A problem better than Monty Hall

[Gendo]

So this was posted on IMDB way back when; I remember it caused quite an argument, with Chx refusing to admit that every logician/mathematician was right and she was wrong.

I've taken this wording from XKCD; there's various versions:

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. 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?

It's an old puzzle; so most of you have probably heard it before. At least, if you're the type of person who cares to read puzzles like this. But it's always been one of my favorites; the answer just seems so intuitively wrong, but then the math/logic proves it to be correct.

[Unvoiced_Apollo]

So I cheated and looked at the solution (at least the xkcd version). I'll just say that I had figured who leaves but not on what day.

though realistically, I'm pretty sure all 201 would just storm the ferry on the first day.

[BruceSmith78]

I guess after a hundred nights of seeing no one leave, or 99 nights, I dunno, the blue-eyed people would all figure out they had blue eyes, and they'd all leave. Cuz they know there's at least 99 people with blue eyes, and at most 100 if they have blue eyes themselves. Then the brown-eyed people the next day? That might be wrong. Maybe it would be sooner. But the guru's fucked. She's never leaving the island, unless she looks at her reflection in the water.

[Gendo]

I guess after a hundred nights of seeing no one leave, or 99 nights, I dunno, the blue-eyed people would all figure out they had blue eyes, and they'd all leave. Cuz they know there's at least 99 people with blue eyes, and at most 100 if they have blue eyes themselves. Then the brown-eyed people the next day? That might be wrong. Maybe it would be sooner. But the guru's fucked. She's never leaving the island, unless she looks at her reflection in the water.

Yeah that's basically right. Well, with the puzzle as stated above; then the brown-eyed people will never leave, because they dunno anything at all (except once the blue-eyed people leave, they learn that they do NOT have brown eyes). But the puzzle is commonly stated as everyone knows that they all have either brown or blue eyes. In that case, the brown-eyed people all leave the next day.

[Pope Bucky]

The riddle, as worded, contradicts itself. It says all the islanders know the rules in the following paragraph. 100 blue eyes and 100 brown eyes but then claims that for all they know it could be 101 and 99. Which is it? Do they know it's 100 and 100 or could it be something else.

I have issues with this riddle.

I have issues.

[BruceSmith78]

It says they know all the rules in "this" paragraph. It's unnecessarily ambiguous, but it's actually referring to the first paragraph, not the second. I made a similar error when registering for this site.

[Pope Bucky]

I'm having a logic issue with the solution provided on xkcd.

On day 99, all the blue eyed people see 99 other blue eyed people. They have no way of knowing if those 99 are it or if they are #100. No one leaves because they don't know.

On day 100, they still don't know whether the 99 are it or if they're #100. No one should leave.

What am I missing?

[Gendo]

Ha, that's funny... didn't notice the ambiguity there; then again, I'm the one who wrote that question for registration on this site. Funny that the wording of the problem makes the same mistake I did.

[Gendo]

I'm having a logic issue with the solution provided on xkcd.

On day 99, all the blue eyed people see 99 other blue eyed people. They have no way of knowing if those 99 are it or if they are #100. No one leaves because they don't know.

On day 100, they still don't know whether the 99 are it or if they're #100. No one should leave.

What am I missing?

So the explanation for why this works....

First off, pretend that there is actually only 1 blue-eyed person and 100 brown-eyed people. In this situation, the blue-eyed person knows 1 of 2 things is true: either there are 0 blue-eyed people, or 1 blue-eyed people (himself). He doesn't know which, but once the guru makes his announcement, then he knows that it is option B, so he must have blue eyes. Thus, he leaves after 1 day.

Now, pretend that there are actually 2 blue-eyed people and 100 brown-eyed people. Each of the blue-eyed people knows that 1 of 2 things is true: A) There is 1 blue-eyed person, or B) there are 2 (including himself). The Guru makes his announcement. Each blue-eyed person knows that IF the truth had been A; then that 1 blue-eyed person would have left the day that the Guru made the announcement (because of what was shown above; when there's only 1 blue-eyed person, he leaves after 1 day). So once no one leaves after 1 day, both blue-eyed people now now that it must be B), not A). So they both leave after day 2.

