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Blue Eyes logic puzzle
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tuxedobob
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PostPosted: Fri Jun 26, 2009 12:45 am    Post subject: Reply with quote

It brings a base case of one. You can't establish another base case with the puzzle as given.
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TFBW
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PostPosted: Fri Jun 26, 2009 1:13 am    Post subject: Reply with quote

tuxedobob wrote:
Logically, wouldn't a group of perfect logicians choose the only option that results in an answer?

Yes.

tuxedobob wrote:
The only base case that the puzzle establishes is that a person has blue eyes.

That is incorrect, because the guru's statement is redundant in context. When the guru speaks, everyone already knows that he can see someone with blue eyes as a matter of observation. By the same reasoning, they all know that the guru can see someone with brown eyes, even though he doesn't say so. A test for blue eyes is thus not the only option.
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tuxedobob
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PostPosted: Fri Jun 26, 2009 4:39 am    Post subject: Reply with quote

In context, perhaps, but not in logical deduction. Razz
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Caldazar
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PostPosted: Sat Jun 27, 2009 3:34 pm    Post subject: Reply with quote

that depends on what logic you're using I guess. Once again the guru (and if I remember the puzzle correctly I think it's actually a she) doesn't really add anything new in a logical matter. I certainly understand why the puzzle is formulated as thus because it throws people off the track. But the conclusion that "one person has blue eyes" leads to "let's all try out this theory that I have blue eyes" isn't a logical one to me, nor is the statement which supposedly also comes from a perfect logician.
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Guset
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PostPosted: Wed Jul 15, 2009 6:11 am    Post subject: Reply with quote

I think something crucial has been overlooked, the 200 people do not know that their eye color is restricted to either blue or brown. So brown is in the category of 'not-blue' as is the other colors that can be logically chosen to be possible. I believe that's why the example of red eyes being a hypothetical possibility was in the original question.
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TFBW
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PostPosted: Wed Jul 15, 2009 7:35 am    Post subject: Reply with quote

Everyone except the guru knows that eye colours are not limited to blue and brown: they can see that the guru has green eyes. That's not important, though. What's important is that there are enough people with blue and brown eyes present that everyone knows about them even without comment from the guru.
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themusicroob
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PostPosted: Thu Sep 03, 2009 1:11 pm    Post subject: Reply with quote

I believe I understand why the Guru's announcement is important. That is to say, I believe I know the reason even if it doesn't entirely make sense to me. What the Guru tells them isn't that there's someone with blue eyes on the island, of course they all know that. It doesn't tell them that everyone else knows that there is someone with blue eyes on the island, because they know that too. It doesn't tell them that everyone knows that everyone knows that someone has blue eyes, this is also obvious. However, if you repeat this 100 times you're closer to the solution.

Remember the logicians on the island are just like us, trying to solve the puzzle: who will leave and when. However, their puzzle is conditional, who leave and when IF I don't have blue eyes. If they don't have blue eyes then their puzzle is slightly different to ours, because they are observing 99 blue eyed people rather than our 100. (Incidentally, if we were actually observing the situation, and no one left on the 100th night, we would know that we had blue eyes.) So we can now restate the puzzle from the point of view of one blue eyed person, who is assuming (for the sake of his test) that he does not have blue eyes. So forget the original puzzle, the case is now that there are 99 blue eyed people, 100 brown eyed and 1 green eyed.

In this puzzle, we think of one blue eyed person who, being a perfect logician, will be testing the case that he does not have blue eyes. Therefore his perception of the puzzle is that there are 98 blue eyed people. Forget that in reality all blue eyed people percieve 99 other blue eyed people, becuase this puzzle is the puzzle as percieved by the first blue eyed person.

Continue this process until we get to a new puzzle, 1 blue eyed person and 100 brown eyed people (and one green). Now when the guru says she can see one blue eyed person, he must leave that night. And if he doesn't, then we know that the assumption that led us to see the puzzle this way is wrong.

Of course all the logicians know that no one actually percieves the puzzle this way. What they don't know is that no one percieves that someone might percieve that someone might percieve (x100) that someone percieves the puzzle like this.

So the important thing is that while each blue eyed person is assuming for the sake of his test that he does not have blue eyes, he sees himself as having an extra piece of information compared to the other blue eyed people as he can see 99 of them, and they can only see 98. And he knows that if his assumption is corrent, and as they are all making the same assumption, they all see themselves as having an extra piece of information, the 98th blue eyed person that they can see.

I'm certain this is correct, and that the gurus statement is not just a psychological revelation but a concrete one. I'm not sure how well I've explained it, or if I've said anything that has not already been said, but I hope this might shed some light for someone.
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Akane
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PostPosted: Tue Mar 16, 2010 11:07 pm    Post subject: Reply with quote

There is a problem with the puzzle as stated. The islanders are all perfect logicians: granted. But it was not stated that each knows all the others are perfect logicians, which is clearly required before any deductions of the form "if A could deduce X, he would do Y, but he has not done Y, so X is not the case" can take place.
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Mephias
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PostPosted: Tue Apr 06, 2010 12:14 am    Post subject: Reply with quote

tuxedobob wrote:
The only base case that the puzzle establishes is that a person has blue eyes.

