The island where eye color can kill you.

February 15, 2008 7:47 AM

A mindbending logic puzzle. A thousand people on the island, 900 brown-eyed and 100 blue-eyed; anyone who learns their own eye color must kill themself the next day; a visitor mentions that there is a blue-eyed person on the island; what happens? Nothing, you say, because they already know that? Wrong. Further details at the Terry Tao post linked above, but don't scroll down below the boxed description unless you want hints and/or spoilers.

I'm not normally one for logic puzzles, but this one baffled and fascinated me to the point that I couldn't get my work done, so I thought I'd wreck your lives as well. If you want a complete spoiler, it's explained at the Wikipedia article on common knowledge. I note that some of the commenters at the Terry Tao post are remarkably stupid. (Via Avva; Terry's blog previously on MeFi in this post.)

posted by languagehat (390 comments total)

On 100th day, all blue eyed people kill themselves?
posted by trol at 7:55 AM on February 15

or possibly 101st, not thinking straight, its too early for me.
posted by trol at 7:55 AM on February 15

I have problems with a logic puzzle that consists of several illogical premises.
posted by Brandon Blatcher at 8:01 AM on February 15 [8 favorites]

I think I have this one solved. I should maybe Mefimail you.
posted by misha at 8:02 AM on February 15

I don't get how the islanders know that there are exactly 100 blue-eyed people on the island.

And 100 suicides? Can we submit this as evidence that religion is dangerous superstition and should be abolished?
posted by Mayor Curley at 8:03 AM on February 15

Argument 3: The tribesmen force the visitor to commit suicide in the town square because he knows the color of his own eyes. They then, of course, play soccer with his head.
posted by milarepa at 8:06 AM on February 15 [38 favorites]

Are they cannibals?
posted by Brandon Blatcher at 8:08 AM on February 15

milarepa wins.
posted by Faint of Butt at 8:08 AM on February 15

Among their many superstitions, notice that these people never look at reflections in water.

This is a fascinating study of an exotic culture. Thanks, Hat man!
posted by rokusan at 8:10 AM on February 15

The islanders do not know their own eye colors, but do they know that there are only two choices, i.e. blue and brown? Do they have any way of knowing, for example, that their own eyes are not green or hazel? Because then I can't answer the question.
posted by misha at 8:10 AM on February 15

Shame, too. The guy had an NSF grant and was about to make tenure.
posted by cog_nate at 8:11 AM on February 15 [4 favorites]

If brown and blue are the only choices, and are known to be so, I say they all die. Way to go, stupid visitor. Another culture wiped from the earth!
posted by misha at 8:11 AM on February 15 [1 favorite]

Why not just kill the foreigner? He admits knowing his own eye colour, and his painful death solves the problem.

On preview, milarepa beat me to it. That's what I get for editing.
posted by Chuckles McLaughy du Haha, the depressed clown at 8:12 AM on February 15 [1 favorite]

This should end soon.
posted by DU at 8:14 AM on February 15

I would wipe the whole place out by praising the beauty of the green eyes I saw on one of the tribesmen.
posted by JeremiahBritt at 8:15 AM on February 15 [11 favorites]

Also, why would 100% logical people kill themselves over eye-color. Alternatively, if eye color is really important for some reason, why would 100% logical people be bound by a taboo not to talk about it?

Also, why would such a rational group be unable to invent a mirror or even look in a pool of water?
posted by DU at 8:15 AM on February 15 [1 favorite]

This is a fascinating study of an exotic culture. Thanks, Hat man!

You're welcome! I visited the island myself, but of course having read the Lonely Planet guide I was sensitive enough not to mention eye color. No point reprinting the guide now, I guess...
posted by languagehat at 8:17 AM on February 15 [1 favorite]

Also, why would such a rational group be unable to invent a mirror or even look in a pool of water?

'Cause the island is in the middle of an ocean of lava. Lava!

Neat post, by the way.
posted by cog_nate at 8:18 AM on February 15

I thought I understood it until i read the answer and now I dont have a clue what its about. Not exactly flash friday fun is it
posted by criticalbill at 8:18 AM on February 15

I'd like to visit this island. Where is it?
posted by parki at 8:19 AM on February 15

It may make more sense to assume that the tribe only comprises vampires, whom we all know don't cast a reflection. Because that would make more sense.
posted by Admiral Haddock at 8:19 AM on February 15 [12 favorites]

I don't get how the islanders know that there are exactly 100 blue-eyed people on the island.

Well, assuming they're attentive, they certainly know that there are no more than 100 blue-eyed people on the island. The lower-bound question is trickier—the point about more than two potential eye colors is valid, I think, and isn't addressed in Tao's formulation.

The plane takes off.
posted by cortex at 8:20 AM on February 15 [1 favorite]

There's a fundamental problem with his logic. If I'm blue-eyed and don't know it but see a single other blue-eyed person, I'll just assume that that is who is being referred to. In addition, if that person does not commit suicide I'll think that they're not devout. If they do commit suicide there's no need for me to think about it anymore.

Or... what am I missing?
posted by dobbs at 8:20 AM on February 15 [2 favorites]

In the induction argument (all of the blue eyed people commit suicide on the 100th day), the day after all the blue-eyed people commit suicide, the rest of the inhabitants also commit suicide, as they now realize that they don't have blue eyes (and therefore have brown eyes).

However, highly logical people would recognize the implicit false dilemma between blue eyes and brown eyes and no one would kill themselves.
posted by graymouser at 8:21 AM on February 15

Having a lava ocean handy makes ritual suicide that much easier to execute. Just sayin'.
posted by cog_nate at 8:22 AM on February 15

I don't think that brown and blue are the only known choices, just the only two represented choices. Even if all the blue-eyed people commit suicide, any observer in the tribe will not know their own eye color, even if they know that everyone else has brown eyes.

I do, by the way, understand after thinking about it the solution in which all blue-eyed people commit suicide.
posted by OmieWise at 8:22 AM on February 15

Post-script: the blue-eyed people each know there's no more than 100 blue-eyes, and no less than 99. The brown-eyed people know there's no more than 101 blue-eyes, and no less than 100. Which is what, in theory, saves the brown-eyes from the mass suicide on Day 100.

And because the brown-eyes don't know that there's not, say, a green-eye among the surviving brown-eyes, no individual surviving brown-eye knows that he is a brown-eye. Ergo, he doesn't know his own eye color and doesn't kill himself.
posted by cortex at 8:23 AM on February 15 [1 favorite]

I'll think that they're not devout

I don't believe that's an option. It seems clear that they all default to devout and default to considering their neighbors devout.
posted by OmieWise at 8:24 AM on February 15

If I'm blue-eyed and don't know it but see a single other blue-eyed person, I'll just assume that that is who is being referred to

But there are 100 blue eyed people on the island and surely you've seen more than 1 blue eyed person.

If they do commit suicide there's no need for me to think about it anymore.

If it was your parents who committed suicide you might wonder if you're blue eyed.
posted by Brandon Blatcher at 8:24 AM on February 15

the blue-eyed people each know there's no more than 100 blue-eyes, and no less than 99.

How do they know this?
posted by Brandon Blatcher at 8:25 AM on February 15 [1 favorite]

But he does repeat himself, apparently, cortex. ; )
posted by misha at 8:26 AM on February 15

what happens if one of the blue-eyed people is blind? what happens then?
posted by criticalbill at 8:28 AM on February 15

How do they know this?

Presumably, they can count.
posted by dobbs at 8:31 AM on February 15

I have problems with a logic puzzle that consists of several illogical premises.

Yeah, I'll say.

The islanders do not know their own eye colors, but do they know that there are only two choices, i.e. blue and brown?
Well, in this case they see everyone else's eyes. They would know that 99 other people have blue eyes, but if they already knew that there were a total of 900 brown eyed people, and 100 blue eyed people, all one thousand would already have killed themselves. So, clearly no one in the tribe knows the exact number of blue or brown eyed people. They only know the number +/- 1.

