| Blue Eyes: The hardest logic puzzle in the world
Posted: 2/18/2008 10:14:40 AM | The previous "blue eyes" post reminded me of this, so I tracked it down to share here. Occupied my mind for several days. :)
A group of people with assorted eye colors live on an island. They are all perfect logicians -- if a conclusion can be logically deduced, they will do it instantly. No one knows the color of their eyes. 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. Everyone can see everyone else at all times and keeps a count of the number of people they see with each eye color (excluding themselves), but they cannot otherwise communicate. Everyone on the island knows all the rules in this paragraph.
On this island there are 100 blue-eyed people, 100 brown-eyed people, and the Guru (she happens to have green eyes). So any given blue-eyed person can see 100 people with brown eyes and 99 people with blue eyes (and one with green), but that does not tell him his own eye color; as far as he knows the totals could be 101 brown and 99 blue. Or 100 brown, 99 blue, and he could have red eyes.
The Guru is allowed to speak once (let's say at noon), on one day in all their endless years on the island. Standing before the islanders, she says the following:
"I can see someone who has blue eyes."
Who leaves the island, and on what night?
There are no mirrors or reflecting surfaces, nothing dumb. It is not a trick question, and the answer is logical. It doesn't depend on tricky wording or anyone lying or guessing, and it doesn't involve people doing something silly like creating a sign language or doing genetics. The Guru is not making eye contact with anyone in particular; she's simply saying "I count at least one blue-eyed person on this island who isn't me."
And lastly, the answer is not "no one leaves." | |
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| Blue Eyes: The hardest logic puzzle in the world
Posted: 2/18/2008 11:29:18 AM | Who leaves the island, and on what night?
The second-last person leaves, at midnight after the guru speaks. As the last person to not figure it out, this person had no way to know until the guru spoke. The guru is the last person, and has no way to deduce her own eye color, so is stuck there unless she decides to leave.
I see no requirement in the question that I explain how the OTHER 199 people figured it out, only that I explain ONE person. | |
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vro312
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Joined: 11/22/2007
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vro312
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Joined: 11/22/2007
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| Blue Eyes: The hardest logic puzzle in the world
Posted: 2/18/2008 7:08:36 PM | | The "standard" answer is a bit more complex and interesting. Having looked it up, I understand it, and it's distinctly different from mine. Given the way this particular version was worded however, I'm comfortable with my answer, even though the logic is drastically different. | |
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| Blue Eyes: The hardest logic puzzle in the world
Posted: 2/18/2008 7:31:34 PM | | I think that's wrong but close. The guru spoke to the islanderS, plural. There were at least two people besides the guru there. If one was a brown and the other a blue, then when the guru spoke they could look at each other and the blue eyed person could see that the blue the guru mentioned wasn't the other person, so it must be them. So I think it was the second last person (not counting the guru themself). | |
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| Blue Eyes: The hardest logic puzzle in the world
Posted: 2/18/2008 11:03:10 PM | The last person with blue eyes who remained there: somehow the others with blue eyes had already worked out their eye colour and left. Only if there was just one person with blue eyes who could look at everyone else and know that they must be the one the guru spoke of would they be able to leave.
If there were still 100 blue-eyed people on the island when the guru spoke then everyone would already know that there was at least one blue-eyed person on the island because they'd all be able to see them so her announcement would add no information.
I guess if there were two blue-eyed people and the first one saw that the second had not left at the first opportunity, then he'd realise he must have blue eyes too so the other one didn't realise he was the only one.
Hm
If there were three blue-eyed people only, I guess the third would expect the two others to use their logic and leave on the 2nd day as described above. When they don't, he'd know that he too must have blue eyes and so all 3 would leave at the 3rd opportunity...
Oh
So all 100 blue-eyed people would be able to leave after 100 days?
But the guru didn't add any information, did she? Or did she?
Would all the brown-eyed people be able to leave at the same time through the same reasoning? It seems like they can't because the inductive pattern seems to need to start with the possibility of there being only one,which everyone knows to be false. But the statement "if there is only one then ...." is not false and the fact that the "then..." doesn't happen is what leads you to know the condition is false and allow you to deduce more...
Looking forward to seeing this one explained. | |
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| Blue Eyes: The hardest logic puzzle in the world
Posted: 2/19/2008 6:34:11 AM | Blue eyes people generally have blond hair. Brown eyed people generally have brown coloured hair.
