Blue Eyes Logic Puzzle • Hacker News Board

Re: Blue Eyes Logic Puzzle

#11

Post by benjohnson » Fri, Jan 03, 2014, 7:06 PM UTC

aolol wrote: post #4

Taking the 'dramatic effect' argument further: if all the blue-eyed people kill themselves, would the brown-eyed people all simultaneously know they have brown eyes and also have to kill themselves?

And do the islanders know that there are 100 blue-eyed people? Remember... they don't talk about eye state.

Each Blue-eyed person may think that there's 99 people with blue eyes and still not know their own state. So basing any logic on the fact that there is 100 people with blue eyes makes sense to an outside observer, but to the blue-eyed people they can't use this fact in any logical conclusion.

Re: Blue Eyes Logic Puzzle

#12

Post by Mindless2112 » Fri, Jan 03, 2014, 7:09 PM UTC

Jehar wrote: post #8

aolol wrote: post #4

Taking the 'dramatic effect' argument further: if all the blue-eyed people kill themselves, would the brown-eyed people all simultaneously know they have brown eyes and also have to kill themselves?

This is the missing element from most explanations I see of this problem. We all look at the n cases from the POV of a blue-eyed person. An outside observer with brown eyes has the same level of information available to him, so it seems to me just as likely that after the first day, each person, regardless of eye color, could reason that nobody left the previous day, so the visitor must have been referring to me. So…

Each blue-eyed person observes 99 other blue-eyed people in the tribe, thus reasoning that he/she has blue eyes on the 100th day. However, each brown-eyed person observes 100 blue-eyed people in the tribe, thus reasoning that he/she has blue eyes on the 101st day (however this does not happen because on the 100th day all the blue-eyed people commit ritual suicide.)

Re: Blue Eyes Logic Puzzle

#13

Post by maxerickson » Fri, Jan 03, 2014, 7:09 PM UTC

Jehar wrote: post #8

aolol wrote: post #4

Taking the 'dramatic effect' argument further: if all the blue-eyed people kill themselves, would the brown-eyed people all simultaneously know they have brown eyes and also have to kill themselves?

This is the missing element from most explanations I see of this problem. We all look at the n cases from the POV of a blue-eyed person. An outside observer with brown eyes has the same level of information available to him, so it seems to me just as likely that after the first day, each person, regardless of eye color, could reason that nobody left the previous day, so the visitor must have been referring to me. So…

In the case that there is at least 1 member of the tribe with blue eyes, all the people with brown eyes can see his eye color. So they satisfy the question of who the traveler is talking about with the blue eyed member.

The blue eyed member doesn't see anyone else with blue eyes.

Re: Blue Eyes Logic Puzzle

#14

Post by maxerickson » Fri, Jan 03, 2014, 7:11 PM UTC

benjohnson wrote: post #11

aolol wrote: post #4

Taking the 'dramatic effect' argument further: if all the blue-eyed people kill themselves, would the brown-eyed people all simultaneously know they have brown eyes and also have to kill themselves?

And do the islanders know that there are 100 blue-eyed people? Remember... they don't talk about eye state. Each Blue-eyed person may think that there's 99 people with blue eyes and still not know their own state. So basing any logic on the fact that there is 100 people with blue eyes makes sense to an outside observer, but to the blue-eyed people they can't use this fact in any logical conclusion.

They are perfect logicians. It is not especially generous to think they can count (that is, if there are 100 blue eyed members, they can each count the other 99).

If each blue eye can see 99 others and nothing happens on day 99, then they know what has to happen on day 100.

Re: Blue Eyes Logic Puzzle

#15

Post by Symmetry » Fri, Jan 03, 2014, 7:17 PM UTC

So, what the visitor is providing is really the coordination, the point at which you can measure 100 or 99 days. But doesn't this setup require that there have always been 100 blue eyed people since forever? Any birth or death or all the islanders being crated at once would serve equally well as a timer. It seems like this problem only works because the blue-eyed islanders all know that there are 99 other islanders with blue eyes, but there was no moment in time where they learned it. And since that is so contrary to our expectations, it's what ends up making the whole scenario seem so unintuitive.