Also, lets assume that the people on this island do follow the path of reasoning laid out and kill themselves. What prevents a brown eyed person from killing themselves? If 100 people do kill themselves, it seems like most of the people who die would be brown eyed. Except, seeing brown eyed people kill themselves would obviously illustrate the flaw in that particular train of thought.

So my answer is that nothing happens, and the argument that all the blue eyed people would kill themselves is nonsense. It's the same kind of "logic" that says an airplane won't take off if it's on a treadmill. It makes perfect sense to the to the thinker, but it is flawed at the foundation.
posted by delmoi at 8:33 AM on February 15 [2 favorites]

I think it helps to make the numbers smaller.

There are ten people on the island, one blue eyed, the rest brown. quonsar says it's nice to see another blue-eyed person. Bluey kills himself the next day because he can see nine brownies.

ten people, two blueys, eight brownies. quonsar stirs shit up. bluey1 thinks q could be talking about bluey2, so does not kill himself. bluey2 thinks the same thing, so does not kill himself. When neither kills himself the next day, they both know that the other bluey must be seeing another bluey, or they would have offed themselves!, which means that since they both see eight brownies (Ummm! Brownies!), they must be the other bluey. Both kill themselves on quonsay day+2. Etc. All the brownies are still alive because no one told the islanders there are only two eye colors.

I could be wrong.
posted by OmieWise at 8:34 AM on February 15 [15 favorites]

If we add to this wacky eye-color religion that the Holy Book informs them there are only two eye colors, then our hapless traveler wipes out the entire population on day 100, right? Am I getting this right?
posted by The Bellman at 8:34 AM on February 15

But there are 100 blue eyed people on the island and surely you've seen more than 1 blue eyed person.

No no no. If they knew that there were a total of 100 blue eyed people on the island, then everyone would already know their own eye color, and killed themselves when they noticed that there were a total of either 99 or 100 other blue eyed people.

So they can't know the total number of blue or brown eyed people.
posted by delmoi at 8:35 AM on February 15

So they can't know the total number of blue or brown eyed people.

No, they aren't TOLD there are 100 people with blue eyes and 1000 with brown, but they can see that there are 99 people with blue eyes and 1000 with brown or 100 blues and 999 browns if they have brown eyes. They just don't know they're the 100th blue eyed or the 1000th brown eyed.
posted by dobbs at 8:37 AM on February 15

How do they know this?

Assume (and given the premises we're dished, this isn't a far cry) that these villagers all know one another. So you have two cases:

1. Villager is brown-eyed. He looks at (or devoutly recalls) the eyes of his 999 friends and neighbors and knows that 100 of them have blue eyes and 899 have brown. His own eyes, he does not know; if he is brown-eyed, there are 100 blue-eyes. If he is blue-eyed, there are 101 blue-eyes.

2. Villager is blue-eyed. He knows that there are 900 brown-eyes and 99 blue-eyes, and him. If he is brown-eyed, there are 99 blue-eyes. If he is blue-eyed, there are 100 blue-eyes.

So browns know there are between 100 and 101 blue-eyeses; blues know there are between 99 and 100.

Following the stated induction solution, any given blue will know that there will be a mass suicide on the 99th day if he's not a blue-eye; and so on the 99th day he doesn't not commit suicide, as he still doesn't know his eye-color. Thereafter, when there is no suicide (because every blue-eye has to this point reasoned thus), he will know there are not 99 but 100 blue-eyes, and that he must therefore be a blue-eye. So he kills himself on day 100.

Each brown-eyes has been, by the same reasoning, sweating the hell out of day 100 and hoping (if hating himself for it) that all the blue-eyes he knows kill themselves that day, because he has reasoned that if day 100 isn't a mass suicide, there must be a 101st blue-eye. And that'd be him, and he'd be killing himself on day 101.
posted by cortex at 8:38 AM on February 15 [2 favorites]

ten people, two blueys, eight brownies. quonsar stirs shit up. bluey1 thinks q could be talking about bluey2, so does not kill himself. bluey2 thinks the same thing, so does not kill himself. When neither kills himself the next day, they both know that the other bluey must be seeing another bluey, or they would have offed themselves!, which means that since they both see eight brownies (Ummm! Brownies!), they must be the other bluey. Both kill themselves on quonsay day+2. Etc.

OmieWise has it. Excellent summary.

Also, why would 100% logical people kill themselves over eye-color. Alternatively, if eye color is really important for some reason, why would 100% logical people be bound by a taboo not to talk about it?

I take it you don't care for logic puzzles. That's fine, but complaining about things like that is like complaining that you can't smell the oceans on a map.
posted by languagehat at 8:39 AM on February 15 [21 favorites]

Assume (and given the premises we're dished, this isn't a far cry) that these villagers all know one another.

And they've all counted each other's eye color? That's a stretch.
posted by Brandon Blatcher at 8:42 AM on February 15 [1 favorite]

If we add to this wacky eye-color religion that the Holy Book informs them there are only two eye colors, then our hapless traveler wipes out the entire population on day 100, right? Am I getting this right?

Not quite; if "if not blue, then brown" was a given, the brown-eyes would kill themselves the NEXT day, having only discovered their brown-eyedness when the blue-eyes killed themselves on day 100.
posted by cortex at 8:42 AM on February 15 [1 favorite]

Well, no because we are assuming that all the blue-eyes figure it out at the same time and commit mass suicide, so why wouldn't the equally clever brown-eyes do so?
posted by The Bellman at 8:43 AM on February 15

And they've all counted each other's eye color? That's a stretch.

There's not much to do. They're highly numerate. The deathly prohibition against eye-color knowledge has turned them into obsessive eye-watchers.
posted by cortex at 8:43 AM on February 15

I approach this problem from a different angle. Suppose the visitor had said "I see 100 blue-eyed people." Obviously, all the brown-eyed people would already know that and sleep well, and all the blue-eyed people, knowing that they can only see 99 blue-eyed people, would figure out that they must have blue eyes and kill themselves that very night.

Now suppose that the visitor had said "I see at least 99 blue-eyed people." Everybody would sleep well the first night, because each of the blue-eyed people knows of exactly 99 blue-eyed people. The second night, though, each of those blue-eyed people would realize that nobody killed himself, and therefore there must be one more blue-eyed person than he can see, and that must be him. So on the second night all the blue-eyed people would kill themselves.

Obviously this can be extrapolated down to the visitor saying what he says in the puzzle. The blue-eyed people will always all kill themselves on the (101-N)th night, where N is the "at least" number of blue-eyed people the visitor says he can see. So, in the puzzle as given, the blue-eyed people will all kill themselves on the 100th night.

I find it much easier to understand the answer approaching it this way.
posted by cerebus19 at 8:44 AM on February 15 [8 favorites]

0.o
posted by ZachsMind at 8:45 AM on February 15

Well, no because we are assuming that all the blue-eyes figure it out at the same time and commit mass suicide, so why wouldn't the equally clever brown-eyes do so?

See above, on the collectively known lower- and upper-bounds. The brown-eyes don't have the same bounds as the blue-eyes; they wouldn't kill themselves (based on a mistaken presumption by each and every brown-eye of his own blue-eyedness) until day 101; they do not know they have brown eyes until the blue-eyes all kill themselves and reveal thereby that the brown-eyes are all, in fact, brown-eyes. And then it's a really dismal stretch until noon the next day before they off themselves.
posted by cortex at 8:45 AM on February 15

There's not much to do.

Nobody needs to catch food or build shelter, etc, etc :)

Anyway, laying out the culture of such a tribe sounds much more interesting than the puzzle.

