Problem in logic

[Kommander]

All Occupants of the island are Perfect logicians.
No Humans are Perfect logicians.
Therefore, no Occupants are Human.

This syllogism commits no fallacies, and the truth table checks out.

There are no people on the island. There are only polar bears and a smoke monster.

 

[Ha'kiv]

Irrefutable logic.

 

[Avon]

so why arent these supposedly logical people talking to each other? or at least looking at their reflections in the sea, etc?

 

[M'Sharak]

Why 100 days, while we're at it? Or was that figure simply pulled out of thin air?

 

[Holdfast]

The answer in this thread is wrong, although it took me a few moments to figure out why it felt intuitively wrong.

It assumes that the fundamental aim of the exercise is to get as many people off the island as possible, regardless of how long it takes. However, each individual person (if a perfect logician) would not see the situation in that way. Let me explain:

Morning of day one. The island's population look at each other. Each person sees either 100 blue/99 brown or 99 blue/100 brown. Crucially however, they don't know what their own eye colour is. It could be blue, but equally, it could be any other colour (they don't know there's 100 of each).

However, because the Guru can only proclaim about blue eyes, they know for a fact that they can only ever leave if they do have blue eyes (per your solution).

Each person has no way of knowing what the odds are that they have blue eyes or not, and therefore have no way of computing their personal odds of their leaving on day 100. We know it's 50% but THEY DO NOT. For each individual it's a binary problem, not a probabilistic one. It's either 100% or 0%, NOT 50% (eye colour as defined in this problem is a discrete variable; you either have it or not. It doesn't vary continuously, in a probabilistic manner).

The logical thing to do under these circumstances is for each person to kill as many non-blue eyed people as possible, as if only blue eyed people are left, they will leave the island that very evening when the ferry comes. If you're lucky and have blue eyes, you leave. If not, tough.

Of course, those being attacked will fight back. Thus, by midday on day one, either only blue eyed people will remain, or one brown eyed person will remain. The guru is only relevant if one blue eyed person is the sole survivor.

If a sole brown eye is left standing, he/she stays on the island

The number of blue eyes leaving on midnight on day 1 will therefore vary from 0-100 in a probabilistic manner (although I lack the mathematical ability to number-crunch it properly). This is not the optimal solution for the population as a whole, but there's no logical reason to prioritise that (that's an ethical choice).

In reality, by following my method, each islander has swapped a 50% chance of escape for something that averages out to be lower than 50%. However, the islanders don't KNOW that their original chances of escape were 50%, they only know they WOULD escape or WOULD NOT escape, with no way of calculating the odds. Might as well find out at the end of day 1 rather than day 100.

Of course, I have made the assumption that each islander has an overriding desire to leave the island rather than stay on the island. But if you remove that assumption, the entire problem is negated at its very core. Still, it demonstrates how important initial assumptions are (for instance, if you extend it from being an overriding desire to leave the island, to feeling that if they have to remain on the island life is not worth living, then the non-blue eyes will consent to being killed, meaning that 50% of the population get out on day one).

 

[Scout101]

The only person who has successfully solved and explained this was Emilia, and you cannot know if she is not familiar with such riddles.

Since I said I understood what was going on, don't think she's the only one. Rojohen also got it, although came at it from a strange direction at first. Since it's a BBS, though, wouldn't it be pretty silly to have a 3rd, 4th person explain basically the same thing? Tradition is to just chip in new info, rather than repeat what the last guy said.

Saying "well this is easy" while missing the very point of the riddle, then claiming you can solve the actual riddle with the same thought, is - to say the least - ridiculous.

As is telling us that it's impossible to do so, after several people have done exactly that. Not sure I understand the arrogant/condescending attitude, honestly. You had a hard time with this one, several others didn't struggle as much. Didn't say it was easy, just said I got the gist of it, and didn't struggle as much as some others.

What was the point of the thread, if you think it can't be solved by anyone other than a genius? Show of superiority? The riddle got solved, which is the POINT of riddles...

 

[sojourner]

There is nothing in the original problem that states that the guru is talking about a different person each day even if the guru "looks at no one in particular".

The guru could easily be talking about blue eyed Susy each day.

 

[Holdfast]

There is nothing in the original problem that states that the guru is talking about a different person each day even if the guru "looks at no one in particular".

The guru could easily be talking about blue eyed Susy each day.

The guru doesn't speak every day, only once ever.

 

[Jadzia]

The answer in this thread is wrong, although it took me a few moments to figure out why it felt intuitively wrong.

Here is my take on it...


The people are stuck on the island because they haven't been able to deduce their own eye colour. Whatever information they currently have must be insufficient to determine their own eye colour, otherwise they would have left by now.

Now realise that the guru isn't telling them anything they don't already know -- all residents can already see someone with blue eyes, so nobody has any new information to work with. Perfect logicians won't be able to deduce anything new, so nothing will change.

To the OP I must ask, where does this answer of 100 days come from? Are you merely quoting it from someone who claims it is correct?