Re: Blue Eyes Logic Puzzle

#16

Post by nchallak » Fri, Jan 03, 2014, 7:19 PM UTC

maxerickson wrote: post #14

benjohnson wrote: post #11

Earlier quoted context omitted.

And do the islanders know that there are 100 blue-eyed people? Remember... they don't talk about eye state. Each Blue-eyed person may think that there's 99 people with blue eyes and still not know their own state. So basing any logic on the fact that there is 100 people with blue eyes makes sense to an outside observer, but to the blue-eyed people they can't use this fact in any logical conclusion.

They are perfect logicians. It is not especially generous to think they can count (that is, if there are 100 blue eyed members, they can each count the other 99). If each blue eye can see 99 others and nothing happens on day 99, then they know what has to happen on day 100.

But why will a non-blue eyed person not think the same then?

Re: Blue Eyes Logic Puzzle

#17

Post by Mindless2112 » Fri, Jan 03, 2014, 7:21 PM UTC

hornd wrote: post #9

I think the foreigners statement is ambiguous enough to render the proof in argument 2 incorrect. "... another blue-eyed person", to me, implies a singular person in the tribe has blue eyes. All of tribespeople will see multiple tribespeople with blue eyes, and therefore assume the foreigners statement was wrong, rendering no effect.

It's not so much that the statement was ambiguous or wrong -- it's that the statement gave no one in the tribe more information than was previously available to him/her.

Everyone in the tribe already observes at least 99 members with blue eyes, and everyone knows that everyone else observes at least 99 members with blue eyes, so the statement should have no effect.

Edit: split to separate comment.

Re: Blue Eyes Logic Puzzle

#18

Post by lisper » Fri, Jan 03, 2014, 7:21 PM UTC

Symmetry wrote: post #15

So, what the visitor is providing is really the coordination, the point at which you can measure 100 or 99 days. But doesn't this setup require that there have always been 100 blue eyed people since forever? Any birth or death or all the islanders being crated at once would serve equally well as a timer. It seems like this problem only works because the blue-eyed islanders all know that there are 99 other islanders w…

No, the key is that the foreigner's statement establishes common knowledge at some point in time. What happened before that point in time is irrelevant.

> the blue-eyed islanders all know that there are 99 other islanders with blue eyes

That's true, but what they don't know (until the 2nd) day) is that all the other blue-eyed islanders know that all the other blue-eyed islanders know that there are 99 other blue-eyed islanders. On the 3rd day they will realize that all the other blue-eyed islanders know that all the other blue eyed islanders know that all the other blue eyed islanders know that... and so on. Then on the 100th day there are 100 iterations of recursive knowledge, and all the blue-eyed islanders realize they themselves must have blue eyes.

UPDATE: note that it is crucial that all the islanders are together when the foreigner makes his statement. If he goes to each islander individually and says "some of you have blue eyes" then it doesn't work. What matters is not the statement, but that all the islanders witness all the other islanders hearing the statement.

Re: Blue Eyes Logic Puzzle

#19

Post by lazyant » Fri, Jan 03, 2014, 7:22 PM UTC

Shouldn't it be better if people had to commit suicide in their own houses? I mean if there are 2 people with blue eyes and on the second day they see each other at the town square about to commit seppuko they'd figure perhaps the other is the only one with blue eyes. Or this whole thing went over my head.

Re: Blue Eyes Logic Puzzle

#20

Post by maxerickson » Fri, Jan 03, 2014, 7:26 PM UTC

nchallak wrote: post #16

maxerickson wrote: post #14

Earlier quoted context omitted.

They are perfect logicians. It is not especially generous to think they can count (that is, if there are 100 blue eyed members, they can each count the other 99). If each blue eye can see 99 others and nothing happens on day 99, then they know what has to happen on day 100.

But why will a non-blue eyed person not think the same then?

A non blue eyed person will see that there are 100 blue eyed people and observe them killing themselves on the 100th day. For the blue eyes, if the 99 blue eyes they can see were all of the blue eyes, they would kill themselves on day 99. So the different eye colors have different information.

As far as I can see, the brown eyes all kill themselves the next day.