Since it's taboo, it makes it sexy, so surely some of them are asking each other what they're eye color is. And there's always going to be some rebel who doesn't follow the custom. There might also be social customs where they dress or paint each other since they can't look in a mirror or the water.

Or maybe they all wear sunglasses.
posted by Brandon Blatcher at 8:49 AM on February 15

No-one has to know how many colours it's possible to have. I think OmieWise above has it -working up from 1 person in the group being blue and killing himself straight off, then thinking what the two people would do etc.
posted by Gratishades at 8:51 AM on February 15

And I'd suggest that none of the brownies would kill themselves the day after as each of them, for all they know, might be a special snowflake with different colour of eyes than everyone in their group.
posted by Gratishades at 8:55 AM on February 15

OK, cortex that makes no sense to me at all. I get the 100 days to blue-eye extinction thing: I'm a blue-eye; I've been waiting around, day after day; day 100 comes along and there has been no mass suicide so I think "well, it's day 100, I only count 99 blue-eyes, I must be number 100"; at that point I off myself and so does every other blue-eye who has been making the same calculation.

At that same instant every brown-eye (who knows he's either a blue-eye or a brown eye) has been making the same calculation, but waiting for day 101, because he or she sees 100 blue eyes. Now he sees the mass suicide and thinks, at first: "Whew! I guess I'm safe."

But then, he thinks: "Wait, all of those people were looking around and killed themselves on day 100 -- they must have only seen 99 blue-eyes. And since 100 blue-eyes just offed themselves, there must be only brown-eyes left. And that makes me a . . . . BRING ME MY SHARPENED CONCH SHELL!"
posted by The Bellman at 8:56 AM on February 15 [1 favorite]

Oh, I see what the argument is. Because no one leaves the island for 100 days, the islanders are able to determine the number of people with blue eyes. And once they learn that number, all the blue eyed people kill themselves. And then the next day all the brown eyed people would kill themselves as well.

On wikipedia, there is a slightly different formulation, where people 'leave' the island rather then being killed. Interestingly that makes it easier to understand the argument, because it's hard for people to imagine other people wanting to kill themselves.

The problem with the wikipedia explanation is that it only goes up to k=2, which is easy to understand.

First, imagine there is only one blue eyed person. When the traveler mentions that, the blue eyed person realizes they have blue eyes. Obvious.

Second, imagine there are two blue eyed people. When the traveler mentions that the blue eyed person sees the other blue eyed person not kill themselves the first day. So the second day, they realize that the two of them have blue eyes, and they both kill each other.

And that's as far as wikipedia takes it.

But what about the third case? If there are three blue eyed people, each of them will simply assume that the traveler was talking about one of those two people. And the argument would say that after the second day, when they don't kill each other they would all realize there were three blue eyed people, and that they were each one of them.

But in order for this to work out, you have to assume that everyone "starts counting" on that start date. What if on the third day one of those blue eyed people thinks that it's possible that one of those other blue eyed people didn't hear the news, or forgot, or something. In the problem explanation, it's given that everyone is hyper-rational and aware that everyone else is.

So, in this setup, the statement "one of you has blue eyes" is equivalent to the statement "Okay, everyone start counting days to determine who has blue eyes and who does not."
posted by delmoi at 8:59 AM on February 15

Gratishades: agreed, with the original problem statement by Tao. Blues kill themselves; Browns don't know they're all Browns, and don't.

The Bellman and I are arguing a variation where knowledge that everyone must either be blue-eyed or brown-eyed is somehow given. And thus:

At that same instant every brown-eye (who knows he's either a blue-eye or a brown eye) has been making the same calculation, but waiting for day 101, because he or she sees 100 blue eyes. Now he sees the mass suicide and thinks, at first: "Whew! I guess I'm safe."

I think the only thing we're disagreeing on is that "same instant" bit. I'm saying that they won't say, "whew, I'm safe." They'll know, as they see the Blues commit suicide, that they're none of them Blue, and hence by our variation they know themselves all to be Browns. But the instant of noon, by my reading, has passed; these are devout people of ritual and aren't about to engage in a late mass suicide at 12:01 or 12:03 or however long it takes them to get their conches together. And so they wait until the annointed time on the next day.

If they're more casual about the timeframe, then, sure: they get their sloppy seconds on as soon as they can cogitate on the Blue massacre.
posted by cortex at 9:02 AM on February 15

But the problem states that there are all different kinds of eye colors, it just so happens that this island only has blue and brown. So the only conclusion that the remaining 900 can make after the 100 blue-eyed islanders is kill themselves is not "there are 900 brown-eyers left" but instead "there are 900 non-blue-eyers left".
posted by turaho at 9:02 AM on February 15

cortex, The Bellman

I think your argument boils down to (or begs the question) "Once one person kills themselves, can someone else use that information to kill themselves now, or must they wait one day to process the information?" Or "how long is 'noon'"

Also (now that I realize it), it's possible that each brown eyed person might think they were the one hazel-eyed motherfucker on the beach, and not kill themselves.
posted by delmoi at 9:02 AM on February 15

What I don't understand is, wouldn't all the brown-eyed people all kill themselves at the same time as the blue-eyed people for the same exact reasoning that the blue-eyed people would?
posted by yeoz at 9:04 AM on February 15

Thanks for this. Friday logic fun!
posted by anotherpanacea at 9:04 AM on February 15

My response was to Bellman before I realized he and cortex were arguing a variation on the original problem. Oops.
posted by turaho at 9:07 AM on February 15

I see a marketing opportunity for coloured contact lenses.
posted by blue_beetle at 9:07 AM on February 15

Got it, cortex. You're right -- it's about the definition of "noon" which I had missed. They all glance down at their perfectly synchronized, non-reflective watches and, realizing it is no longer noon, go back to their ordinary lives for 23 hours and 59 minutes. Then, conch shells for everyone! Oh yes, there will be blood!
posted by The Bellman at 9:09 AM on February 15

Through the recessive genetics of the blue-eyed, some of the grandparents will eventually kill themselves. Though that would be outside the scope of this puzzle.
posted by Blazecock Pileon at 9:10 AM on February 15

Those tribesmen aren't logical at all. The logical thing would be to not think about eye colour and thus spare yourself the necessity of suicide. Giving the matter further consideration beyond "That guy's a douche!" is extremely illogical.
posted by Pope Guilty at 9:10 AM on February 15

I see a marketing opportunity for coloured contact lenses.
posted by blue_beetle at 12:07 PM on February 15 [+] [!]

I see a very short-lived and ill-conceived marketing opportunity for mirror shades.
posted by The Bellman at 9:11 AM on February 15 [2 favorites]

What I don't understand is, wouldn't all the brown-eyed people all kill themselves at the same time as the blue-eyed people for the same exact reasoning that the blue-eyed people would?

No, because their assessment of how many blue eyed people there are is always one higher than that of blue-eyed people.
posted by Pope Guilty at 9:13 AM on February 15

And while we're arguing premise-breaking variations, I think a little bit of colorblindness in the mix would be hilarious.
posted by cortex at 9:13 AM on February 15

((Pope Guilty just made me lose the Game. Damn you, PG, I was on a roll!))
posted by The Bellman at 9:15 AM on February 15

I believe that there is a user named "You just lost the game."
posted by Pope Guilty at 9:19 AM on February 15

Thanks cortex for taking the time to explain that.
posted by saladin at 9:19 AM on February 15

I think I got it l-hat! Nice puzzle!
posted by vronsky at 9:22 AM on February 15

I believe that there is a user named "You just lost the game."

Aye, there is.
posted by cortex at 9:24 AM on February 15

i think in OmniWise's version Bluey1 thinks it is possible that he himself is a green eye or red eye or whatever and he is okay with the fact that Bluey2 didn't kill himself because Bluey1 assumes Bluey2 has no idea that he (Bluey2) is a blue-eye.

