Blue Eyes - The Hardest Logic Puzzle Ever

[Kodilynn]

Blue Eyes
The Hardest Logic Puzzle in the World​

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. Any islanders who have figured out the color of their own eyes then leave the island, and the rest stay. 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?

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There are no mirrors or reflecting surfaces. 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."

I've done my best to make the wording as precise and unambiguious as possible (after working through the explanation with many people), but if you're confused about anything, please let me know. A word of warning: The answer is not simple. This is an exercise in serious logic, not a lateral thinking riddle. There is not a quick-and-easy answer, and really understanding it takes some effort.

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Borrowed from the wonderful folks of http://www.xkcd.com

Yes I know the answer, no I won't PM it to you. 30 Day Gamecard in it to whomever gets it right (and if you try googling the answer or wiki'ing it, i'll know because of the nature of the answer and the "common answers" which aren't right might I add).

You can PM me your answer or you can post it. First right (as close to right as you can be, there is a correct answer) will win!

Good luck!

 

[teeroyoyort]

a brown eye'd person leaves the island on the 199th night?

 

[teeroyoyort]

nm too tough!

 

[kukmuk]

It isn't fair for me to answer because my Discrete Structures teacher used an example extremely similar to this as an introduction to Mathematical Induction back in college. If you start simple and work your way up it actually isn't all that complicated.

If this comment reveals too much about the solution, feel free to remove it.

 

[r0xx0r]

!

reminds me of Search for the Holy Grail.

There the other side you see! but you must first- answer me these questions, three!



BLUE- NO GREEN AAAAIIIIIYYYYYYEEEEEEEE!!!!!!

 

[VyralPandemic]

Too smart

Ok, While I do not posses any previous knowledge of the question, I too will refrain from posting my answer. I am just too @#$-ing smart and don't want to ruin it for everyone else.

Happy Pondering and Best Wishes to the Mathematically Challenged

With a tip of my cap, and a wave of my finger,

-Vyral

 

[Argo]

100 days after the guru speaks all of the blue eye'd people leave.

 

[Sadge]

100 days after the guru speaks all of the blue eye'd people leave.

Yes, but can truly explain why?

That's the real point of this puzzle, not the answer itself.

 

[r0xx0r]

yes

already sent in pm to K-man, before my reply with explanation of my answer, so i'll go ahead and let it out.



100 blue eyes leave on 100th day/night, simply because when the guru says i see one person with blue eyes, instantly (given that all the ppl know everyone's eye color, which they do as explained in the question) everyone is thinking these 99 blue eye'd ppl i see are gonna leave on the 99th day/night. And if they do, that means i'm foked because i have some other color'd eyes and i'm doomed to be on these islands with these perfectly logical mimes. But if they don't leave, that means 100 ppl have blue eyes, so that means my eyes are blue too! So I leave on the 100th day/night with all the other blue eyes. It would be non-logical for any of them to wait one day past 100 because of the simple fact that they only see 99 blue eye'd ppl.


/claps

 

[Argo]

Yes, but can truly explain why?

That's the real point of this puzzle, not the answer itself.

Lets say that that there is one blue eye'd person total. When the guru says this he knows he is the only blue eye'd person cause he see's all different color'd eye people. If there is two blue eye'd people. No one will know that they are blue eye'd for sure on the first day and no one will leave. On the second day they will know that there has to be at least 2 blue eye'd people because no one left on the first day. He looks and sees that there is only 1 other blue eye'd person so he can conclude that he has blue eyes. Same if there were 3 and so on all the way up to 100.

 

[Sadge]

already sent in pm to K-man, before my reply with explanation of my answer, so i'll go ahead and let it out.



