How do they know?
And, what is the answer to the question posed in the logic puzzle: Who leaves the island, and on what night?
they’re all a bunch of cloned zombies and they all think the exact same thing so when each one imagines a binarical kind of branch of situations always assuming that he’s simply watching these others follow the same binarical system they all assume that they have no chance of leaving and are right until the binarical system they followed leads them back up the same exact binarical system knowing that the path at the the bottom which was that there were only 1 possible blue eyed person was no longer possible so the possibility of the binarical system they had investigated downward til the bottom is day by day cleared out of all impossibilities following the same binarical system upward again until they finally get to the very top and final possibility that they were blue eyed.
hope I didn’t confuse anyone who reads this i’m trying to find a good way to explain it also.
See, I don’t see why they can’t make an assumption. Sure, they might be odd one out, but how likely is that? Is it perfectly logical not to make an assumption when an assumption might lead to a favorable outcome?
Sure, the blue eyed people have motivation to make sure that every other person can get out, because it gives the greatest chance for the greatest number of people to escape. But after the blue eyed people all leave, is there any reason not to try to break the rules and claim you are sure even though you are not?
Granted, unlike the other problem, there’s no guaranteed number of people or amount of time of people who can be saved, but I do think it foolish to think that people who want to be able to leave (as the problem states) would just stay together because the rules say they have to. Not if they were perfectly logical.
The only exception is if there is someone who will punish them not only if they are wrong, but also if he knows they don’t know for sure.
nevermind that they ALL left on the same day. just think about it as if each ONE of them left on that day because they happened to figure it out. It’s not even that they all left it’s that none of them stayed. if one of them figured it out and decided to stay that’s cool but none of them happened to do that. hope I’m not confusing anyone…
Awww, son of a b. I thought we were done here.
Interesting true-life case study from two nights ago: The Chessic family is all playing Clue. For those that have forgotten, you place cards in the center envelope and then deal out the remaining cards. Any extra cards go in the middle, face down, and can be looked at by moving to that space on your turn.
So in this game, it’s four players and there’s one card in the middle. I move to it and see Colonel Mustard, secretly, of course. Everyone saw me do this. So:
Me: I know it’s Colonel Mustard.
Dad: Doesn’t know. Knows I know.
Mom: Doesn’t know. Knows I know.
Wife: Doesn’t know. Knows I know.
Then on their turns, Dad and Mom both move to the middle and look at the card. So:
Me: Knows it’s Mustard. Knows Dad and Mom know. Knows Wife doesn’t know.
Dad: Knows it’s Mustard. Knows Mom and I know. Knows Wife doesn’t know.
Mom: Knows it’s Mustard. Knows Dad and I know. Knows Wife doesn’t know.
Wife: Doesn’t know. Knows Dad, Mom, and I know. Knows we all know about the others.
Now here’s where it gets tricky. I get up to get a new beer. Wife moves into the square and looks at the card. I was in the kitchen, watching through the pass-through, but they all thought I was at the fridge and wasn’t looking. So:
Me: Knows it’s Mustard. Knows Dad, Mom, and Wife know.
Dad: Knows it’s Mustard. Knows I, Mom, and Wife know.
Mom: Knows it’s Mustard. Knows Dad, I, and Wife know.
Wife: Knows it’s Mustard. Knows Dad, Mom, and I know.
Does everyone know it’s Mustard? Yes.
Does everyone know that everyone knows? Yes.
Does everyone know that everyone knows that everyone knows? NO! They all thought I was ignorant of my wife’s looking. But I wasn’t. I knew about it. If you were to ask Dad the second question above, he’d say “No, CS is unaware of Wife’s knowledge of Mustard.” But that’s false. I do know about her last move/peek, but they’re all unaware that I was still watching.
So as you all can see, “Everyone knows that everyone knows” is different that “everyone knows that everyone knows that everyone knows”.
Anyone want to go to a fourth layer? OK. Eventually, I revealed that I saw my Wife’s turn after all. So then “everyone knows x3” became true. But they don’t know that I told everyone in this thread about the story. So now:
Does everyone (including posters in this thread) know it’s Mustard? Yes.
Does everyone know x2? Yes.