If there is the possibility of more than 2 colors then any one villager could assume they are the 1 person of that non-blue or non-brown eye color.

If there is the dictum that "All villagers know that there is only brown and blue" then I believe the outcome is that they will all kill themselves according to the common knowledge logic.
posted by rlef98 at 9:30 AM on February 15 [1 favorite]

I don't think the induction is valid. In the inductive step, the islanders are assuming the truth of the theorem, which means they must have proved it somehow. And "our" proof depends on them actually committing suicide, which they only do if they've proved it. But our proof can't depend on itself; therefore it's only valid if there's another proof. So our proof is useless. The theorem may be true, but this isn't a proof of it.

But I don't see how the theorem can be true. The traveler's statement is redundant; everyone already knows that there's at least one blue-eyed islander. If they were going to kill themselves, they'd still do it without the traveler.
posted by equalpants at 9:33 AM on February 15

I'm unclear why the correct answer is "the entire island converts to Christianity."
posted by dw at 9:36 AM on February 15

Once upon a time, on an island far, far away, there was a native named Idiot Jed. Idiot Jed had bulging muscles, greasy hair, shining blue eyes, and was (secretly) not obsessed with logic puzzles. When anyone brought up a logic puzzle (which was all the damn time), Idiot Jed would shout "Kirsten sits next to Lord Bickerstaff who does not own a pony!" and run off to rotate the coconuts.

One day, a stranger showed up on the island, and said some crap about blue eyes. "He's talking about my friend Doug", thought Idiot Jed. "Doug has blue eyes. Doug is my friend." For the next few days, everyone on the island ran around looking worried and keeping logs of eye colors, while Idiot Jed worked on his car stereo. Then on day 100, the other 99 blue-eyed people on the island killed themselves. On day 101, most of the brown-eyed people's heads exploded, and the rest began randomly stabbing each other while sobbing "Does not compute". Eventually, Idiot Jed took control of the demoralized survivors and started a new religion based on driving cars very fast.

And today, we call that island... North Carolina!
posted by ormondsacker at 9:37 AM on February 15 [3 favorites]

I have problems with a logic puzzle that consists of several illogical premises.

damn right.

the problem i'm having is that the islanders are all highly "Logical" AND highly "Devout." Those two just don't mesh well.

Logical implies reasoning. It would mean that each and every islander would have attempted to reason out exactly who had blue eyes or brown eyes long ago, before the foreigner ever even arrived. They'd have counted. They'd have figured it out.

But they're also Devout. And they're religion seems to practice from a point of Ignorance, being that they should NOT attempt to discover anything about eye color.

So either they're instincts are to discover their eye colors or to ignore them. Not both. I realize this is a hypothetical, but its a poorly drawn one. It would have been better expressed only mathematically. I have a problem when logic problems get expressed as word problems improperly and cause the premises to not make sense any longer. I could conclude anything I wanted, really.

So sure, after 100 days the islanders have a big orgy, drink some Evan Williams, crown the foreigner king, and fuck off to Narnia.
posted by mr_book at 9:38 AM on February 15

why the correct answer ISN'T.

Blew the joke. DAMN YOU MATHOWIE FOR NOT LETTING US EDIT OUR COMMENTS!!!!! I WILL GET YOU YOU BROWN EYED BASTARD!
posted by dw at 9:38 AM on February 15 [1 favorite]

I got the answer and accept the conclusion, but then I started thinking: "wait, say I'm a villager and heard the pronouncement. What did I just learn?" I don't get the appeal to common knowledge. I knew there was a blue-eye, and knew that everyone else knew that. So what's the piece of common knowledge that the community gained?

Then I loaded up the wikipedia page languagehat linked to and saw that there was a formalization of common knowledge in modal logic. This is great. I'm going to puzzle over it for a while.
posted by painquale at 9:38 AM on February 15

the problem i'm having is that the islanders are all highly "Logical" AND highly "Devout." Those two just don't mesh well.

You must be a hoot at parties.
posted by dw at 9:39 AM on February 15

cortex: The plane takes off.

Indeed. Previously discussed in the conveyor belt thread. [**CONTAINS SPOILERS**]
posted by goodnewsfortheinsane at 9:40 AM on February 15

I would have to agree with the poster at the site who said: "There’s no new information here. The foreigner merely acknowledges that there are people of blue eye colour on the island, and this is a fact previously established by the islanders through observation."

Given this and the induction, the blue eyed people would already have killed themselves, which means the puzzle could not exist - even as a hypothetical.
posted by dances_with_sneetches at 9:40 AM on February 15

equalpants: The traveler's statement is redundant; everyone already knows that there's at least one blue-eyed islander. If they were going to kill themselves, they'd still do it without the traveler.

From the wikipedia page: "What's most interesting about this scenario is that, for k > 1, the outsider is only telling the island citizens what they already know: that there are blue-eyed people among them. However, before this fact is announced, the fact is not common knowledge."
posted by painquale at 9:40 AM on February 15

This isn't mindbending, it's consfusing.
posted by Stonestock Relentless at 9:42 AM on February 15 [1 favorite]

OmieWise writes...
ten people, two blueys, eight brownies. quonsar stirs shit up. bluey1 thinks q could be talking about bluey2, so does not kill himself. bluey2 thinks the same thing, so does not kill himself. When neither kills himself the next day, they both know that the other bluey must be seeing another bluey, or they would have offed themselves!, which means that since they both see eight brownies (Ummm! Brownies!), they must be the other bluey. Both kill themselves on quonsay day+2.

Excellent summary of the two blue situation, but then you wrote...

Etc.

Which is the same mistake that Terry Tao makes when he tries to use induction.

So -- ten people, four blueys, six brownies. quonsar stirs shit up.

Day one rolls around and everyone on the island can see at least see at LEAST three blueys. Let's call them A, B, and C. Any given observer will reach the following conclusions:

A can see both B and C so he has no reason to suspect that he's a bluey
B can see both A and C so he has no reason to suspect that he's a bluey
C can see both A and B so he has no reason to suspect that he's a bluey

Because of this mutual deadlock nobody is expected to commit suicide and nobody does.

Day two rolls around and the same deadlock exists.
Day three ....
Day four ...
Day five ...
Etc.

In short, when you have at least four blueys, you immediately reach the point where no one is committing suicide and *everybody knows why that is*. And it doesn't matter how many days pass.
posted by tkolar at 9:46 AM on February 15 [8 favorites]

On day 101, wouldn't all the brown eyes people then know they had brown eyes, and be forced, by their weird tribal beliefs, to commit mass suicide.

There's my solution: everyone's dead. Everyone. Even you, you brown eyed freak.
posted by Astro Zombie at 9:48 AM on February 15

It's counter-intuitive, but I do understand the logic behind the second argument. In reality, however, most humans have trouble dealing with numbers higher than 5, or 10, making the first argument much more practical, if logically incorrect.

The second argument assumes that every islander knows exactly how many *other* blue-eyed and brown-eyed people are on the island--100 blue and and 899 brown or 99 blue and 900 brown, correct? It also assumes that the tribal members have an interest in determining *exactly* how many members have blue eyes and are willing to *sustain* that interest for nearly four moons.

Many primitive/ancient civilizations had words for 1, 2, and then "many" (ref) - I don't think it's in the realm of formal logic (of which I know very little) to account for cultural issues like that, but in all honesty, the first conclusion (though not logical) makes much more sense in reality. (Move the number down to a graspable set, like 5 or 10, and the second argument becomes more probable.)

My guess is self-preservation issues would cause 99.99% of humans to ignore the formal logic, put their fingers in their ears, and say "LA LA LA LA!!! I don't know exactly how many other blue-eyed tribe members there are!"

