Problem in logic

[boobatuba]

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?

Here is the problem with your logic...

The guru does tell them something they don't know. The guru gives each and every person on the island their possible eye color. She gives them specific information about someone that might be them. No one knows exactly who the guru is talking about when she says "I see someone with blue eyes." But they do know that she is either talking about them or someone else. This introduces a binary logical situation that makes the problem solvable.

Look at it this way. Assume you have blue eyes. For your entire time on the island, you have seen 99 people with blue eyes, 100 people with brown eyes, and the guru with green eyes. This new information from the guru tells you that either 99 people have blue eyes (if she wasn't talking about you) or 100 people have blue eyes (if she was).

All of the 99 people you see that have blue eyes either see 98 people who do (if you don't) or 99 people who do (if you do). If you don't have blue eyes, they will figure out when the 98 people they see with blue eyes DON'T leave on the 98th day that they also have blue eyes. Then, if you don't have blue eyes, they will leave together on the 99th day. Since they DON'T leave together on the 99th day, you will determine on the 100th day that you, too, must have blue eyes. All of your fellow blue-eyed island dwellers use the same logical deduction and leave with you on the 100th day.

The guru's information simply "starts the countdown." Also, the poor folks with brown eyes can never leave because all they will ever know is that their eyes are NOT blue.

 

[Ha'kiv]

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...

I assumed that would be the only part of my reply you would comment on. I said "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." and I stand my ground. Perhaps you missed it, because we clearly agree on this, judging by your explanation. There is a difference between understanding the logic of this situation and actually understanding, or imagining the situation - the entire proccess taking place. I apologize again for my attitude which apparently bothers you. I base my opinion not on the fact that I supposedly believe no one could solve this faster than me, but on my limited knowledge and understanding of how the human brain works.

My opinion has not changed. I still oppose your first claim, that you have solved this riddle in the same way even without the previously mentioned piece of information. If my attitude is still a problem, I will stop addressing you.

 

[Ha'kiv]

Here is the problem with your logic...

The guru does tell them something they don't know. The guru gives each and every person on the island their possible eye color. She gives them specific information about someone that might be them. No one knows exactly who the guru is talking about when she says "I see someone with blue eyes." But they do know that she is either talking about them or someone else. This introduces a binary logical situation that makes the problem solvable.

Look at it this way. Assume you have blue eyes. For your entire time on the island, you have seen 99 people with blue eyes, 100 people with brown eyes, and the guru with green eyes. This new information from the guru tells you that either 99 people have blue eyes (if she wasn't talking about you) or 100 people have blue eyes (if she was).

All of the 99 people you see that have blue eyes either see 98 people who do (if you don't) or 99 people who do (if you do). If you don't have blue eyes, they will figure out when the 98 people they see with blue eyes DON'T leave on the 98th day that they also have blue eyes. Then, if you don't have blue eyes, they will leave together on the 99th day. Since they DON'T leave together on the 99th day, you will determine on the 100th day that you, too, must have blue eyes. All of your fellow blue-eyed island dwellers use the same logical deduction and leave with you on the 100th day.

The guru's information simply "starts the countdown." Also, the poor folks with brown eyes can never leave because all they will ever know is that their eyes are NOT blue.

I agree that is "starts the countdown", but it doesn't give them any other information, does it? Everyone is already aware of the fact that their eyes might be blue. How do they know that everyone else also agrees that her statement "starts the countdown"?

 

[boobatuba]

I agree that is "starts the countdown", but it doesn't give them any other information, does it? Everyone is already aware of the fact that their eyes might be blue. How do they know that everyone else also agrees that her statement "starts the countdown"?

Yes, the guru's statement does give everyone other information. It tells everyone on the island that they might have been the one the guru was talking about.

Before that, at the beginning of this song-and-dance, you notice that the 99 people you see with blue eyes don't leave on the 99th day. But you still don't have any specific information about your own eye color at that point, so you can't be sure that yours are also blue and get on the ferry with your 99 buddies on the 100th day.

See the difference?

ETA: For an easier look at it, try it with only four people and the guru on the island.