Does everyone know x3? Yes.
Does everyone know x4? No! My parents and wife are unaware that I’ve brought you all up to speed. Only some of us (me and you guys) know that everyone knows that everyone knows that everyone knows the card is Colonel Mustard.
Yes, some could have, and some wondering if they have brown or chartreuse eyes…and none have any way of knowing what the other may be considering. Once the guru makes her announcement, though, they are all wondering the same thing, and know that they are wondering the same thing, and they also all know when this coordinated pondering began.
With everyone able to see either 99 or 100 pairs of blue eyes, then everyone knows there is a blue eyed person, and knows everyone knows this, and knows that everyone knows that everyone knows this…ad infinitum. They also have the same information about the brown eyes. I do not see any way that any islander could possibly think that there is an islander who doesn’t know about a blue eyed person…or think that another islander could think that…or think that another islander could think that another islander could think that…etc.
What I am not getting is how each layer of recursion might remove any islander from the pool of people who are not ignorant of the existence of a blue-eyed islander. (which is how I interpret the argument) As far as I can see, all the islanders stay in that pool at each level of recursion, so there is never a point at which the recursion can unwind due to the one possible ignorant person having been informed…which is how I read what people are saying the guru’s pronouncement allows.
The pronouncement makes no change to this, but it provides a reason to focus on the blue eyes, and also a known day on which each islander knows that every other islander started paying attention to this. What the guru did was not to add new facts, but to add a tiny amount of communication between the islanders: “Hey everyone, from this day forward, only consider that your eyes are blue or not blue.” Because they logicians, and they know this about each other, it also imparts a means of additional communication through simply waiting.
Looked at in this way, knowing that everyone heard that is sufficient and you don’t need to dig down all the layers of knowing.
The blue-eyed islanders know this: “I can see 99 pairs of blue eyes, so each of them can see either 98 or 99 thus if my eyes are blue, nobody can leave until the 100th day. If my eyes are not blue, all the blue eyes will leave on day 99 and I will know my eyes are not blue.”
Brown eyed islanders know this:“I can see 100 pairs of blue eyes, so each of them can see either 99 or 100 thus if my eyes are blue, nobody can leave until the 101st day. If my eyes are not blue, all the blue eyes will leave on day 100 and I will know my eyes are not blue.”
You can’t just start counting from, what may as well be then some, random event. How do you know some people aren’t counting backwards from 200? Or counting all the even numbers first and then the odds? Either are just as logical.
The best advice I have for starting to understand the puzzle is finding the smallest group of blue eyed people for which you think they’d leave the island. (Surely one would leave on the first night. It’s pretty easy to see why two would leave on the second…) Then put yourself on the island. You’re watching that number of blue eyed people (3, say). And they don’t leave on the night they’re supposed to. In that case, why wouldn’t you think you have blue eyes?
If that Guru said only that, nobody would ever leave. The simplest case, of one blue-eyed islander, breaks down:
A: “Hmm, I never thought of that, but now that you mention it, I wonder if I have blue eyes? I can’t see anybody else with blue eyes, but the Guru didn’t say that anyone necessarily did, so I don’t know.”
Next day, A does not leave, because he doesn’t know that he has blue eyes.
Likewise, the cases with higher numbers of islanders no longer work.
The problem states no such thing. For all we know, the islanders want to stay on the island and what you call “escape”, they call exile; what you call “be saved”, they call “be punished”.
Not that any of that’s relevant to the puzzle. I’m just amused that more than one person has assumed the islanders hate their paradise and want to come live in Manhattan or something.
False. As we’ve said before, t’s not ad infinitum, it’s ad 99. The islanders don’t know that everyone is able to see 99 or 100. You only know that because we postulated it. The islanders have no way of knowing that everyone knows that.
Missed the edit window, but the above got me thinking.
If they were allowed to say, “We’re going to pick today and start counting nights. When we get to the correct number of nights for blue eyed people they can leave.” Other than waiting, they are still not allowed to tell anyone about their eye color. They were told when arriving on the island somebody had blue eyes.