100 blue eyes leave on 100th day/night, simply because when the guru says i see one person with blue eyes, instantly (given that all the ppl know everyone's eye color, which they do as explained in the question) everyone is thinking these 99 blue eye'd ppl i see are gonna leave on the 99th day/night. And if they do, that means i'm foked because i have some other color'd eyes and i'm doomed to be on these islands with these perfectly logical mimes. But if they don't leave, that means 100 ppl have blue eyes, so that means my eyes are blue too! So I leave on the 100th day/night with all the other blue eyes. It would be non-logical for any of them to wait one day past 100 because of the simple fact that they only see 99 blue eye'd ppl.


/claps

Yes I have already PM'd the answer as well.

You don't really explain why the person knows that 99 people would wait 99 days (and therefore all 100 people would leave on the 100th day).

 

[VyralPandemic]

I don't buy it.

*** double post see below***

 

[VyralPandemic]

I don't buy it.

Lets say that that there is one blue eye'd person total. When the guru says this he knows he is the only blue eye'd person cause he see's all different color'd eye people. If there is two blue eye'd people. No one will know that they are blue eye'd for sure on the first day and no one will leave. On the second day they will know that there has to be at least 2 blue eye'd people because no one left on the first day. He looks and sees that there is only 1 other blue eye'd person so he can conclude that he has blue eyes. Same if there were 3 and so on all the way up to 100.

Cough wrong Cough Perfect logicians wait 99 nights to see what you can see after a few hours?

Ok,
I worked something like this for a friend in college. the wording was different but the problem was the same... his was with cows /shrug. What's really pissing me off right now is that I never found out if my answer was right or wrong. The prof had his answer and we (he) had ours. Prof said it was wrong. I didn't believe him then, and don't now.
We decided on A, professor said is was B ( one of those nights)

A. all blue eyed people leave on the second day.
or
B. All blue eyed peeps leave on the 99, 100, or 101 day.

I argued it then and I argue it now.

As a "Perfect Logician" if I were sitting on an island watching all the other blue eyed people to see if they knew they had blue eyes and would be leaving it wouldn't take me all that long to sort it out.
* If no one is watching me, then "Logically" I do not have blue eyes, as everyone else is watching all the blue eyed people. BEFORE night 1 is over, all the brown eye people have deduced that they have brown eyes. No questions there.
* IF someone, blue or brown eyed, is watching me to see if I talk to the fairy LOGICALLY I have blue eyes.
* After the fairy comes and goes on the first night; I imagine a fairly intense and uncomfortable situation with 100 people starting at each other and 100 more starting at the first 100 - I imagine even the fairy would feel a bit out of place.... anyways lets more forward.
Fairy leave alone on the first night... Therefore I KNOW that the other 99 do not know for certain that they have blue eyes, but after everyone watching only the blue eyed people, and watching me I MUST have blue eyes.
*SINCE I am a "perfect logician" this was not that difficult to see. I must assume that since they are all perfect logicians, that all blue eyed people come to the same conclusion:

I was watched, I must have blue eyes.

*Second night all 100 blue eyed people speak up and get off the island.
Don't believe what you were told, follow the logic. My answer is better.

 

[Argo]

Cough wrong Cough Perfect logicians wait 99 nights to see what you can see after a few hours?

Ok,
I worked something like this for a friend in college. the wording was different but the problem was the same... his was with cows /shrug. What's really pissing me off right now is that I never found out if my answer was right or wrong. The prof had his answer and we (he) had ours. Prof said it was wrong. I didn't believe him then, and don't now.
We decided on A, professor said is was B ( one of those nights)

A. all blue eyed people leave on the second day.
or
B. All blue eyed peeps leave on the 99, 100, or 101 day.

I argued it then and I argue it now.

*Second night all 100 blue eyed people speak up and get off the island.
Don't believe what you were told, follow the logic. My answer is better.

I believe this would be considered communication and it says

they cannot otherwise communicate

I consider eye contact like you mention communcation

 

[VyralPandemic]

/shrug

I believe this would be considered communication and it says

they cannot otherwise communicate

I consider eye contact like you mention communcation

/ponder... hmm.