You see 1 blue-eyed person, 2 brown-eyed people, and the green-eyed guru. You know that the blue-eyed person sees 2 brown-eyed people, the green-eyed guru, and you. You know that the brown-eyed people both see 1 blue-eyed person, 1 brown-eyed person, the green-eyed guru, and you.

Without the statement "I see someone with blue eyes," no one can leave since they have no way of knowing if their eye color is the same as someone else's or completely different. However, as soon as the guru makes her statement, you immediately know that either 1 person has blue eyes (the person you see) or 2 people do (that person and you). You can then wait to see if the blue-eyed person leaves to deduce whether or not you have blue eyes.

The same scenario plays out with 100-100, it just takes longer.

 

[Ha'kiv]

I agree that is "starts the countdown", but it doesn't give them any other information, does it? Everyone is already aware of the fact that their eyes might be blue. How do they know that everyone else also agrees that her statement "starts the countdown"?

Yes, the guru's statement does give everyone other information. It tells everyone on the island that they might have been the one the guru was talking about.

Before that, at the beginning of this song-and-dance, you notice that the 99 people you see with blue eyes don't leave on the 99th day. But you still don't have any specific information about your own eye color at that point, so you can't be sure that yours are also blue and get on the ferry with your 99 buddies on the 100th day.

See the difference?

I'm afraid not.

 

[boobatuba]

I'm afraid not.

Look at my ETA above. The guru's statement tells everyone that they might have blue eyes and might be the one she's talking about. They can then deduce logically whether they do or not if the number of people they see with blue eyes (N) leave in N days (since all of those people can make the same logical deduction).

It's interesting that one idiot would make the whole plan fall apart.

 

[Scout101]

My opinion has not changed. I still oppose your first claim, that you have solved this riddle in the same way even without the previously mentioned piece of information. If my attitude is still a problem, I will stop addressing you.

You're aware that I'm not the same poster as Rojohen, yes? He was the one that went off on a different tangent. I pretty much went down the same path as Emilia, and broke it down and worked up.

Insisting that I couldn't have done that (when you admitted the way I explained it worked just fine) is a pretty interesting disconnect. And you can't base it on your knowledge of the human brain, because surely you came across in your studies the fact that they all process things uniquely, which is why we aren't all painters, or rocket scientists. Actually a pretty good example in this thread, as various posters are explaining how they understood the problem, and you're not understanding what they are saying (see above this post). That they didn't follow your exact path doesn't make their answer wrong, as they got the right answer using the right facts.

Either way, not going to make it here very long with that attitude, but good luck.

 

[Lindley]

The guru does tell them something they don't know.

Inductively speaking, the guru provides the base case. Consider what happens when there is only one person with blue eyes.

This person sees no other blue-eyed people, but also has no information about his own eyes. When the guru speaks, suddenly he knows his eyes must be blue since no one else's are.

 

[Ha'kiv]

My opinion has not changed. I still oppose your first claim, that you have solved this riddle in the same way even without the previously mentioned piece of information. If my attitude is still a problem, I will stop addressing you.

You're aware that I'm not the same poster as Rojohen, yes? He was the one that went off on a different tangent. I pretty much went down the same path as Emilia, and broke it down and worked up.

Insisting that I couldn't have done that (when you admitted the way I explained it worked just fine) is a pretty interesting disconnect. And you can't base it on your knowledge of the human brain, because surely you came across in your studies the fact that they all process things uniquely, which is why we aren't all painters, or rocket scientists. Actually a pretty good example in this thread, as various posters are explaining how they understood the problem, and you're not understanding what they are saying (see above this post). That they didn't follow your exact path doesn't make their answer wrong, as they got the right answer using the right facts.

Either way, not going to make it here very long with that attitude, but good luck.

Oh, indeed you are, I'm sorry.
In the post above, we are discussing if the guru gives new information or not.
After this, I'm going to stop repeating myself. One individual might be a lot more successful in quantum mechanics, but this doesn't mean he can actually visualize or understand certain quantum events beyond the mathematics. Indeed some could not even understand the math of it - but no one is capable of imagining this, because we do not need to and so evolution didn't reward us with this power. So my opinion is based on what I think the human brain is capable of - not how different individuals perceive things. Surely you agree that situations exist which are too complex for us to fully grasp? Not to complex to solve, to explain or to understand the logic of it - but to fully grasp the situation. I don't know how else to explain this.