After a week, even the taste of that blue-eyed stranger's flesh would be forgotten. ;)

there are two separate and seemingly valid arguments which start with the same hypotheses but yield contradictory conclusions

Welcome to the universe.
posted by mrgrimm at 9:52 AM on February 15

Also, why would 100% logical people kill themselves over eye-color. Alternatively, if eye color is really important for some reason, why would 100% logical people be bound by a taboo not to talk about it?

Please produce the chain of logic that refutes either of these notions.

I know that the common meaning of the word "logical" is "makes sense (to me)" and I know that the Mr. Spock use of the word "logical" is too inconsistent to define, but here it just means that the islanders always reason clearly from premises to conclusions.

To put it in logical terms, the puzzle says the islanders always believe something if it can be logically deduced. It does not say that the islanders always believe something if and only if it can be logically deduced.
posted by straight at 9:55 AM on February 15 [2 favorites]

I can question the unstated-but-generally-agreed-upon conventions of a logic puzzle! I am so smart! S-M-R-T smart!
posted by DevilsAdvocate at 9:56 AM on February 15 [1 favorite]

Cortex's reasoning above lays out the logic of it exactly.

They can't kill themselves without the traveller to start their simultaneous induction process. Imagine being a blue eyed person on the island. You spend each and every day wondering if there are 99 or 100 blue eyed people. You never get a date to refer back to 99 or 100 days ago, when everyone could synchronise their logical processes. You have to wonder though how the islanders ever got through their first 99 or 100 days on the island without dying.
posted by roofus at 10:00 AM on February 15

I see a lot of people tearing out their own eyes on day 99....
posted by illuminatus at 10:03 AM on February 15

There's also the pirate puzzle:

There are five rational pirates, A, B, C, D and E. They find 100 gold coins. They must decide how to distribute them.

The Pirates have a strict order of seniority: A is superior to B, who is superior to C, who is superior to D, who is superior to E.

The Pirate world's rules of distribution are thus: that the most senior pirate should propose a distribution of coins. The pirates should then vote on whether to accept this distribution; the proposer is able to vote, and has the casting vote in the event of a tie. If the proposed allocation is approved by vote, it happens. If not, the proposer is thrown overboard on the pirate ship and dies, and the next most senior pirate makes a new proposal to begin the system again.

Pirates base their decisions on three factors. First of all, each pirate wants to survive. Secondly, each pirate wants to maximize the amount of gold coins they receive. Thirdly, each pirate would prefer to throw another overboard, if all other results would otherwise be equal.
posted by tksh at 10:13 AM on February 15 [1 favorite]

How did the visitor get to the island if his plane was stuck on the treadmill and couldn't take off? The island remains stationary, people!!!! I'm quoting SCIENCE!!!!
posted by The World Famous at 10:15 AM on February 15

Yes, yes... but I have green eyes.
posted by markkraft at 10:20 AM on February 15

Really, though... Unless one of the villagers told me that I had blue eyes, then how would I know I *didn't* have green eyes? I would only know that 99 of you had blue eyes, and that 900 of you had brown eyes.

Maybe I'm a special little flower.
posted by markkraft at 10:24 AM on February 15

Just think if the traveler lied and said "Wow, only one green-eyed tribesman amongst ya. Think of the odds!"

Everyone would be dead the next day, having erroneously assumed they were the one.
posted by klangklangston at 10:25 AM on February 15 [1 favorite]

I could look at the villagers around me and think it's more statistically likely that I would be either brown or blue-eyed, but probability is not proof, and, therefore, not something I would act upon.
posted by markkraft at 10:28 AM on February 15

Everyone can see at least 99 blue-eyed people.

So everyone knows that there is a blue-eyed person on the island.

And because everyone can see more than one blue-eyed person, everyone also knows that everyone else on the island knows that there is a blue-eyed person on the island.

So, when the traveller arrives and tells everyone there is a blue-eyed person on the island...

He provides no new information.

To anyone.

He cannot affect anything.

The premise of the puzzle is flawed, because under the puzzle's own logic, the stable situation described could not have come about. The people of the island would have already committed suicide, or if they had not, they would be due to soon, and the traveller's appearance would not speed their departure.
posted by East Manitoba Regional Junior Kabaddi Champion '94 at 10:32 AM on February 15 [2 favorites]

Damn your logic you green blooded pointy eared blue eyed hobgoblins!
posted by tkchrist at 10:35 AM on February 15

tkolar: John Armstrong's comment from the Terry Tao thread may help:

n=2. Each blue-eyed person says: “If I am not blue-eyed, then there is one blue-eyed person on the island, and he will kill himself tomorrow because of what the traveller just told us.”

“But then the next day comes and the other doesn’t kill himself! That must mean there are two blue-eyed people, and I am the other one!”

n=3. Each blue-eyed person says: “If I am not blue-eyed, then there are two blue-eyed people on the island. They are each looking at each other and thinking (”If I am not blue-eyed, then there is one blue-eyed person on the island, and he will kill himself tomorrow because of what the traveller just told us.”). But neither will kill himself tomorrow, and they will realize their mistake tomorrow and kill themselves on the day after.”

“But then the third day comes and they don’t kill themselves! That must mean there are three blue-eyed people, and I am the third!”
posted by languagehat at 10:35 AM on February 15 [3 favorites]

Yep, that's why it has to be n=4 for the deadlock to be reached immediately. If no one is expecting anyone to kill themselves tomorrow, no new information is gained when nobody does.
posted by tkolar at 10:40 AM on February 15

I'm approaching this puzzle on a completely different level:

Heh-Heh, you said "brown-eyes".
posted by Horace Rumpole at 10:40 AM on February 15

Or, I should say, n >= 4.
posted by tkolar at 10:41 AM on February 15

But, tkolar, why do you think that for n=4 the rules change. The day of suicide is equal to n, so for larger values of n the day of reckoning just comes later.
posted by OmieWise at 10:43 AM on February 15

*gouges out eyes*
posted by chugg at 10:45 AM on February 15

Ok, for all those who are having trouble with the people being logical and devout, think of them as robots with or without spots on their heads. They have the the instruction set that they move themselves into a corner if they have a spot on their head, they have working visual sensors and also start with the knowledge that there is at least one robot with a spot on its head in the room.

Same problem, leaves out the human factor.
posted by Hactar at 10:49 AM on February 15

An interesting twist exists for the special case where the islanders know that the only two eye colors are brown and blue. If the rules of their religion were they were never allowed to know the color of their eyes, they'd kill themselves on day 1, since once the tourist mentioned seeing a single eye color, they were all doomed to know their eye color. In other words, they might not know at that moment in time what their eye color was, but they knew that it was only a matter of time, and being such devout islanders would kill themselves for that very reason.

Of course, if the suicidal edict is interpreted to only be in force when they absolutely know the color of their eyes, then the blues would kill themselves on day 100, and the browns on day 101 as mentioned above.
posted by forforf at 10:50 AM on February 15 [1 favorite]

Of the 1000 islanders, it turns out that 100 of them have blue eyes and 900 of them have brown eyes, although the islanders are not initially aware of these statistics

And they still are unaware as the foreigner did not tell them of these statistics, he only says that ONE of them is blue eyed, which they already know.
posted by Pollomacho at 10:50 AM on February 15

I'm sorry, all this discussion is reminding me of Vizzini debating Westley while trying to decide which cup holds the poison.
posted by etaoin at 10:52 AM on February 15 [3 favorites]

This island's religion makes cargo cults look positively not goofy. Although, I guess their rules are no less arbitrary than "God hates shrimp".
posted by DecemberBoy at 10:54 AM on February 15

Okay, I think I see what's going on. Here's the inductive step, as given by Tao and by the Wiki page:

Inductive step: if n-1 blues commit suicide after n-1 days, then n blues commit suicide after n days.
Proof:
1. Assume inductive hypothesis: n-1 blues die after n-1 days.
*2. Consider an island with n blues, in a world in which it is true that n-1 blues die after n-1 days.
3. Each blue on this island sees n-1 blues; knows that there are either n-1 or n blues overall.
*4. Each blue knows that n-1 blues will die after n-1 days.
5. Each blue sees n-1 days go by without death, concludes that there are n blues.
6. Each blue now knows his own color; dies the next day.