Just because I made eye contact as I passed someone in the isle of the grocery store and didn't run flat out into while staring at my feet doe not really constitute communication. ... though making eye contact with a cute girl does count as 'flirting' to my fiance.. so if she wrote this question... maybe I buy that argument.

So 200 perfectly logical, crooked necked, mimes on an island. Hard to know who all has blue eyes if you are always staring at the ground.
I don't buy his answer either: http://xkcd.com/solution.html

Kodi's question : Kodi's solution. I'll stand by his decision even if I don't agree with it.

 

[r0xx0r]

yeah that does explain why it would be the hundred night, i see 99 sets of blue eyes, if i don't have blue eyes, they'll all leave on 99th day/night. cuz if i don't have blues eye's all the blue eye'd ppl are seeing 98 sets of blue eyes, which means they'll leave on night 99.

 

[Argo]

/ponder... hmm.

Just because I made eye contact as I passed someone in the isle of the grocery store and didn't run flat out into while staring at my feet doe not really constitute communication. ... though making eye contact with a cute girl does count as 'flirting' to my fiance.. so if she wrote this question... maybe I buy that argument.

So 200 perfectly logical, crooked necked, mimes on an island. Hard to know who all has blue eyes if you are always staring at the ground.
I don't buy his answer either: http://xkcd.com/solution.html

Kodi's question : Kodi's solution. I'll stand by his decision even if I don't agree with it.

I also do not agree that ONLY the blue eye'd people would be watched. Everyone would be watched... blue eye'd or not to confirm whether or not they have blue eyes. If people did not watch you they would have no idea your eye color. So that way everyone would think they would have blue eyes.

 

[Sadge]

Ok -- maybe this would help ...

A lot of people like to start their answer with "imagine there were 2 people with blue eyes". While that helps, I think it's better to start with 1 blue-eyed person.

Fact: We know that there is at least 1 blue-eyed person on the island (because the guru said so, and for this case we will assume the guru never lies).

Now, if there is only 1 blue-eyed person on the island, he immediately knows it's him, right? He cannot see another blue-eyed person among the 199 other people on the island (and it is 199 people -- the guru does not count for 2 reasons: the main one being that the guru (we assume) cannot see her own eye color). So in this case, the 1 blue-eyed person leaves that night.

Now let's go to 2 blue-eyed people. Person A (the first blue-eyed person) can see 1 other blue-eyed person (Person B). But remember that neither person A or B know that their own eyes are blue. So person A can assume that person B will leave the island that night if they do not see any other blue-eyed people (because of the first example), because person B would not be able to see any other blue-eyed people. But after the first night, person B does not leave because he knows that he sees person A with blue eyes (mathematically speaking, there can only be number of people seen with blue eyes+ the person looking ... or seen+1). Since person A did not leave the first night, person B knows that A cannot logically deduce how many people have blue eyes because he sees at least 1 other person with blue eyes. Since person B cannot see any more people with blue eyes, he must have blue eyes (person A reaches the same conclusion). So on the second night, both person A&B leave being the only 2 people with blue eyes.

You can follow that logic all the way up to 100 blue-eyed people. Each person can see that there are 99 other blue-eyed people. So he simply has to wait 99 days to see if they leave and if they don't, he can assume that he himself has blue eyes and leaves on the 100th day (and of course the other 99 people are reaching the same conclusion after 99 days).

I can go further if you want.

 

[Argo]

Lets say that that there is one blue eye'd person total. When the guru says this he knows he is the only blue eye'd person cause he see's all different color'd eye people. If there is two blue eye'd people. No one will know that they are blue eye'd for sure on the first day and no one will leave. On the second day they will know that there has to be at least 2 blue eye'd people because no one left on the first day. He looks and sees that there is only 1 other blue eye'd person so he can conclude that he has blue eyes. Same if there were 3 and so on all the way up to 100.

 

[VyralPandemic]

edited severly

Now I can see what you were hinting at ARGO, though SADGE put it much more clearly for me.