Red and blond hair colours generally show less pigment in the skin as brown colours ergo if your hair is blond or red, then your eyes will be blue, if it is brown, black or any other brownish colour, then your eyes will be brown.
Simple! | |
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| Blue Eyes: The hardest logic puzzle in the world
Posted: 2/19/2008 7:32:36 AM | Yeah, rune, you got it. However many blue eyed people are there, that's how many days it would take all of them at the same time to realize they have blue eyes. Then they'd all leave together.
Me, I got as far as recognizing that the view would be the same for all the blue eyed people, so there'd be no way that one of them would leave unless all of them left. Then I had to cheat and look up the answer to get the rest.
/tess | |
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| Blue Eyes: The hardest logic puzzle in the world
Posted: 2/19/2008 7:33:15 AM | Maybe this one then?
Everyone has a tendancy to pick and choose favourites. So one person will think "My eyes are brown" if they leave once they figure out their eye colour, then they would have left. When they didnt leave, they will have then gone to thinking "My eyes are blue" which is correct and they would leave. This is assuming that the immediate midnight after they figure it they leave.
You said they are perfect logicians. Therefore they will instantly realise once the Guru says that someone has blue eyes then the key to getting off the island is the colour of their eyes. So they would have gone through trial and error and once they picked the correct colour of their eyes, then they leave. | |
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| Blue Eyes: The hardest logic puzzle in the world
Posted: 2/19/2008 9:34:04 AM | I don’t see where the Guru spoke more than once: “… is allowed to speak once … on one day in all their endless years …”.
After “endless years”, I expect that 198 islanders had died of old age, and the last 2 raced for the boat when one flinched, after the Guru finally spoke; with at least one still having blue eyes.
(Even if the Guru spoke every day, he could be referring to the same individual with blue eyes over and over again, and therefore adding no new information.)
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| Blue Eyes: The hardest logic puzzle in the world
Posted: 2/19/2008 12:48:23 PM | Interesting puzzle, hdsa0irse. 
Great answer, rune. Incredibly ingenious.
However, if I may add something:
So all 100 blue-eyed people would be able to leave after 100 days? If there is only 1 blue-eyed person, he can see no other blue-eyed people, so he must leave on the night after the day of the Guru's statement.
If there are 2 blue-eyed people, they must leave on the 2nd night after the day of the Guru's statement.
If there are 3 blue-eyed people, they must leave on the 3rd night after the day of the Guru's statement.
Since there are 100 blue-eyed people, they must leave on the 100th night after the day of the Guru's statement.
So on the 100th day after the day of the Guru's statement, there are no blue-eyed people on the island. But the brown-eyed people and the Guru would still be there.
But this would all be true only if no-one wanted to leave the island.
However...if they all wanted to leave the island, all the blue-eyed people and the brown-eyed people could all leave on the night after the day of the Guru's statement.
A simple way to determine your eye colour would be to form everyone into groups of the same eye colour. You couldn't say what colour the other people's eyes were. But you would have to start somehow, so you approach someone and stand next to him, say someone with blue eyes. At this point, you would not know what colour your eyes were, so you could be in a pair of 2 blue-eyed people, or 1 blue-eyed person and 1 brown eyed person (you).
Then, if someone else approached you, then it would be no advantage to approach one blue-eyed person and one brown-eyed person, so that person would only approach 2 blue-eyed people, or 2 brown-eyed people. Then, if the person standing next to you has blue eyes, and you can see that someone is approaching you, then you would have blue eyes. If that person has blue eyes, you would encourage that person to join you, to make groups, so you would walk towards that person. So would the person next to you, because they have figured this out too. So the both of you would approach that person, and now you would have 3 blue-eyed people. If the person approaching you had brown eyes, then you would back off, and so would the person next to you, so you would both back off, and now he knows that he has brown eyes. So he would seek out 2 people with brown eyes.
If you had brown eyes, and you were standing next to the blue-eyed person, you would have no-one approach you. But that is not a guarantee. But everyone else would think the same, so eventually everyone else would pair off, just to get things moving.