 

[Gryffindorian]

From my point of view, the riddle, by the way, was poorly written and incoherent. In any case, those so called perfect logicians would've figured out a way to get out of the island and not wait 90 days by (A) hijacking Charles Widmore's submarine; (B) blackmailing Richard Alpert to reveal Jacob's location; or (C) threatening to kill Lapidus unless he fixed the Ajira plane.

 

[Holdfast]

The answer in this thread is wrong, although it took me a few moments to figure out why it felt intuitively wrong.

Here is my take on it...


The people are stuck on the island because they haven't been able to deduce their own eye colour. Whatever information they currently have must be insufficient to determine their own eye colour, otherwise they would have left by now.

Now realise that the guru isn't telling them anything they don't already know -- all residents can already see someone with blue eyes, so nobody has any new information to work with. Perfect logicians won't be able to deduce anything new, so nothing will change.

To the OP I must ask, where does this answer of 100 days come from? Are you merely quoting it from someone who claims it is correct?

No, the OP's logic on this part is sound.

Until the guru speaks, NONE of the population can SHARE the knowledge that anybody has blue eyes and so solve the problem. That added knowledge is what lets them solve the puzzle over 100 nights. Of course, this assumes that when the poster says that "... any islander who knows their eye colour then leaves...", they actually mean "... any islander who knows their eye colour MUST leave".

That's a bit of a stretch, but it's the only way to make the problem fit the OP's logic.

However, I still maintain my answer is more optimal, from an individual islander's perspective.

 

[Trekker4747]

{ Emilia }
Yes indeed. It's impossible to imagine this, though.
Luckily they're all perfect logicians. If I were one of them, I'd ruin the whole thing.

Reading through this thread I believe I have crossed eyes!

Maybe I should read it after a night's sleep?

 

[sojourner]

There is nothing in the original problem that states that the guru is talking about a different person each day even if the guru "looks at no one in particular".

The guru could easily be talking about blue eyed Susy each day.

The guru doesn't speak every day, only once ever.

Doh! Now I feel like digging some holes.

 

[Holdfast]

It's actually a harder problem than it needs to be, because the OP's phrasing is very inexact and ambiguous. In particular, I didn't understand the solution touted in this thread until I realised that the OP meant that you MUST leave if you know you have blue eyes, not just that you CAN leave.

For those still struggling, reduce it first to there being one person with blue eyes among 3 brown eyes. Then (and this is the crucial step), try it with two people with blue eyes and still 3 people with brown eyes and see what happens. Also, test out what happens if you don't KNOW that the other person KNOWS that there's at least one person with blue eyes out there (ie in the second scenario I've outlined, they each KNOW that EVERYONE knows there are only two solutions: 1 blue/4 brown or 2 blue/3 brown). Try viewing it from each person's perspective in turn. Once you understand it for two blue-eyed people among 3 brown, it's easily scaled up to 100 blues and 100 browns.

However, I still think that this assumes that killing people is not an option. If it is (and there's no reason why this should be closed off to logicians), since you cannot know whether you will get off the island or not in advance of the 100th day, it makes sense to speed up the process and solve the matter by the end of day 1 and so get off the island 99 days earlier (or not at all, but that would be the case anyway). Killing off non-blues does this.

 

[Ha'kiv]

Each person has no way of knowing what the odds are that they have blue eyes or not, and therefore have no way of computing their personal odds of their leaving on day 100. We know it's 50% but THEY DO NOT. For each individual it's a binary problem, not a probabilistic one. It's either 100% or 0%, NOT 50% (eye colour as defined in this problem is a discrete variable; you either have it or not. It doesn't vary continuously, in a probabilistic manner).

The logical thing to do under these circumstances is for each person to kill as many non-blue eyed people as possible, as if only blue eyed people are left, they will leave the island that very evening when the ferry comes. If you're lucky and have blue eyes, you leave. If not, tough.

Of course, those being attacked will fight back. Thus, by midday on day one, either only blue eyed people will remain, or one brown eyed person will remain. The guru is only relevant if one blue eyed person is the sole survivor.

If a sole brown eye is left standing, he/she stays on the island

The number of blue eyes leaving on midnight on day 1 will therefore vary from 0-100 in a probabilistic manner (although I lack the mathematical ability to number-crunch it properly). This is not the optimal solution for the population as a whole, but there's no logical reason to prioritise that (that's an ethical choice).

In reality, by following my method, each islander has swapped a 50% chance of escape for something that averages out to be lower than 50%. However, the islanders don't KNOW that their original chances of escape were 50%, they only know they WOULD escape or WOULD NOT escape, with no way of calculating the odds. Might as well find out at the end of day 1 rather than day 100.

Of course, I have made the assumption that each islander has an overriding desire to leave the island rather than stay on the island. But if you remove that assumption, the entire problem is negated at its very core. Still, it demonstrates how important initial assumptions are (for instance, if you extend it from being an overriding desire to leave the island, to feeling that if they have to remain on the island life is not worth living, then the non-blue eyes will consent to being killed, meaning that 50% of the population get out on day one).