That is incorrect, because the guru's statement is redundant in context. When the guru speaks, everyone already knows that he can see someone with blue eyes as a matter of observation. By the same reasoning, they all know that the guru can see someone with brown eyes, even though he doesn't say so. A test for blue eyes is thus not the only option.[/quote]

A test for blue eyes IS the only option. Suppose that one person is doing a "test for blueness." This person will assume that he has brown eyes, and that Person B will assume that he has brown eyes, and that Person B will assume that Person C will assume that he himself has brown eyes,etc. Let's call the last person in this chain of assumptions Person Z. You came to conclusion in a previous post that if Person Z sees that everybody else has brown eyes, he will determine that he himself has blue eyes, and leave the island. But this is only the case if Person Z has the information given by the Guru - that at least one person has blue eyes!!!

Otherwise, Person Z will be unable to determine what color eyes he has. Thus, tests for any color besides blue will not work. Being perfect logicians, everybody will be "testing for blueness."
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Mephias
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PostPosted: Tue Apr 06, 2010 12:15 am    Post subject: Reply with quote

^^^


Wrong quote. I meant to quote this:

TFBW wrote:

That is incorrect, because the guru's statement is redundant in context. When the guru speaks, everyone already knows that he can see someone with blue eyes as a matter of observation. By the same reasoning, they all know that the guru can see someone with brown eyes, even though he doesn't say so. A test for blue eyes is thus not the only option.
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JustPokingAround
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PostPosted: Mon Jun 07, 2010 3:27 am    Post subject: Reply with quote

There is a simple answer to this puzzle.
The first depends on the statement made at the beginning of the puzzle; 100 have blue eyes, 100 have brown eyes. That means that nobody on the island has heterochromia. (two differently colored eyes) Therefore, one eye will be the same color as the other. Therefore, any blue-eyed person stoic enough to gouge out one of their own eyes and look at it will be able to leave.
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Bezman
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PostPosted: Mon Jun 07, 2010 10:45 am    Post subject: Reply with quote


Very Happy
Muahahahaha, they be DEAD before they leave, MUAHAHAHAHAHAAAAAA!!! Twisted Evil Twisted Evil Twisted Evil Twisted Evil
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Bezman
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PostPosted: Thu Sep 02, 2010 5:45 pm    Post subject: Reply with quote

Last night I couldn't sleep, so I thought I'd give the ole' Blue Eyes Logic Puzzle another go. I did the "eliminative" approach that tuxedobob already wrote down, and came to the solution that it both holds up and that the number of non-blue eyeholders is irrelevant to the case.
Anyone disagree?
1 Blueeye (BL) + 0 to UNLIMITED others => BL leaves day 1
2 BL + 0 to UNL others => 2 BL leaves day 2
3 BL + 0 to UNL others => 3 BL leaves day 3
...etc...
n BL + 0 to UNL others => n BL leaves day n

I don't think that's been said before.

I won't post my entire test, since it resembles tuxedobob's on page 1 a lot, but if anyone wants it, PM me.
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TFBW
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PostPosted: Fri Sep 03, 2010 3:06 pm    Post subject: Reply with quote

My question is not about the validity of the mathematics, but the underlying assumptions. What did you have to assume in order to make those calculations?
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Bezman
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PostPosted: Mon Sep 13, 2010 12:34 pm    Post subject: Reply with quote

Oh. I just went with the assumptions given in the problem description. And then I set up an exhaustive test scheme like tux-bob did.

I think the guru is put there to maintain the matter that none of the logicians can communicate with each other. It would be rather awkward if one of them could do that, for just one phrase, and then no more.
IMO, they could have dropped a bottle-mail or a stone dais or whatever from space aliens just as well to establish the fact that "at least one of you have blue eyes". And like they say, we need a "day 1" to start making the calculations;

If you narrow down the problem to 2 or 1 person (1 BR + 1 BL), they cannot ever secure the color of their own eyes, unless this piece of info is given to them. Even if it is obvious to one of them (or to 199 of them) that at least one person has blue eyes. Let me know if you get my point.

But I agree, it could just as well have been "brown eyes" in the message.
Or plucked out eyes... no, scratch that one! Laughing
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thedave
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PostPosted: Mon Sep 20, 2010 8:32 am    Post subject: Reply with quote

The thought process behind it works with a recursive type of thought.

We'll start at 5:

As someone with blue eyes, they see 4 and consider what the group of 4 would do if they are the only 4. If they don't do it, the person will leave the next day with them.

If there were 4 people, they would consider what a group of 3 would do, if it is not done they will leave the next day.

If there were 3 people, they consider what a group of 2 would do.

If there's a group of 2, they know a single person would leave on day 1, so they leave day 2.

If there's a group of 1, they see zero others and leave on the first day.




The reason the same thing does not work with the brown eyed islanders is that there is no base case established. Each amount could consider what the lower amount would do, but the base guy (1) would have no clue what color his eyes could be. He would be stuck on the island indefinitely, just as the full group must if the guru does not say anything.
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