At that point, you would not know what colour your eyes were, so you would split from your partner, look for a pair of blue-eyed people (or brown-eyed people) and try and join them, and so you would know what colour your eyes were and so would they. So then if you had blue eyes, there would be a trio of blue-eyed people (including yourself) and if you had brown eyes, you would look for a pair of brown-eyed people, who would figure out from your approach that they both have brown eyes, so they would accept you with your brown eyes, and there would be a trio of brown-eyed people. Eventually everyone would now be in trios of blue-eyed people and trios of brown-eyed people.
Then those trios would pair up in a similar fashion, and by midnight, there would be only 3 groups of people: 100 blue-eyed people, 100 brown-eyed people, and the Guru on her own.
The blue-eyed people would know that they all have the same colour, and by seeing the other people in their group, know they have blue eyes. The brown-eyed people deduce their eye colour similarly. The Guru is the only one who can never leave, because she is the only one with green eyes. She knows that she doesn't have blue eyes or brown eyes, because the other groups move away from her. But she doesn't know if she has red eyes, and no-one can tell her. So she cannot leave. | |
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| Blue Eyes: The hardest logic puzzle in the world
Posted: 2/19/2008 9:39:43 PM | I thought that the brown-eyed ones would be able to leave the day after all the blue eyed ones, but they don't know that their eyes aren't green.
What has me puzzled is that the guru doesn't actually tell them anything they don't know. Everyone on the island can see at least one blue-eyed person so they already know the truth of the guru's statement before she makes it. So how does her making the statement trigger a chain of reasoning that couldn't happen before? If the brown-eyed people all said to themselves "she sees at least one brown-eyed person" couldn't they all escape due to the same kind of reasoning? It sees like they can't and that confuses me -- is there a flaw in the induction?
Scorpio, your method is ingenious but it amounts to sign language, which isn't allowed. I thought of the people standing in a circle when the guru spoke and looking at the person to their left and nodding in agreement with the guru if that person had blue eyes. But that would be sign language too. | |
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| Blue Eyes: The hardest logic puzzle in the world
Posted: 2/20/2008 9:43:53 AM | The reason the guru has to speak is because, like any logic problem, this one has to be distilled down to its most simple form in order to be tested or proved. This answer must apply whether there's one or 100 blue eyed people on the island. If there's only one blue eyed person on the island, then the only way she knows she has blue eyes is because the guru sees someone with blue eyes and all the eyes she sees are brown.
Without the guru, the solution fails in it's most simple form (one blue eyed person).
/tess | |
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| Blue Eyes: The hardest logic puzzle in the world
Posted: 7/5/2008 9:32:07 PM | It's not that difficult to understand, but I will admit it is difficult to figure out, so I did look up the solution as well.
It's easier to understand if you bring it down to scale. Imagine if only 2 people on the island had blue eyes and you're one of them. The guru speaks with her fine-@$$ sexy green eyes :) and gorgeous dark brown hair. Ahem...sorry. She speaks and you see the other person with blue eyes that has also heard the guru speak, but remains.
Why is he not leaving? He must see someone else with blue eyes and so believes the guru might be talking about this other person. You've seen every other person on the island and none of them have blue eyes. Who could he be looking at? He must be looking at YOUR blue eyes.
However, he must WAIT until after midnight to realize that YOU are not leaving either and therefore must see one other person with blue eyes. Using the same logic that you used, he correctly deduces that HE must also have blue eyes as well. So you waited one midnight and didn't see him leave. He also noticed that you didn't leave on that midnight either. Consequently, you both realize that you have blue eyes on the 2nd midnight and leave together.
It works for 3 people, 4 people, 5 people, and so on until you reach 100. They all leave on the 100th midnight because each person has to realize that all of the other 99 people must see one other with blue eyes.
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| Blue Eyes: The hardest logic puzzle in the world
Posted: 7/6/2008 2:23:47 AM | Okay, I understand the anwer, but this seems like a lot of non-sense to me. If I had brown eyes and I knew and watched the blue eyed people begin to dwindle down from 100 to 1, all the while I knew that there were 99 other brown eyed people I would have to come to the conclusion that people with brown eyes are dumb if all of them stayed.
Also the primary question doesn't seem to be completely answered, "and on what night?" No one is answering that and there seems to be no frame of referance to do so unless brown eyed people are dumb and don't leave. I find this offensive to brown eyed people...I would leave on Friday....
Thats suppose to be a joke, but take it as it comes from this brown eyed gent...
Or 100 brown, 99 blue, and he could have red eyes.
Also, how does this work? If we have 100 brown and 100 blue, why is there 99 blue and a guru with red eyes?