 

[Ha'kiv]

I agree that is "starts the countdown", but it doesn't give them any other information, does it? Everyone is already aware of the fact that their eyes might be blue. How do they know that everyone else also agrees that her statement "starts the countdown"?

Yes, the guru's statement does give everyone other information. It tells everyone on the island that they might have been the one the guru was talking about.

Before that, at the beginning of this song-and-dance, you notice that the 99 people you see with blue eyes don't leave on the 99th day. But you still don't have any specific information about your own eye color at that point, so you can't be sure that yours are also blue and get on the ferry with your 99 buddies on the 100th day.

See the difference?

ETA: For an easier look at it, try it with only four people and the guru on the island.

You see 1 blue-eyed person, 2 brown-eyed people, and the green-eyed guru. You know that the blue-eyed person sees 2 brown-eyed people, the green-eyed guru, and you. You know that the brown-eyed people both see 1 blue-eyed person, 1 brown-eyed person, the green-eyed guru, and you.

Without the statement "I see someone with blue eyes," no one can leave since they have no way of knowing if their eye color is the same as someone else's or completely different. However, as soon as the guru makes her statement, you immediately know that either 1 person has blue eyes (the person you see) or 2 people do (that person and you). You can then wait to see if the blue-eyed person leaves to deduce whether or not you have blue eyes.

The same scenario plays out with 100-100, it just takes longer.

Before the guru speaks, I see two people with brown eyes, one person with blue eyes and one person with green eyes. I assume my eyes could be blue, green, brown, purple, black, white, yellow, pink. After the guru speaks, I know she could or could not be talking about me. I assume my eyes could be blue, green, brown, purple, black, white, yellow, pink. Nothing has changed.

 

[Holdfast]

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.

Look at it this way, which simplifies my solution to its most basic:

You are on the island. You know you will either escape, or not. There is nothing at all you can do to improve your chances of escape, but there is something you can do to improve your speed of escape. There is no logical reason to consider the welfare of the population as a whole as being important.

Slaughtering anyone you see with non-blue eyes reduces the set remaining to people who KNOW they must have blue eyes (as they're still alive, and no longer being attacked), and can therefore leave on the very first night, as opposed to on night 100. You have not affected your personal odds of leaving at all, merely increased the speed with which you do so. Therefore, it's worth doing it.

The only snag is if the people being attacked fight back. If that is the case, they would surely end up killing some blue-eyed people, reducing your personal odds of leaving, and raising the theoretical - albeit unlikely - possibility of everyone fighting to just one survivor, who might not have blue eys.

However, if the primary aim of the exercise is to leave the island (as opposed to survival on the island being an acceptable outcome), then actually the non-blue eyed people would have no reason to fight back as they would know that if they are being attacked, they do not have blue eyes, and therefore can never leave anyway so may as well let themselves be killed.

(Put another way, essentially my solution simplifies the set to only one type of content right from day one, therefore the problem can be solved on night one, in conceptually just the same way it would be if there was only one blue eyed person among any number of brown-eyed people).

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?

To know that the other parties know other people have blue eyes requires a leap of faith (it's inductive, not deductive). Humans like you or I can do this, but as Kommander jokingly - but correctly - pointed out, these people on the island are not human like you or I; they work solely on the basis of fact - perfect strong logic - and can't make that inductive leap of faith. Therefore it's only when actively provided with the absolute factual knowledge that other people know that other people have blue eyes, that then can solve the problem using their perfect deductive logic.

THAT'S why they need the Guru under the original solution; to provide 100% truthful data which they cannot gather objectively via any other means. Without that fact, the problem ends up being insoluble as they won't induct it for themselves. You or I (being capable of induction) would not require the guru, you're correct... but we could be wrong in our inductive logic (whereas deductive logic cannot ever be internally wrong under any circumstances) and therefore we're not being perfect logicians in the way the islanders are being defined as. Incidentally, it points out how humans solve complex problems quicker and better, with less data, than an algorithmic computer program would need... at the admitted expense of being wrong sometimes.