The trick is in steps 2/4. We're giving the islanders in our hypothetical world access to our inductive hypothesis. We're saying, "Consider a world in which fact X is true; the islanders in this world know that X is true, since they know everything." We're not specifying how the islanders know X. Can they possibly have a proof of X? Yes, but not this proof, because that would be an infinite regression.

The key is in the statement of the problem, where we assume that the islanders know the truth of everything, even things which can't be proved. We don't provide the proof that the islanders are using; instead we assume that they don't need one, so it doesn't matter whether or not they have one.

So I think the wording of the problem is the source of the confusion. This is really a knights-and-knaves-type problem, where the inhabitants are oracles, but Tao just says that they're "highly logical". Is it reasonable to infer "oracle" from "highly logical"? Well, that's a philosophical question.
posted by equalpants at 10:54 AM on February 15

But, tkolar, why do you think that for n=4 the rules change. The day of suicide is equal to n, so for larger values of n the day of reckoning just comes later.

The induction argument is intimately tied to the idea that you as an islander are expecting someone else to commit suicide, and that you make a new deduction when you see that they don't.

If you you have no expectation of anyone committing suicide (as I layed out above for n=4) the inductive argument falls apart.
posted by tkolar at 10:55 AM on February 15

But I don't think you do away with the expectation, that's what I'm asking you about. I think you're ignoring the constants, which are the day of the revelation and the number of people on the island.

The day of reckoning is visible blueys + 1.

The expectation is that no matter how many visible blueys there are, if they don't commit suicide on day=visible blueys, then you know there's one more bluey than there are visible blueys.

Since i islanders=(visible brownies+visible blueys-you), the only other possible bluey is you.
posted by OmieWise at 11:01 AM on February 15

after going through this problem several times i have come to the inescapable and inevitable conclusion that this island must have been located in the middle east
posted by pyramid termite at 11:02 AM on February 15

Paging asavage...
posted by thirteenkiller at 11:04 AM on February 15

If Omie's explanation still doesn't work for you, here's another one.'

Somewhere in my investigation of this problem—alas, I don't remember where—I ran across a comment to the effect that a farsighted islander could short-circuit the whole thing by killing a blue-eyed islander immediately after the announcement, preventing further violence.
posted by languagehat at 11:06 AM on February 15

1. This assumes that no one kills themselves for any other reason.
2. It also assumes that people would deliberately seek out knowledge that might end up with them having to kill themselves. Which is silly because they would avoid this just like mirrors. I don't know many people's eye colors and knowing it doesn't kill me. I think islanders would avoid inventoring other people's eye colors just as they would avoid looking into a pool of standing water.
posted by I Foody at 11:06 AM on February 15

For n = 4, each person knows that there are 3 other persons with blue eyes.

The foreigner makes his comment but doesn't specify the person or an upper-bound.

So each could simply think none of the other three would be sure of their own status (analogous to our subject) and hence, no suicide expected, and no suicides.
posted by Gyan at 11:11 AM on February 15

The expectation is that no matter how many visible blueys there are, if they don't commit suicide on day=visible blueys, then you know there's one more bluey than there are visible blueys.

On the contrary -- if the blueys that you can see (and it only takes three) are in a logical deadlock where none of them have any reason to suspect a problem, then you have no reason to suspect anything at all about yourself. You know that there are at least three blue eyed people on the island and you know why they're not killing themselves.
posted by tkolar at 11:11 AM on February 15

"And they still are unaware as the foreigner did not tell them of these statistics, he only says that ONE of them is blue eyed, which they already know."

But no one has discussed it. That's the new information.
posted by klangklangston at 11:15 AM on February 15

Gyan wrote...
For n = 4, each person knows that there are 3 other persons with blue eyes.

The foreigner makes his comment but doesn't specify the person or an upper-bound.

So each could simply think none of the other three would be sure of their own status (analogous to our subject) and hence, no suicide expected, and no suicides.

Exactly so.
posted by tkolar at 11:15 AM on February 15

Man, it is really hard to see what new information the visitor brings. He has to bring something new, otherwise everyone would have killed themselves long ago, as many people in this thread have pointed out.

But it can't be the case that he brings no new info, because everyone accepts (or should accept, anyway) the following argument: if there are only two blue eyed people on the island, then they'll both kill themselves after one day.

Consider the case where n=1 (that is, there's one blue-eyed villager). When the visitor makes his statement, the blue-eyed guy gains knowledge of proposition p: "someone has blue eyes". He kills himself the next day.

Now consider where n=2. What the each blue eyed person learns from the visitor's statement is not p, because everyone knows that someone has blue eyes (everyone can see a blue-eyed person). What they learn from the visitor's statement is proposition p2: "everyone knows that p". So after day 1 rolls around, the blue-eyed villagers can deduce from the lack of deaths and from p2 that they are blue eyed.

Now consider n=3. After 2 days, no one has killed themselves, but they kill themselves the next day because they learned from the visitor's statement p3: "everyone knows p2".

So, it looks like in order for the argument to go through, the visitor had to bring not just the information that someone has blue eyes (everyone in cases of n > 1 knew that already), but that everyone knows that someone has blue eyes, and that everyone knows that everyone knows that someone has blue eyes, and so on. However, to make it such that the visitor really did offer new knowledge, and the villagers wouldn't have all killed themselves beforehand, they couldn't have known this prior to the visitor's speech.

That's what is surprising to me. Intuitively, I would think that if I can see 99 other blue eyed people who can see each other, then I know that any other person knows that someone is blue eyed, and they know that I know that, and I know that they know that, etc. But this has to be denied for the visitor to actually bring new information. And that's weird. It's obvious how it all works out when n=2, but make the number of blue-eyeds any larger and it seems like I shouldn't deny it.

So I'm a little stymied. How is it that they didn't have this sort of recursive meta-knowledge beforehand? And what is it about the visitor's statement such that they gain this recursive meta-knowledge?
posted by painquale at 11:17 AM on February 15 [2 favorites]

I understand that you think that, I don't understand why you think there is a logical deadlock. I didn't understand your explanation of that deadlock.

If x blueys don't kill themselves on day n-1 then they didn't do it because they see n-1 blueys. They can only not kill themselves because they are at a day equal to or less than n-1. Once day=n, and no one dies, any observer o knows that it means that the number of blueys is not n-1 (the number he can see), but n, which means he's a bluey.

Why do you think this changes as n grows larger?
posted by OmieWise at 11:17 AM on February 15

More explicit version:

Theorem: In any world where the islanders are oracles (they know whether or not any statement is true), n blues die after n days.

Base case: Only one blue; easy proof. The proof of this case does not require the islanders to be oracles; this case would be true even in the real world.

Inductive step: Assume that, in any world where islanders are oracles, it is true that n-1 blues die after n-1 days. In any such world, n blues will die in n days, because the blues know that the n-1 case is true, discover their own color, etc.
posted by equalpants at 11:18 AM on February 15

I think there's something very intuitively appealing about tkolar's argument. Omiewise, if you start with his perspective -- what precisely do you see that's wrong with it?

In addition, I think the puzzle assumes without stating it that there is a particular time point (e.g. midnight each night) by which the villagers have made a discrete decision to kill themselves or not and can evaluate whether the others know something by whether they've acted or not.

Without that assumption, it seems like the villagers could as easily deduce from the others' lack of suicide that they simply had not yet decided to act rather than that they had decided NOT to act. As a result, the decision not to commit suicide could happen for the wrong reasons. This pollutes the information stream and screws up everyone's decisionmaking.

