Omnimaga

General Discussion => Other Discussions => Math and Science => Topic started by: Builderboy on November 27, 2012, 07:25:21 pm

Title: Blue Eyed Islanders
Post by: Builderboy on November 27, 2012, 07:25:21 pm

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; as far as he knows the totals could be 101 brown and 99 blue.

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:

"Among you is at least 1 person with 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."

Title: Re: Blue Eyed Islanders
Post by: alberthrocks on November 27, 2012, 07:47:46 pm

Hmm...

Spoiler For My Answer:

Spoiler For This might be right... so final warning!:

Is it that because the leader made eye contact with someone, and said, "I can see someone who has blue eyes," that a person with blue eyes leaves the island? As for the night, it would be the night on the day she said her sentence.

If no eye contact is made, it is assumed that the "who, me?" effect applies - the crowd turns towards the direction of her gaze, landing on a person that realizes that his/her eye color was revealed, and leaves that night.

Title: Re: Blue Eyed Islanders
Post by: Builderboy on November 27, 2012, 07:48:57 pm

No, there is no trickery of any kind, just logic.

Title: Re: Blue Eyed Islanders
Post by: alberthrocks on November 27, 2012, 07:51:00 pm

Spoiler For Hidden:

Oh... wait, I thought my response wasn't trickery? Just that she can see someone with blue eyes, and when that person realizes it's him/her, logically deduces that it is him/her that has blue eyes and leaves the island.

Title: Re: Blue Eyed Islanders
Post by: Builderboy on November 27, 2012, 07:51:50 pm

I suppose that could be logic, but no, she is not looking at anybody specific. 

Title: Re: Blue Eyed Islanders
Post by: merthsoft on November 27, 2012, 07:59:51 pm

Spoiler For Spoiler:

On the 100th day after the announcement all the blue-eyed people leave. Then all the brown-eyed.

Title: Re: Blue Eyed Islanders
Post by: Builderboy on November 27, 2012, 08:01:14 pm

That is correct!  I am curious, did you manage to deduce the answer or did you have to look it up?

Title: Re: Blue Eyed Islanders
Post by: merthsoft on November 27, 2012, 08:03:32 pm

I was given this problem a few years ago by a professor in college, and worked through it then.

Spoiler For Spoiler:

It was given to us as an exercise in reducing sets to make sense of problems (that terminology is wrong, but you get the idea :)). For example, this is super simple if you just think "what if there's just one blue-eyed person", and the solution follows pretty easily from there.

Title: Re: Blue Eyed Islanders
Post by: pimathbrainiac on November 27, 2012, 08:07:16 pm

How is it deduced? I think I have a clue, but I don't think it's right

Title: Re: Blue Eyed Islanders
Post by: leafy on November 27, 2012, 08:08:33 pm

Spoiler For Spoiler:

Start with the base case, where there's only one blue-eyed islander, and work your way upwards.

Title: Re: Blue Eyed Islanders
Post by: Hayleia on November 28, 2012, 01:18:43 pm

Lol, a friend of mine asked me a varient of that question 4 years ago :P

Spoiler For Spoiler:

If there was only one blue eyed guy, he would see everyone with brown eyes, so when the Guru says "there is at least one with blue eyes", seeing no one with blue eyes, he would guess that he is the one that must leave.
Now of they are 2, then, they both think of each other "ah, the Guru is talking about that guy, he'll leave this night". But the day after, they see that the other one is still here, so they deduct that he sees someone else with blue eyes. So they both guess that they are the ones that must leave.
Etc :)