# The blue eye problem

#### Merick

##### Diabloii.Net Member
The blue eye problem

Apparently it's supposed to be a really hard and not cheap logic problem.
I really don't have any idea, anyone want to give it a shot?

http://xkcd.com/blue_eyes.html

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; it 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, 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.

#### {KOW}Spazed

##### Banned
EDIT: Link was slightly different in the information that the islanders had, which made this whole thing pointless.

I don't understand it completely, but it seems to work out nicely.

#### Merick

##### Diabloii.Net Member
This has nothing to do with volcanoes. Are you sure this is the same problem?

#### {KOW}Spazed

##### Banned
If blue eyed people = 1 then he leaves the night the Guru tells them.

If blue eyed people = 2 then seeing that only one other blue eyed person is there and that no one left the previous night will leave.

That goes on forever so the answer is the amount of blue eyed people minus one days.

At least that is how I think it should work. My link confused the hell out of me and I think I am actually going against what it says. The link told the islanders that there was at least one blue eyed person, so the guru actually wouldn't be disclosing any information and all the blue eyed people would have already left. It was a faulty version of the riddle.

#### Xynrx

##### Diabloii.Net Member
Hey, that's a really slick problem. I like it.

The solution involving volcanoes and people leaving on ships are very similar. Here's the jist of why:

Basically, the mathematical induction from the volcano link says that since everyone is a master logician, they will all know what the only solution is.

Now, with that said, here is the answer to the two similar but different problems:

Volcano:
- Blue eyed people must die.
- If you have BROWN eyes you see X number of people with blue eyes.
- If you have BLUE eyes you see X-1 number of people with blue eyes.
- the magical logic that everyone is expected to understand without discussion is that everyone with blue eyes jumps into the volcano when the number of days past equals the number of blue eyed people... that's it

Here are the two scenarios assuming there are 4 blue eyed people (X=4)
i) You have blue eyes, and you see 3 blue eyes. If they all kill themselves on day 3 you have brown eyes, if they don't, logically, you must also have blue eyes. You die on day 4.

ii) You have brown eyes, and you see 4 blue eyes. The blue eyed people go through the same logic as you did in part i). They only saw 3 people with BLUE, and you saw 4. They all kill themselves for the same reason as in part i) and you therefore must have brown eyes.

Ships:
- everyone must know their eye colour before leaving
- the same magical logic occurs as above with the exception that everyone questions whether or not they have that 100th blue eyes or 101st brown eyes, or vice versa brown/blue
- all blue see 99 blue and 100 brown, and all brown see 100 blue and 99 brown
- same deal, if you are blue, and think you may be the 101 brown, you assume each blue only sees 98 other blues plus themselves, thus they all leave on night 99. vice versa with brown/blue
- since no one will leave on night 99 since they see AT LEAST 99 of blue and 99 of brown (plus one), they have to wait one more day because no one had boarded and thus all leave on night 100.

#### Dondrei

##### Diabloii.Net Member
Xynrx said:
Hey, that's a really slick problem. I like it.

The solution involving volcanoes and people leaving on ships are very similar. Here's the jist of why:

Basically, the mathematical induction from the volcano link says that since everyone is a master logician, they will all know what the only solution is.

Now, with that said, here is the answer to the two similar but different problems:

Volcano:
- Blue eyed people must die.
- If you have BROWN eyes you see X number of people with blue eyes.
- If you have BLUE eyes you see X-1 number of people with blue eyes.
- the magical logic that everyone is expected to understand without discussion is that everyone with blue eyes jumps into the volcano when the number of days past equals the number of blue eyed people... that's it

Here are the two scenarios assuming there are 4 blue eyed people (X=4)
i) You have blue eyes, and you see 3 blue eyes. If they all kill themselves on day 3 you have brown eyes, if they don't, logically, you must also have blue eyes. You die on day 4.

ii) You have brown eyes, and you see 4 blue eyes. The blue eyed people go through the same logic as you did in part i). They only saw 3 people with BLUE, and you saw 4. They all kill themselves for the same reason as in part i) and you therefore must have brown eyes.