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| Blue Eyes: The hardest logic puzzle in the world
Posted: 7/6/2008 8:13:09 AM |
Also, how does this work? If we have 100 brown and 100 blue, why is there 99 blue and a guru with red eyes?
No one on the island actually has red eyes. The original post states that as far as each blue-eyed person knows, it is possible for him to have red eyes even though we already know that he doesn't based on what we are told.
The color red is arbitrary choosen out of all possible colors that aren't brown, blue, or green. This part of the riddle was included to destroy the idea that he could deduce his eye color using information about the distribution of brown-eyed or blue-eyed people on the island (100 of each). Knowledge of the fact that there are 100 brown-eye people and 99 blue-eyed people CANNOT therefore be used to solve the problem.
Incorrect: Hey! There are 100 people with brown eyes and only 99 people with blue eyes. If I had blue eyes that would make 100 people with blue eyes, which is a nice even, comfortable number. I must have blue eyes.
As far as that one blue-eyed person knows, it is possibile that HE might have red eyes or even purples eyes for that matter.
From the persepctive of a blue eyed person, only 99 people with blue eyes would be visible to him even though in fact, there are 100 blue-eyed persons on the island. | |
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| Blue Eyes: The hardest logic puzzle in the world
Posted: 7/8/2008 2:47:47 AM | Couldn't they just all get together and let the guru divide everyone into 2 groups, blue eyes and brown eyes.
Then when the guru got to speak it would be, " All you people are blue and all you people are brown ".
Then everybody leaves, except the guru who has no way of knowing what his eye color is. | |
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| Blue Eyes: The hardest logic puzzle in the world
Posted: 7/9/2008 7:07:59 AM | ok this is what i found
It is common to introduce the idea of common knowledge by some variant of the following logic puzzle:[2] On an island, there are k people who have blue eyes, and the rest of the people have green eyes. There is at least one blue-eyed person on the island (k >= 1). If a person ever knows herself to have blue eyes, she must leave the island at dawn the next day. Each person can see every other persons' eye color, there are no mirrors, and there is no discussion of eye color. At some point, an outsider comes to the island and makes the following public announcement, heard and understood by all people on the island: "at least one of you has blue eyes". The problem: Assuming all persons on the island are truthful and completely logical, what is the eventual outcome?
The answer is that, on the k dawns after the announcement, all the blue-eyed people will leave the island.
This can be easily seen with an inductive argument. If k = 1, the person will recognize that he or she has blue eyes (by seeing only green eyes in the others) and leave at the first dawn. If k = 2, no one will leave at the first dawn. The two blue-eyed people, seeing only one person with blue eyes, and that no one left on the 1st dawn, will leave on the second dawn. So on, it can be reasoned that no one will leave at the first k-1 dawns if and only if there are at least k blue-eyed people. Those with blue eyes, seeing k-1 blue-eyed people among the others and knowing there must be at least k, will reason that they have blue eyes and leave.
What's most interesting about this scenario is that, for k > 1, the outsider is only telling the island citizens what they already know: that there are blue-eyed people among them. However, before this fact is announced, the fact is not common knowledge. | |
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| Blue Eyes: The hardest logic puzzle in the world
Posted: 7/9/2008 7:12:17 AM | the last 2 lines are very interesting
i think our perfect logicians are able to reason that an outsider "could" arrive and make the statement, hence they wouldn't need the outsider at all to begin their countdown
wow!!!! that was fun  | |
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| Blue Eyes: The hardest logic puzzle in the world
Posted: 7/9/2008 8:47:13 AM |
On this island there are 100 blue-eyed people, 100 brown-eyed people, and the Guru (she happens to have green eyes). So any given blue-eyed person can see 100 people with brown eyes and 99 people with blue eyes (and one with green), but that does not tell him his own eye color; as far as he knows the totals could be 101 brown and 99 blue. Or 100 brown, 99 blue, and he could have red eyes.
There are, then, 201 people on the island. Logically, the only way that the communication, "I can see someone who has blue eyes" can be effective is if only one individual is presenting himself for inspection. That individual then learns what color his eyes are, and that individual leaves at midnight that night, taking 200 days for completion. One never knows until tested by the guru what color their eyes are.
If, however, she can only speak generally, she is never offering new information. Unless islanders can hide themselves for the day, there's no way to determine eye color. | |
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