Or that's my sense, anyway.
posted by shivohum at 11:18 AM on February 15

Some of you people are reeeeeeeeeeally hung up on trying to inject human psychology into a puzzle that is pure logic. I understand why you would do this, since the agents in question are people and not widgets, but you have to understand it's the pure formal logic of this that's under analysis. Earth-shattering pronouncements like "you can't be both highly logical and highly religious to the point of suicide over eye color" are perfectly true and yet could not be further from the point of this exercise. See languagehat's comment above, if you missed it the first time around.

Pretend they're not humans. Pretend they're aluminum balls colored a certain color, and they have a chip inside them that both (a) allows them the power of logic for solving puzzles, and (b) requires them to roll off of a high cliff if they ever figure out what color they are.
posted by middleclasstool at 11:18 AM on February 15

That was to tkolar.
posted by OmieWise at 11:20 AM on February 15

The foreigner doesn't specify the upper bound, but the islanders themselves do. Look, I'm one of the islanders. I see 99 blueys and 900 brownies. I know for a fact that there are either 99 or 100 blue-eyes. It's one of the two. There are no more or less than either 99 or 100, because my own eyes tell me this.

If there's 99, they'll figure it out by the 98th day and kill themselves on day 99, following the inductive logic. If they don't do that, that means they're seeing exactly the same number of blue eyes I do, and we all go bang-bang on day 100.
posted by middleclasstool at 11:25 AM on February 15

If eye color was such a taboo, wouldn't they avoid looking at each other's eyes?

The last person can't kill himself, because they're supposed to commit a ritual suicide "for all to witness."
posted by kirkaracha at 11:25 AM on February 15

So each could simply think none of the other three would be sure of their own status (analogous to our subject) and hence, no suicide expected, and no suicides.

Except let's say there are four blue eyed islanders A, B, C, and D. Islander A knows there at at least 3 blueys. Now let A assume he's brown eyed. In that case, B would see two blueys. But then when islander B sees C and D not kill themselves on day two, B would know that he is blue eyed and kill himself on day three. When this doesn't happen, A concludes that his assumption that he is brown eyed must be false and must off himself right away.
posted by Schismatic at 11:26 AM on February 15

Come to think of it, even with n=3, there ought to be no suicides.

You have A, B, C as blue-eyes.

A doesn't commit suicide as he sees B & C with blue-eyes.
A expects that B expects C to commit suicide (A knows that B can see C).
A expects that C expects B to commit suicide (A knows that C can see B)

A hence realizes that neither will do so, and not because A himself is blue-eyed.

Similar for B & C.

The key is that the foreigner's remarks don't specify the person nor an exact number.
posted by Gyan at 11:30 AM on February 15

And the new information that the foreigner adds is more subtle than the existence of a single blued eyed person. Everyone knows that there is are at least 99 blue eyed folks, but what the statement does is inform the islanders that if one were to look around and see no pairs of blue eyes, he must kill himself. This is new information that affects how they interpret the behavior of the other islanders.
posted by Schismatic at 11:38 AM on February 15 [2 favorites]

EVERYONE would kill themselves on the 100th day.

EVERYONE would see on the 99th day that no one killed themselves.

EVERYONE, since they don’t know the color of their own eyes, would assume that *they* were the 100th.

And EVERYONE would die on the 100th day.

If you’re making the assumptions that lead to the second conclusion, then the real answer is the whole tribe is dead, on day 100.
posted by MythMaker at 11:38 AM on February 15

Gyan: But there are multiple days on which they can commit suicide. You can't just say that someone 'commits suicide' without specifying when.

He provides no new information.

Not for the n=100 case, but without the outsider, the base case for the induction (n=1) does not hold. If there were exactly one blue-eyed person on the island, he/she would not know for certain that there were any.
posted by obvious at 11:40 AM on February 15

More like this:

A doesn't commit suicide right away because he sees B&C.
A expects that no one will commit suicide tomorrow, because they can each see at least one guy with blue eyes. A knows he sees two, and he knows that B&C each see either one or two.
If B&C each see one, then they will kill themselves on the second day afterward, following the logic.
If B&C see two, then they won't, meaning the only possible remaining blue eye is A.

Similar for B&C.

Again, the remarks don't specify the exact number, but their eyes confirm only two possible numbers, n-1 or n. If it's n-1, they'll figure it out on day n-2 and kill themselves on day n-1. If not, then the nth person knows he's blue-eyed and must kill himself on day n.
posted by middleclasstool at 11:41 AM on February 15

MythMaker: No, because on the 100th day everyone but the blue-eyed person would notice that there was still one blue-eyed person left.
posted by obvious at 11:42 AM on February 15

Another inherent premise: the islanders overthink everything, almost with the aim of killing themselves.
posted by supercres at 11:47 AM on February 15

Can someone help me out with the common knowledge effect here?

I can understand how it works in the k=2 case. Before the visitor arrives, each of the two blue-eyed people know that there is at least one blue-eyed person, but they don't know that the one blue-eyed person they can see knows that there are any blue-eyed people. So each blue-eyed person thinks, "I see that guy has blue eyes. That means there are either one or two people on this island with blue eyes. Since he hasn't killed himself, it is possible that my eyes are brown and that (1) he doesn't realize there are any blue-eyed people on this island. Alternately, (2) my eyes are blue, but using the reasoning of proposition (1), he believes his eyes might be brown. Either way, I cannot deduce my eye color from his behavior." Upon the arrival of the visitor, everyone knows that there is at least one blue eyed person, so proposition (1) fails, and the countdown begins.

How does it work for k>2, though?
posted by mr_roboto at 11:49 AM on February 15 [3 favorites]

It works for k>2 because the third man realizes there are two cases, k=2 or k=3. If k=2, then it goes down as you describe, both the first two men kill themselves two days after the speech, so the third man knows he doesn't have blue eyes. If they don't kill themselves, then the only other possibility is that the first two men see a third blue-eyed man. Since the third man cannot see another blue-eyed man, then he must logically be it, so all kill themselves on day 3. Same for 4 and on up, following this same chain.
posted by middleclasstool at 11:53 AM on February 15 [3 favorites]

If the suicide thing is really bothering you (which it shouldn't, because this is a logic puzzle, not an anthropological study), then you could say instead that if someone learns the color of their own eyes, then the next day they must announce this fact to the whole island. As far as I can tell, the problem would be the same.
posted by obvious at 11:55 AM on February 15

OK, hang on one more time. If they have perfect logic, wouldn't they have already killed themselves?

I'm still not understanding why the traveler saying there's a person with blue eyes creates a problem. No, they don't know the stats, but observation would tell all these people that:

-- there are brown eyed people
-- there are blue eyed people
-- the observer has eyes and they're probably a color

It's that last thing that's throwing me. The traveler points out that he sees one blue-eyed person. He never says who it is. And he doesn't point out there are ONLY TWO OPTIONS.

So, a logical person, knowing that he/she sees a blue eyed person, can see that he might have been referring to that person. BUT THE OBSERVER STILL DOESN'T KNOW WHAT COLOR HIS/HER EYES ARE, BECAUSE HE/SHE DOESN'T KNOW ALL THE OPTIONS.

I mean, how do they know there are only brown and blue? It's likely, but that doesn't mean it's 100% true. And highly logical people aren't going to kill themselves without 100% certainty.

Seems to me the only logical solutions to this are to either commit mass suicide (because if one person has blue eyes, then everyone has blue eyes, even if they don't) or to kill the traveler (for offending their religion, and anyway, blue eyes he said so himself).