Additionally, they would like to leave as quickly as possible so they want to start the count as close to the actual number of blue eyed people as possible. But because they cannot tell each other anything about anybody’s eye color, they must arrive at that number algorithmically (example: The count starts at the closest multiple of 5 lower than the number of blue eyed people you see. So if you 99, start the count at 95).
For reasons similar to why the guru is needed to start the count in the original puzzle, I don’t believe there’s any way for them to devise a system where they leave any quicker than starting the count at 0.
you can’t really explain an idea that’s why it hasn’t been explained…
implied task?
“I see at least 1 person with blue eyes”
that statement is matter, it’s fact, it’s data, useless without the idea “implied task”
the implied task is what turns it into information for me as an islander
if you don’t understand after this whole thread just think implied task.
when your dad tells you son the dog just took a crap on the couch… he said the dog took a crap on the couch
but from that sentence he said you know that you need to get to clean it up… he didn’t tell you to clean it though
Oh! I’ve been lurking in this thread in confusion for a little bit and I think this post just gave me a lightbulb moment. The islander’s thought should continue:
…Right?
Why did I make that statement? Not because I doubt the inductive process, which certainly does work, but just because it’s interesting that a statement, which on the face of it provides no information to anyone, can be used to draw a correct conclusion.
I agree that the 100 brown-eyed people are left stuck on the island, but let’s say there were more of them, say 120. Each of them finds out on the 100th day that their eyes are not blue. The thought process of each one should go like this:
- My eyes are not blue.
- All the other 119 people realize now that their eyes are not blue.
- 100 days ago the guru could have said at least one person has brown eyes.
- Everyone else knows this too, because everyone can see at least 118 with brown eyes, and possibly one (me) with some other color.
- Therefore, 19 days from now all of them will leave, and I’ll be stuck here, or 20 days from now, we’ll all leave together.
Reading back some of the posts that were confusing me before, I think they were basically saying the same thing in a slightly different way. I think I got it now though.
in your scenario 100 days ago the guru could have said pink eyes. he could have said green just like in the real blue eyed logic puzzle the author could have said brown instead of blue. the “could” matters, which one you say determines the answer the scenario. so until you come up with a specific scenario we can’t come up with a specific count of people that leave.
now if you take out the “could” of your scenario and replace it with “did”.
then what would really happen is that out of those 119 people left 19 days from then 18 of those islanders (the perfect logicains) would have figured it out using the logic in the thread and rest left after that didn’t leave because of one of 2 reasons
-
they weren’t perfect logicians using this logic
-
or they were perfect logicians and used the logic and stayed because the logic led them to believe their eyes weren’t brown
Oh ok, yes I see. I got it for if there were only 2 blue eyed islanders, that the remaining guy wouldn’t be able to leave, but didn’t follow the logic through correctly for 3, and came to a poor conclusion. I see now that in the case of 3 blue eyed islanders it would go something like this…
Three blue eyed islanders, one struck dead, leaving person A and person B. A imagines that he has brown eyes and thinks, “perhaps B sees me with brown eyes, and B himself may believe that he has brown eyes, so maybe he thinks C was the only blue eyed person on the island, who is now dead, therefore B will have no way of knowing when to leave the island.”
Very cool!!!
It’s so weird that the day makes such a difference.
Doh! OK, I am getting it. Everyone else is in each individual’s initial sample space, but it is the knowing what is known that reduces the sample space of the hypothetical lower level quotes.
The point is that everyone knows the day the puzzle is solved is based on how many blue-eyed people there are. So if it gets solved on Day X, they know there are X amount of blue-eyed people. If it isn’t solved on Day X (because people don’t leave) then everyone knows there are more than X amount of blue-eyed people (specifically X+1).
OK, to reiterate, my scenario was 100 islanders with blue eyes, and 120 with brown eyes, and the guru still saying that at least one islander had blue eyes. So the guru could not have said pink eyes since that would have been false. He could have said green if you count him among the islanders, but that would imply he knew his own eye color and should be gone already, again if you count him among the islanders. He didn’t say brown in my scenario, so there’s no question of substituting “could have” with “did”. Even so, the islanders with blue eyes would leave after 100 days, and with brown eyes after 120, assuming they all use flawless logic and trust each other to do the same. It doesn’t work though if there are only 100 with brown eyes.