Now pretend that there are actually 3 blue-eyed people. Each of them knows that there's either 2 blue-eyed people, or 3 (including himself). After day 1, no one leaves. After day 2, no one leaves. Well, IF there had only been 2 blue-eyed people, then both of them would have left on day 2. The fact that they didn't leave lets all three of them know that there must be 3, not 2... so they all leave after day 3.

Continue this pattern.... with X blue-eyed people, the blue-eyed people will all leave on day X.

[Islandmur]

Don't get it, if the guru says I see someone with blue eyes, why do they assume after day 2 or 3 that it is them? Since they also see someone with blue eyes? wouldn't they just be "why doesn't the fool leave? Or does the guru say I see 3 people with blue eyes? Or i see 100 people with blue eyes? but if he just says "someone" in singular how the heck does that help?

[Gendo]

Don't get it, if the guru says I see someone with blue eyes, why do they assume after day 2 or 3 that it is them? Since they also see someone with blue eyes? wouldn't they just be "why doesn't the fool leave? Or does the guru say I see 3 people with blue eyes? Or i see 100 people with blue eyes? but if he just says "someone" in singular how the heck does that help?

Are you asking this after reading my last post, or did you not see it yet?

[Pope Bucky]

I leave the first night because my survival trait (owning guns) is stronger than having blue eyes (a trait I also possess).

Nyah!

[Islandmur]

Gendo i did see your reply hence my question, does the guru only ever ask "I see SOMEONE with blue eyes" or does he say " I see ten people with blue eyes".

[Gendo]

He only says "I see someone with blue eyes". If he said "I see 100 people with blue eyes", then they would have all left on day 1, not day 100. The answer to "why do they assume after day 2 or 3 that it is them?" is my entire other post... First off, it's not that someone is thinking "oh, the Guru meant ME when he said he sees someone with blue eyes". Rather he is simply thinking "well there's either 99 or 100 people with blue eyes. If there were 99 people, then those 99 would have left after 99 days. Since they didn't leave after 99 days, there must be 100 people with blue eyes."

[Islandmur]

But someone is singular, so if he is saying I see one person with blue eyes and I also see several (99) with blue eyes what's the big deal for me to be wandering that there are 99 people or 100 with blue eyes?

[Gendo]

Not sure I understand your question... let's go back to the situation where there's only 2 blue-eyed people. Pretend you one of the two blue-eyed people, because it's easier to think of it that way. You look around and think "I see 1 blue-eyed person. Either he's the only one, or he and I are the only 2. But I don't know which. And I don't know if he can see any blue-eyed people, because if he's the only one, then he can't see any blue-eyed people." After the announcement is made, now everyone knows that there's at least 1 blue-eyed person, and more importantly, everyone knows that everyone knows.

So now you think "well if that other guy is the only blue-eyed person, then he should now know for a fact that he has blue-eyes, because he can't see any blue-eyed people but he knows someone here has blue eyes. So if he's the only blue-eyed person, then he will leave tonight."

But no one leaves that night. You'd already determined that IF there were only 1 blue-eyed person, he would have left. But he didn't leave, so there must be 2 blue-eyed people, not 1. And since you can only see 1 blue-eyed person, you must be the second.

[Dr_Liszt]

I did know that everyone with blue eyes would leave. But couldn't figure out when.

Mostly you have to take into account how long will stupid people take to figure it out.

[Islandmur]

Yeah that still doesn't answer my question. The Guru says he sees "someone" "one" "singular" person with blue eyes on an island with 100 blue eyed people... so what?

blue or brown eyed I also see "one" person or more with blue eyes, so I haven't learned anything new.

Besides these people are logicians right? so i'm on an island with one guru with purple eyes and 100 brown eyed people and 99 blue eyed people...know what... I do the math and decide right off that I"m number 100 with blue eyes.

I mean really it's waiting 100 nights to pass before accepting the obvious?