(At least, that's the rationale you have to use to make the solution correct... I'm not necessarily saying it's a good problem... and it certainly was a too loosely phrased one).

 

[Ha'kiv]

I see, you are right, however this would most likely still be considered a type of communiaction between the islanders - they are giving each other vital information by killing only the brown-eyed people. Perhaps the instructions should be more clear on this. Afterall even not leaving is a way of communicating something to someone else.

 

[Lindley]

Before the guru speaks, I see two people with brown eyes, one person with blue eyes and one person with green eyes. I assume my eyes could be blue, green, brown, purple, black, white, yellow, pink. After the guru speaks, I know she could or could not be talking about me. I assume my eyes could be blue, green, brown, purple, black, white, yellow, pink. Nothing has changed.

There is one change in this scenario. If your eyes are not blue, then the knowledge of the sole blue-eyed person in your description has changed as a result of the statement.

 

[Ha'kiv]

Before the guru speaks, I see two people with brown eyes, one person with blue eyes and one person with green eyes. I assume my eyes could be blue, green, brown, purple, black, white, yellow, pink. After the guru speaks, I know she could or could not be talking about me. I assume my eyes could be blue, green, brown, purple, black, white, yellow, pink. Nothing has changed.

There is one change in this scenario. If your eyes are not blue, then the knowledge of the sole blue-eyed person in your description has changed as a result of the statement.

That is true, I forgot to mention that my eyes are for a fact blue in this situation. It only changes if there are two or more blue-eyed persons. In the 100-100 situations, there quite obviously is more than one blue-eyed person.

 

[Scout101]

but no one is capable of imagining this, because we do not need to and so evolution didn't reward us with this power.

What do you base this on? Some people CAN, in fact, visualize complex things, when others can't remember why they entered a room. Sometimes, those both apply to the same person! One guy can hit a baseball 500ft, another couldn't hit it if you taped it to the bat. You're throwing around a lot of absolutes, and i don't think they are justified, or correct. You keep insisting no one could solve this riddle without a ton of time and effort, but since several people DID, in fact, do that, your conclusion seems to be in error.

So my opinion is based on what I think the human brain is capable of - not how different individuals perceive things.

Welcome to whatever opinion you like, but it seems silly to keep repeating it in the face of contrary evidence. The BBS isn't exactly the best and brightest of human minds, but seems like at least a few people were able to see the solution without tons of scratch paper and quantum computers. When you boil it right down, it's really just a math problem with a couple conditions, and some people are pretty good at visualizing that.

Surely you agree that situations exist which are too complex for us to fully grasp?

Sure, given our current level of understanding of the universe. Are you arguing that your logic puzzle is an example of such situations? Seems a TAD easier to sort out than the Big Bang or the meaning of life. Wait, that one's easy: 42!

Not to complex to solve, to explain or to understand the logic of it - but to fully grasp the situation. I don't know how else to explain this.

What's the difference between solving, explaining, and understanding the logic of your puzzle, and fully grasping it? Seems like you're creating that dichotomy in order to tell us we don't fully grasp the puzzle that we were able to reduce to simple math and work through.

What am I missing?

 

[Tobin_Dax]

How can the villagers know that their eyes are either blue or brown when they see three different eye colors on the island?

For that matter, how can anybody be absolutely sure that their eyes aren't green, say, even if they only see brown eyes and blue eyes?

 

[Gruß vom Krampus]

^ Not factoring the assumption that the islanders or the guru could be blind or color-blind. Then no one would leave the island in this case.

 

[Ha'kiv]

You keep insisting no one could solve this riddle without a ton of time and effort, but since several people DID, in fact, do that, your conclusion seems to be in error.

The fact that several people have managed to solve this riddle in a short period of time does not.. Oh for the love of, I'm sorry, I am not going to repeat myself for the tenth time and I can only stand such scornfulness for so long.

 

[Ha'kiv]

The difference? How many mathematical concepts do you understand and are able to explain? How many are you able to imagine? Can you imagine logarithms, wave functions, infinity?

 

[Mr Silver]

Can you imagine logarithms, wave functions, infinity?

Can you?