed people see that the other two blue eyed people have not left. They therefor deduce they have blue eyes themselves and were the reason the other two had not left. All three therefore leave that night.

You can expand this by adding another blue eyed person to the scenario. The blue eyed people will always count the other blue eyed people. In our scenario, a blue eyed person would see 99 other blue eyed people. They would conclude that if the 99 other blue eyed people are still around on the 99th day that means that they also have blue eyes, because if they didn't have blue eyes then the blue eyed people would have left on the 98th day. The only reason the other blue eyed people didn't leave on the 98th day was that they also counted 99 blue eyed people and weren't therefore sure on the 98th day what colour their own eyes were. They were only able to deduce that they had blue eyes when the others hadn't left yet.

I hope this makes sense. That was the mathematical answer. However, Mathematicians never take into account the true nature of people. Even if they were perfect logicians they would still not want to be turfed off the island - here is my answer to the riddle: -

My own opinion is that the 200 citizens would actually otherthrow the guru and make her leave on the first night at midnight with the ferry. They are all perfect logicians and they would assume that the guru would know what sharing such information to the rest of the tribe would mean. It would mean destruction of the blue members of the tribe. They would know the guru doesn't have blue eyes and that she wants to get rid of all blue eyed people. Since the whole tribe has no idea what their own eye colour is they would be upset that they might have to leave the island. The blue eyed people would individually deduce that if on the 99th morning the blue eyed people hadn't left then they would have blue eyes themselves and would leave at midnight on the 99th night. The brown eyed people would individually deduce that if on the 100th morning the 100 blue eyed people hadn't left then they had blue eyes too etc. However, if they waited until the 100th day to find out then it would be too late for them, so I expect they would all rally around and overthrow the guru and make her leave - 200 to 1.

Why does the guru have green eyes? Well, she knows exactly the colours of every tribe member apart from her own. She is the only unique person there, a sort of outsider. She is the only one who can give the whole tribe a common knowledge of a single fact. The single fact is "One of you has blue eyes". Even though each member of the tribe individually knows that someone else has that eye colour, they never had the knowledge that everyone else on the island understood the same principle.

Take one of my above examples but with 9 tribe members: -

4 have blue eyes
4 have brown eyes
1 guru

They are all living together. The guru comes along as she does and says something to ruin everything. This time she doesn't do a bad thing. She just reminds them that "anyone who figures out their own eye colour has to leave at midnight". There is no way any of them could deduce there own eye colour from this information. Try and do it, not possible. But if the guru came along and said "one of you has brown eyes" it starts the chain. The cataylst is the shared common notion whereby each individual member can begin deducing what the others are thinking.

Thanks for the answer

there's at least one hole in the puzzle. although they can't talk about eye colour, there is nothing stopping them talking about the boat - they could start saying to each other "if nobody else left before, would you leave on the boat in 100 days time" or something like that, and deduce their own eye colour that way.

i've never seen a proper, rigorous proof of the answer. i had a look but the solutions i saw were like yours, of the 'imagine there were 5 people on the island' variety - i didn't find that convincing (shows sufficiency but not neccessity) unfortunately i can't do any better myself.

All the blue-eyed people would leave on the midnight after ninety-nine days had passed.