Blue Eyes – The hardest logic puzzle in the world

9 01 2009

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 blue_eyesat 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?

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”.

Thanks to xkcd for this logic puzzle, I have never heard of it before.  http://www.xkcd.com/blue_eyes.html

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13 responses

13 01 2009
chadabshier

If anyone would like a hint… try using smaller numbers and see what would happen logically.

14 03 2009
M

On day 100 – all people with blue eyes leave
On day 101 – guru and people with brown eyes leave

17 03 2009
chadabshier

Your first statement is correct, but your second … isn’t. How would the guru and the brown eyed people ever know their own eye color?

1 04 2009
M

solution was based on the fact that when the guru sees all the people preparing to leave it’s bound to know he has a different eye color

Answer you probably wanted:
On day 100 – all people with blue eyes leave
On day 101 – all people with brown eyes leave
On day 102 – guru leaves

1 04 2009
chadabshier

Um, still no. The guru and the brown eyed people … how could they ever deduce their own eye color? How would the guru know that her eyes are green not purple?

2 05 2009
Veiko

1. Day, all blue-eyed people leave.
2. Day, all brown-eyed people leave.
3. Guru leaves when all other people are gone, because of the fact that she is a guru.

14 08 2009
Boris

Still wrong.

THE BROWN EYED PEOPLE CAN NEVER LEAVE.

EVER.

17 10 2009
Chris Pallotta

If these are all logical people doing logical things, wouldn’t they just ask “What color is my eyes” and everybody leaves on Day 1?

26 10 2010
jack

The correct answer seems to have to be that no one leaves. Can anyone explain why another answer is correct?

Here is my reasoning:

Everyone on the island can see lots of blue-eyed people, and lots of brown-eyed people. Even though the guru openly stated she could see a blue-eyed person, that information is of no consequence. Everyone knows there are multiple people with blue eyes on the island. The information is not new to anyone, and therefore, since they are perfect logicians they would already have deduced anything that could have been deduced from that information. However, nothing can be deduced from it, so no one does anything.

29 10 2010
Bammer

The answer is: all 100 blue eyed people will leave on day 100. I will give you this example to make it a little bit simpler. You are on an island and you see 99 blue eyed people, 99 brown eyed people, and one person with red eyes. The guru says “i see a person with red eyes.” You expect the red eyed person to look around and see all blue eyed and brown eyed people, figure out he’s the one with red eyes, and leave on night one. But then you see the red eyed person still there on day two, so he must see someone else with red eyes, which leaves you as his only option. So you will both leave on night two, knowing you both have red eyes. Scale that up to 100 blue eyed people and they will all leave on night 100.

30 08 2011
Steavy

But part of the riddle says that they don’t know how many people have what color of eyes. If there are 100 people with blue eyes and they are all still there on day two, then there is still no way for you to tell whether you have brown eyes or blue eyes, no matter how many days you wait because no one has left.

7 11 2012
M D RAJAN

The puzzle can be only solved if the “GURU” sees only a group of people and makes the statement. IThat is not specifically mentioned.If he calls all the people he cannot make that statement and even if he makes statement the problem cannot be solved.

Rajan

31 07 2014
Yho

I have another theory.
I think most blue-eyed people will go, maybe with some brown-eyed ones you were mistaken.
What happens when someone says “I see Roger !” ? If Roger is next to you you look at him.
So when the guru says “I see someone with blue eyes” everyone will look at a blue-eyed neighbor. When you notice someone is looking at you, you know you are blue-eyed (because you are yourself looking at a blue eyed people, so you know what it means if someone is staring at you).

I thought that the brown-eyed ones would quit next. But as above comments stated, even if all the blue-eyed are gone, and there is only brown-eyed people left. They don’t know if they have brown eyes themselves or any other color.

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