Question

A group of people with assorted eye colors live on an island. No one knows the color of their own 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 explained in this paragraph.

Answer 1

On this island there are 100 blue-eyed people, 100 brown-eyed people, and the Guru (she happens to have green 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?

Answer 2

i think the blue eyed people leave on the 100th day or something. it's an inductive proof that i no longer remember the details of.

Answer 3

I actually didn't think of this one myself X)
If you want the link to where I got this:
http://xkcd.com/blue_eyes.html
So yeah...xkcd is pretty amazing.

Answer 4

if there were only 1 blue-eyed person they'd know they were the only blue-eyed person so they'd leave on the first night, since they could see everyone else was non-blue but the guru said someone was blue, thus they'd conclude there was only 1 blue person and it was themself. so they'd leave on night 1.

if there were two blue-eyed people, they'd see the other person has blue eyes (but not yet knowing they themselves have blue eyes) so they'd think "that person must be able to conclude they're the blue eyed person and will leave tonight." but that night they don't leave. the only way they could fail to leave is if there's at least one other blue-eyed person on the island since that's the only thing that can cause the paragraph 1 reasoning to fail. so each would know they, themselves, must have blue eyes since they don't see a 2nd blue-eyed person on the island. therefore, on the 2nd night, both blue eyed people leave since they now know their eye color.

this works inductively; if there are n blue-eyed people on the island they all leave on the nth night.

Answer 5

i learned this in college, although the variant i learned in college is wives who know if other wives' husbands are unfaithful, but don't know if their own husband is unfaithful. they agree to kill their unfaithful husbands the first night after they're sure of the infidelity. they learn there is at least one cheating husband.

i guess the eye-color version is the more PC version of the problem.