[Gendo]

"Besides these people are logicians right? so i'm on an island with one guru with purple eyes and 100 brown eyed people and 99 blue eyed people...know what... I do the math and decide right off that I"m number 100 with blue eyes"

But you don't know that there are 100 people with blue eyes. There could be either 99 (the 99 you see), or 100. You don't know which.

"The Guru says he sees "someone" "one" "singular" person with blue eyes on an island with 100 blue eyed people... so what? "

It sounds like you might be thinking that they already know that there are 100 people with blue eyes. They don't.

"blue or brown eyed I also see "one" person or more with blue eyes, so I haven't learned anything new."

What you have learned is that everyone else heard this announcement, so all of those people now know that everyone else knows that there's at least 1 blue-eyed person.


Again, forget about the whole "100 people" thing. Just focus only on the situation when there are 2 blue-eyed people. You, and 1 other. Think only about that one particular situation, and see if you can understand why, in that situation, both you and the other guy leave after the second day.

[Islandmur]

What you have learned is that everyone else heard this announcement, so all of those people now know that everyone else knows that there's at least 1 blue-eyed person.

But they already KNEW THAT. they can see others with blue eyes. Saying there is ONE person with blue eyes does not alter their reality. they already know there is a least 1 person with blue eyes, heck they know there is at least 99 people with blue eyes.

And yes i know they don't know how many have blue eyes, but they can count. they see 100 brown eyes and 99 blue eyes... so logically because of the even number of brown eyes they see, logic would dictate also 100 blue eyes.

had they been able to see 55 brown eyes and 144 blue ones... that would have kept them guessing but 100 and 99? naww

[Gendo]

And yes i know they don't know how many have blue eyes, but they can count. they see 100 brown eyes and 99 blue eyes... so logically because of the even number of brown eyes they see, logic would dictate also 100 blue eyes.

had they been able to see 55 brown eyes and 144 blue ones... that would have kept them guessing but 100 and 99? naww

Now you're just gendoing... The number of brown eyed people has nothing to do with the puzzle or the solution. It could have been 8 brown eyed people and 877 blue eyed people. Or vice-versa. There are NO assumptions being made by the natives (other than the assumption that all other natives are perfect logicians). They aren't going to just "guess" that they have blue eyes because having an equal number of each would make more sense than having 99 and 101.

As for how the announcement gave them any new information... Yes, they knew that there was at least 1 blue-eyed person already, but they didn't know that everyone knew that they knew it. The announcement makes it so that not only does everyone know that there's a blue-eyed person, but it makes it so that everyone knows that everyone knows it. See http://en.m.wikipedia.org/wiki/Common_knowledge_(logic" onclick="window.open(this.href);return false;).

Once more, you really need to focus only on the situation of having 2 blue-eyed people to start with. Ignore everything else. In this case, both people knew that there was at least 1 blue-eyed person. But neither of them knew that the other person also knew that. Not until after the announcement.

[CashRules]

The real problem with this problem is it takes too long to explain. Any good math/logic problem can be explained in one paragraph, six sentences at most. Otherwise it's just nerds being far too impressed with their own nerdability.

[Dr_Liszt]

The real problem with this problem is that you have to think. Who has time for that shit?

[CashRules]

The real problem with this problem is that you have to think. Who has time for that shit?

Nerds.

[Islandmur]

I have no clue what gendoing is, so i give up.

[Unvoiced_Apollo]

So this was posted on IMDB way back when; I remember it caused quite an argument, with Chx refusing to admit that every logician/mathematician was right and she was wrong.

I've taken this wording from XKCD; there's various versions:

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. 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?

It's an old puzzle; so most of you have probably heard it before. At least, if you're the type of person who cares to read puzzles like this. But it's always been one of my favorites; the answer just seems so intuitively wrong, but then the math/logic proves it to be correct.

It's a trick question. If their only ability is to see and hear but can't figure out how to communicate, then they clearly don't have the cognitive capacity to be perfect logicians.

[Gendo]

I have no clue what gendoing is, so i give up.

It's a long-running joke here... probably started by Brandon. It basically means either being overly pedantic, or distracting from the real issue with pointless other facts. Something that I've had a tendency to do, thus the name.

[Blade Azaezel]

I like to think it was me who started it