Ships:
- everyone must know their eye colour before leaving
- the same magical logic occurs as above with the exception that everyone questions whether or not they have that 100th blue eyes or 101st brown eyes, or vice versa brown/blue
- all blue see 99 blue and 100 brown, and all brown see 100 blue and 99 brown
- same deal, if you are blue, and think you may be the 101 brown, you assume each blue only sees 98 other blues plus themselves, thus they all leave on night 99. vice versa with brown/blue
- since no one will leave on night 99 since they see AT LEAST 99 of blue and 99 of brown (plus one), they have to wait one more day because no one had boarded and thus all leave on night 100.

That didn't make any sense.

{KOW}Spazed said:
If blue eyed people = 2 then seeing that only one other blue eyed person is there and that no one left the previous night will leave.
That is a very confusing sentence.

#### Cannon Fodder

##### Diabloii.Net Member
Man I see this one pop up more and more- first ran into it about 4 years ago. Stumped me for a good 15 minutes before I got it the first time.

#### UserMathias

##### Diabloii.Net Member
Not sure I get it. If the assumption is that

On this island there are 100 blue-eyed people, 100 brown-eyed people, and the Guru (she happens to have green eyes).
and no one else, then there can't be a red-eyed person, or less than 100 blue-eyed people and so on.

#### {KOW}Spazed

##### Banned
The people don't know that, they only know what is in the first paragraph and the one thing the Guru said.

That is why I took my link down, it gave them that information. . .so no need for the Guru.

#### Dondrei

##### Diabloii.Net Member
I don't get it at all. The idea of induction is impossible, no-one knows their eye colour until people start leaving, and no-one can leave until they know their eye colour. So no-one leaves in that case.

#### Johnny

##### Banned
The funny part is someone probably spent days and countless effort making that up and the best it can do is raise a few eyebrows and then people lose interest.

Then in 15 min someone poops out a ytmnd thats 100 times as amusing.

Ah, the classic.

#### nosignal

##### Diabloii.Net Member
Just scale the numbers down to three a side or something. It'll be easier to figure out, I guess.

#### ffejrxx

##### Diabloii.Net Member
everyone should be able to figure it out by the second day if the people dont know how many people on the island have the same eye color (it would have been easier if they said there was a random population of blue and green eyed people totaling 200)

everyone except the guru should be gone by the second night

the first night everyone could guess and those that guessed correctly would leave

at noon the second day the guru would say "I can see someone who has blue eyes." unless everyone with blue eyes guessed correctly

the second night everyone left could change their answer to the correct one
the only ones left after the second night would not have green or blue eyes

#### {KOW}Spazed

##### Banned
ffejrxx said:
everyone should be able to figure it out by the second day if the people dont know how many people on the island have the same eye color (it would have been easier if they said there was a random population of blue and green eyed people totaling 200)

everyone except the guru should be gone by the second night

the first night everyone could guess and those that guessed correctly would leave

at noon the second day the guru would say "I can see someone who has blue eyes." unless everyone with blue eyes guessed correctly

the second night everyone left could change their answer to the correct one
the only ones left after the second night would not have green or blue eyes
Guessing doesn't count, they have to use logic to figure out their eye color.

#### Johnny

##### Banned
On the 2nd day I arrive in a speed boat with a dagger then just for the fun of it shank all 200 of them. Because none of the unarmed blind and deaf people could figure out a logic way to defend themselves they all die and then I take the foxy green eyed guru babe with me in the speed boat back home. And after that all she ever saw was my blue eyes... And stars ofcourse.

#### ffejrxx

##### Diabloii.Net Member
{KOW}Spazed said:
Guessing doesn't count, they have to use logic to figure out their eye color.
then they would all figure have it out the first night since there so logical

#### {KOW}Spazed

##### Banned
ffejrxx said:
then they would all figure have it out the first night since there so logical
No, it takes time for the information to get out.