Brain Teasers
Brown Eyes, Blue Eyes
On an otherwise deserted and isolated island, 200 perfect logicians are stranded. The islanders are perfectly logical in every decision they make, and they will not do anything unless they are absolutely certain of the outcome. However, they cannot communicate with each other. They are forbidden from speaking with one another, or signing, or writing messages in the sand, else they be smited by the god of the island.
Of the 200 islanders, 100 have blue eyes, and 100 have brown eyes. However, no individual knows what color their own eyes are. There are no reflective surfaces on the island for the inhabitants to see a reflection of their own eyes. They can each see the 199 other islanders and their eye colors, but any given individual does not know if his or her own eyes are brown, blue, or perhaps another color entirely. And remember, they cannot communicate with each other in any way under penalty of death.
Each night, when a ship comes, the islanders have a chance to leave the barren and desolate spit of land they have been marooned on. If an islander tells the captain of the ship the color of his or her own eyes, they may board the ship and leave. If they get it wrong, they will be shot dead.
Now, there is one more person on the island: the guru, who the islanders know to always tell the truth. The guru has green eyes. One day, she stands up before all 200 islanders and says:
"I see a person with blue eyes."
Who leaves the island? And when do they leave?
Of the 200 islanders, 100 have blue eyes, and 100 have brown eyes. However, no individual knows what color their own eyes are. There are no reflective surfaces on the island for the inhabitants to see a reflection of their own eyes. They can each see the 199 other islanders and their eye colors, but any given individual does not know if his or her own eyes are brown, blue, or perhaps another color entirely. And remember, they cannot communicate with each other in any way under penalty of death.
Each night, when a ship comes, the islanders have a chance to leave the barren and desolate spit of land they have been marooned on. If an islander tells the captain of the ship the color of his or her own eyes, they may board the ship and leave. If they get it wrong, they will be shot dead.
Now, there is one more person on the island: the guru, who the islanders know to always tell the truth. The guru has green eyes. One day, she stands up before all 200 islanders and says:
"I see a person with blue eyes."
Who leaves the island? And when do they leave?
Hint
What would happen if there was only one blue-eyed person on the island and the rest of the islanders were brown-eyed? What about two? What about three?Answer
All 100 blue-eyed people leave on the 100th night. First, we propose that two blue-eyed people are on the island. They would both leave on the second night, because they would each look at the other blue-eyed person on the second morning and realize that the only reason the other blue-eyed person wouldn't leave on the first night is because they see another person with blue eyes. Seeing no one else with blue eyes, each of these two people realize it must be them. Now consider the case of three blue-eyed islanders. It has been established that if only two people had blue eyes they would leave on the second night. You, the third blue-eyed person in our example, know this, as do the other two blue-eyed people. So when you wake up on the third morning, and the two other blue-eyed people have not left, you know that they must see another person with blue eyes, and you can see that no one else on the island has blue eyes, so you know it must be you. All three of you realize this simultaneously, and all three leave on the third night. If a fourth blue-eyed person is on the island, he will reason the scenario with three blue-eyed people on the island, and when he sees that they have not left the island during the fourth morning, he/she/it will leave the island on the fourth day, and so will all the other blue-eyed people. If we follow this pattern, we will eventually reach the answer of 100 people leaving the island after 100 days.Hide Hint Show Hint Hide Answer Show Answer
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Comments
I knew this classic would someday make it onto this site!
Well, I'm sure glad it did! I've never seen this before and I love it. I found it very counterintuitive. It seemed that the guru gave no useful information. Everyone could see there were blue-eyed people, so what?
Obviously, I did not solve this myself, and I had to work through the slightly confusing answer to see how the logic works. Once I understood, I was sold. This is a thing of beauty.
Going on my faves list.
Obviously, I did not solve this myself, and I had to work through the slightly confusing answer to see how the logic works. Once I understood, I was sold. This is a thing of beauty.
Going on my faves list.
The so-called solution makes no sense at all.
The problem asserts that there are 100 people with blue eyes.
Then the so-called solution states that "we propose there are two people with blue eyes."
Inconsistent. Illogical. Downvoted.
The problem asserts that there are 100 people with blue eyes.
Then the so-called solution states that "we propose there are two people with blue eyes."
Inconsistent. Illogical. Downvoted.
The solution needs to contain reasoning to reach the solution. Creating another hypothetical scenario helps readers understand the solution, which then leads to the actual solution that is also provided.
Just to let you guys know I just got my first math teaser accepted.
Awesome teaser!
I don't see where the people on the island KNOW there is 100 of each eye colour. The riddle was very clear about them no being able to communicate with each other but not clear on them knowing that there was 100 of each. It states there is 100 of each so the reader knows this but it does not say the people on the island know this. And why wouldn't each go look in the ocean on a calm day?
Are you not interested, what happens with the other 100 people? Next day they leave, too.
Great teaser, though the hint kind of spoiled it for me since I could solve it shortly after.
howhardcanitbe:
It's not necessary for the islanders to know beforehand how many people have blue eyes; they can deduce it from the number of blue-eyed people they see and the number of nights that go by without anyone leaving, as explained in the solution. We can assume the water around the coast is too murky to reflect clear images since the islanders would obviously just leave otherwise.
unlur:
That would only happen if they knew there were only blue-eyed and brown-eyed people on the island. They could have green eyes, for example.
howhardcanitbe:
It's not necessary for the islanders to know beforehand how many people have blue eyes; they can deduce it from the number of blue-eyed people they see and the number of nights that go by without anyone leaving, as explained in the solution. We can assume the water around the coast is too murky to reflect clear images since the islanders would obviously just leave otherwise.
unlur:
That would only happen if they knew there were only blue-eyed and brown-eyed people on the island. They could have green eyes, for example.
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