But this seems to fall apart in assuming there are only 2 choices in eye color. And in reality, there are N choices. And at no point are these highly logical people let in on what N is.
posted by dw at 11:56 AM on February 15

mr_roboto: See here. And to restate for anyone who hasn't gotten it: the new information that the foreigner brings is not that there are blue-eyed islanders, which they knew, but that the existence of such is common knowledge. Read the Wikipedia article and think about it again.
posted by languagehat at 11:57 AM on February 15

But how is the system stable prior to the arrival of the visitor for k>2? If the third man sees two blue-eyed people at any point in the history of the island, must he not assume that each of those people can see at least one other blue-eyed person? Doesn't the killing start at the moment of this realization, which does not depend on the arrival of the visitor?
posted by mr_roboto at 11:57 AM on February 15 [1 favorite]

Oops, I mean see here. Or just read middleclasstool's comment.
posted by languagehat at 11:58 AM on February 15

Hey languagehat, I've been struggling over the Wikipedia article, and hoping that someone could explain k>2 for dummies...
posted by mr_roboto at 11:58 AM on February 15

Some of you people are reeeeeeeeeeally hung up on trying to inject human psychology into a puzzle that is pure logic

Then stop using humans in pure logic puzzles :)
posted by Brandon Blatcher at 11:58 AM on February 15

dw: Everyone's eyes are either blue or non-blue, so in that sense there are only 2 choices.
posted by obvious at 12:02 PM on February 15

Paradoxes like this fascinate me for a couple of reasons: there's the paradox/problem itself, but there's the varied reactions to it. I've noticed that while one camp gets really into it, another is scornful.

When I was in my 20s, I became really fascinated with a story-problem puzzle. I told it to everyone I knew (I can't imagine do that now), and one of my friends surprised me by getting really irritated.

Me: I read this really great paradox. It goes like this: there's an island inhabited with two kinds of people, one that always lies and another that always tells the truth. A visitor walks up to an islander and says.... [I explained the rest of the problem.]

Friend: That's silly. No one is completely truthful all the time.

Me: No... No, I know. I'm just saying, if there was such an island...

Friend: But that's just the point. There couldn't be.

Me: Okay, but just supposing there was... I know there couldn't really be such an island, but let's just pretend. If there was an island like that...

Friend: No! It doesn't make sense. You want me to do a logic problem with an illogical premise!

Me: Well, the premise is fictional, but you can still deduce things from it. I mean, people can't fly, but if I said, "Imagine people could fly: what would happen if one crashed into the blades of a helicopter?", you could answer the question.

Friend: Fine. What was the question, again?

Me: There's this island, and... [I explain it again.]

(Pause.)

Friend: You know what I think?

Me: No. What?

Friend: I think people make things like that up ON PURPOSE!

(Pause.)

Me: Um... Well... yeah. I mean, it is made up. A logic professor made it up.

Friend: Jesus Christ! Aren't there enough problems in the world without people adding more?!?

I've had many discussions like this, and there seem to be two major disconnects:

1) It's pointless to derive logic from an illogical premise.
I think that's wrong, but I wonder if the other view is a confusion or a matter of taste. It might be a confusion. As others have pointed out, we use the word "logic" to mean more than one thing. The danger is using it in multiple ways at once. But if it's a matter of taste, my friend might have just been saying, "Why waste brain cells solving a problem that could never actually arise in the real world." I can't answer that, other than to say that, for me, it's fun. It's fun in the same way that fictions is fun. One could certainly say, "Why bother watching 'Jaws'? It's all made up?"

2) It's pointless and irresponsible to make up problems where none exist.
Again, I can't refute this. It's a matter of taste. Why play baseball? You don't have to.

I do think that logicians and math teachers could help by being really clear why they make up nonsensical islands and unexpected hangings. It's not to make up a crazy scenario. It's because they're trying to make a complex bit of math or logic understandable to laypeople via an analogy.
posted by grumblebee at 12:04 PM on February 15 [9 favorites]

But this seems to fall apart in assuming there are only 2 choices in eye color. And in reality, there are N choices. And at no point are these highly logical people let in on what N is.

Instead of thinking in terms of blue and brown (and red, green, taupe, whatever), you can always get a 2 choice system in terms of blue and not blue. This'll work just as same as the blue/brown thing in the wording of the question, so long as people are told that there is one blue eyed person.
posted by Schismatic at 12:06 PM on February 15

HI I'M ON METAFILTER AND I COULD OVERTHINK AN ISLAND OF LOGICIANS.
posted by empath at 12:06 PM on February 15 [1 favorite]

Ok, yeah, it's death all around.

Let me explicate in detail for n = 4

Day 0
Foreigner's State of the Union address
A sees B,C,D.
A expects B to see C&D and avoid suicide. Similar for others.
A expects B to expect of C & D that each will expect the other to commit suicide.

Day 1
No suicides.
A expects B to expect C & D to be surprised of the other's existence and to commit suicide thereafter.

Day 2
No suicides.
A expects B to be surprised and reason that B himself is blue-eyed and commit suicide.

Day 3
No suicides.
Oops!
posted by Gyan at 12:11 PM on February 15

let me see if I understand this solution:

if one blue eyed islander existed, he would not previously be aware that any blue eyed islanders existed at all, so when the foreigner says that there is one, he realizes it must be him since he knows that everyone else is not blue eyed. he kills himself.

if there are two, then each blue eyed person will assume that the other one should have killed himself by the next day. when the other one does not, they both think "he must think there's another blue eyed islander. since I know no one else has blue eyes, that islander must be me!" they both kill themselves on the second day.

if there are three, then they each think "okay, those two guys don't realize they have blue eyes because they think the foreigner is talking about the other one. so they're going to expect the other one to kill himself on the next day. when that doesn't happen, they're each going to realize that it's because the other one sees another blue eyed islander that they assume the foreigner is talking about and that since no one else has blue eyes it must be them. so on the day after THAT realization, the second day since the announcement, they'll both kill themselves." when that doesn't happen, each of the three islanders realizes that it's because they were each expecting the two islanders they knew about to kill themselves on the second day. when that doesn't happen, they each realize that it's because there was a third islander, namely themselves, that they each didn't know about that was affecting the other two islanders' expectations, and they all kill themselves.

here's where I differ:

if there are 5 islanders, then things change. because the previous four all hinge on the idea that from the perspective of a blue-eyed islander that there is one islander who has a reason to kill himself. to be clearer, it all depends on what each islander THINKS another islander sees, but ONLY one blue-eyed islander iteration lower than their own. so if we're talking about one of 5 blue eyed islanders, it all depends on what each of those islanders would think a blue eyed islander sees if there are only 4 blue eyed islanders, NOT what they would think if there were only 3, 2 or 1 blue eyed islander. This is because each islander knows the total number of blue eyed islanders can only vary by one islander from what they see, if it varies at all.

If you have a 5th blue eyed islander, he knows that the other 4 all see either 3 or 4 other blue eyed islanders. he is aware of the possibility of his being blue eyed, but not its certainty. More importantly, each islander knows what all of the other islanders see and what they can expect the OTHER islanders to see and that they cannot see less than 3 islanders. if the 5th islander thinks the 4th islander only sees 3 islanders, he knows that the third islander ALSO sees at least 3 islanders. since the 5th blue eyed islander sees the same thing all the other blue eyed islanders see, all of the blue eyed islanders know that each OTHER islander sees no fewer than 3 blue eyed islanders. And that's what's important. They all know that no one sees 2 islanders, and no one sees 1, and no one sees 0. Every single islander knows that every single OTHER islander already knew there were blue eyed islanders. No one is expected to commit suicide. without that expected suicide (which depends on at least one islander thinking only one or no other blue eyed islanders exist), there can be no massive blue eyed funeral.
posted by shmegegge at 12:12 PM on February 15 [2 favorites]

I think that's wrong, but I wonder if the other view is a confusion or a matter of taste.

Nah, just brains being wired differently. As soon as you start applying the logic concepts in this puzzle, it's immediately going to throw some people, 'cause they're not wired for logic, full stop. It's just not gonna fly.
posted by Brandon Blatcher at 12:17 PM on February 15