Why would only -one- brown-eyed person remain, or many blue-eyed? And does this idea of yours, which I don't understand by the way, also apply to someone with red, orange, purple eyes? Who kills who and why is everyone aware of this being the most logical decision?

I fail to see any logic in this. By killing brown-eyed people, you would find out your own eye color if a blue-eyed person tried to kill you. This, to me, is a way of communicating between the islanders, which is forbidden. And murdering that many people for a chance no one leaves does not seem very logical to me. It is more logical to wait a 100 days, when all 100 people leave.

 

[Ha'kiv]

To the OP I must ask, where does this answer of 100 days come from? Are you merely quoting it from someone who claims it is correct?

I am not, however people agree on this answer.
Imagine this same situation with two blue-eyed and two brown-eyed islanders (and the Guru). This is easy to imagine. Assuming you are blue-eyed. You see the man in front of you with blue eyes, and he can see you. He can also see the two with brown eyes. So you can conclude that if you had brown eyes too, he would stand up and leave on the first night (because the Guru said she can see at least one person with blue eyes, so this must be him). Since he does not do this on the first night, you realize you must have blue eyes. He is thinking the same thing. You both leave on the second night.
Then do the same for 3-3, and 4-4. Although the logic becomes obvious and perfectly clear, as does the final answer, it becomes more and more difficult to actually imagine and understand the actual situations as the number of people increases. Now, as it has been so kindly pointed out for me, this is, apparently, my fault and other people are, apparently, able to do it without skipping a beat. However, if I may add, I have my doubts.

 

[Jadzia]

Until the guru speaks, NONE of the population can SHARE the knowledge that anybody has blue eyes and so solve the problem. That added knowledge is what lets them solve the puzzle over 100 nights. Of course, this assumes that when the poster says that "... any islander who knows their eye colour then leaves...", they actually mean "... any islander who knows their eye colour MUST leave".

I understand that, and I wrote my post understanding that, so I still stand by my answer.

Suppose you are one of the people on the island, and you can see N people with blue eyes.

Let N=10 for sake of this argument. Whatever colour someone's eyes are, you know that they will also be able to see someone with blue eyes. So everyone on the island can see someone with blue eyes, and everyone knows that everyone can.

So the guru isn't adding anything. Before the guru speaks, everyone on the island already knows that the guru can see someone with blue eyes. So what they say is already a fact known to all. Nobody has any new information to work with. Perfect logicians won't be able to deduce anything new, so nothing will change.

If my answer is wrong, then what's wrong with my logic?

I also want to add that I can follow the induction offered by others, but I feel that something is wrong with that induction ~ an error in the inductive step that's easy to make but hard to notice.

 

[Ha'kiv]

I understand that, and I wrote my post understanding that, so I still stand by my answer.

Suppose you are one of the people on the island, and you can see N people with blue eyes.

Let N=10 for sake of this argument. Whatever colour someone's eyes are, you know that they will also be able to see someone with blue eyes. So everyone on the island can see someone with blue eyes, and everyone knows that everyone can.

So the guru isn't adding anything. Before the guru speaks, everyone on the island already knows that the guru can see someone with blue eyes. So what they say is already a fact known to all. Nobody has any new information to work with. Perfect logicians won't be able to deduce anything new, so nothing will change.

If my answer is wrong, then what's wrong with my logic?

I also want to add that I can follow the induction offered by others, but I feel that something is wrong with that induction ~ an error in the inductive step that's easy to make but hard to notice.

This has been pointed out before. One site says, I quote:

And yet, this seems to be a paradox. Surely the Guru has given us no new information. She told us someone has blue eyes - we can clearly see that. So for extra credit, what new information is the guru providing?
One reader suggested the guru emparts a theory of mind. Sally knows Jane has blue eyes, but after the guru speaks, Sally knows that Jane knows that someone has blue eyes. This makes sense to me, but if you try to ratchet it up to n blue-eyed girls, your brain will ache.


This doesn't make sense to me. It is true if there are two blue-eyed people, but I fail to see how it is true if there are a hundred.

 

[Avon]

logically these 200 people would storm the ferry on the first night and kill their captors.

 

[Scout101]

Now, as it has been so kindly pointed out for me, this is, apparently, my fault and other people are, apparently, able to do it without skipping a beat. However, if I may add, I have my doubts.

This would be the part I was talking about. Why the continuing condescension, that because you struggled before understanding it, no one else could possibly have done better, or they must be lying? It's been shown that people understand this, you've been told repeatedly, and it's even been explained a couple different ways. Outside of assuming you're smarter than everyone else, why would you continue to doubt something that's been demonstrated to you?

As you just explained, it's really not THAT hard once you boil it down to the basics and then slowly add people back in. The solution that works for 1 vs 1, or 2 vs 2 is just as good at 100 vs 100, you just can't start full scale or it's tougher to think of the solution. And you never run into problems with the brown-eyed people, because their math is always 1 higher than everyone else, so everyone always leaves before a brown-eyed person is forced to guess. If everyone got drunk and missed the ferry on the 100th day, though, all 200 would be forced to leave on day 101, though...