The difficulty with “perfect logician” problems is that logic and linguistics do not make a perfect match. Languages differ in their efficiency, and a statement that is unambiguous in one language may not necessarily be in another. What, then, defines “a single statement”? That would be a linguistic definition, not a logical one.
As an illustration, in English, the word “cousins” are the same people, to me, but to my father, they are “nieces and nephews”. In a group of people in a room, my father can, in a single statement, narrow down one of them as his niece, but I have no language with which to do so, as they are all my cousins.
So it would be possible to construct a problem requiring “a single statement” that in another language, or even another person speaking the same language, cannot be covered in a single statement.
This brings to mind the one about the two tribes of natives, both perfect logicians, but one tribe always tells the truth and the other lies, and you don’t know how to distinguish them. If you come to a fork in the road with a native standing there, how can you ask with a single question which way is the village? The answer is “If I were to ask you if this is the way to the village, would you say yes?” But this works only in English, using the principle of a double negative makes a positive. But might fail in Spanish, where the double negative remains a grammatical negative (No tengo nada).
You could also ask “Do you know that they are serving free beer in the village?” and then ignore the answer and follow the native.
No, there is an algorithm to determine if you have the stated colored eyes or not. In other words, once the guru says I see someone with green eyes you will be able to deduce whether or not you have green eyes. If you don’t have green eyes, you still don’t know what color they are.
People are making their explanation a little bit too complicated.
The first thing you need to realize here is that your statement is factually incorrect. There is absolutely new information that was added to the system.
Your problem is that you’re looking for this new information in the wrong place.
The person spoke a sentence, and what you’re trying to do is parse the literal meaning of the sentence. You look at the literal message, decide that the literal message of the sentence is not a revelation for any of the concerned parties – which is totally true, you are absolutely correct about that. But that’s the reason why you miss the actual underlying information that was really there. We need to start much simpler.
On what day did the visitor to the prison speak?
When you say that “no information” was added, you’re ignoring this little bit of info. And this is absolutely information. You can’t deny that. Every single prisoner knows what day the announcement was made. That is a piece of knowledge that they did not have the day before the announcement was made. So when you say there is no new information, you are incorrect. The question is not whether there is no information. There is. That is already proven beyond any doubt. The question now is whether the new information was sufficient for them to make a clever logical deduction. Was it?
That’s the next step. Is yet more information contained in the message, beyond the literal meaning of the message? Yes, absolutely there is. We’ll get to that.
But before we get to that step, it’s best if you acknowledge this first step. You need to start looking for information outside its most obvious place.
I don’t see why everyone has to wait 100 days. If you have 4 people, wouldn’t you know by the second or third day? Why does 4 days have to pass before you’re sure?
Because you need that part of the information. Once you know all four people see three people with green eyes, that means all of them must have green eyes.
But how does that translate into days? It’s not like only one person can go a day. If I see three people with green eyes and nobody leaves after the first day then i know that:
A. I’m the only person with off-color eyes and that is confusing the situation.
or
B. We all have green eyes.
That doesn’t change as time passes. Those are the only choices.
No, each day provides more information. If there’s only one person with green eyes, he won’t see anyone else and leave as soon as he can. If there’s two with green eyes, they’ll both see one other person with green eyes and when he doesn’t leave on the first night, they know they must be the other green-eyed person. And if you see two people with green eyes and they both don’t leave after the second night, you know you’re the third one. Only if everyone sees three green-eyed people will everybody still be there on day 4.
If there are 3 people on the island and you know for a fact that one has green eyes:
You see 2 people with green eyes. That leaves two conclusions: A. Everyone has green eyes or everyone has green eyes but you.
Looking at their thought processes vicariously, I’d imagine that they would either see 2 green eyed people and be going through the same process as myself, or they’d see one green-eyed, one blue-eyed. Giving them the choice of A: Is Bob the only green eyed person? Or do I have green eyes as well?
On day 1, nobody leaves because they’re all wondering if they’re the odd man out.
On day 2, nothing has changed. They’re all left with the same set of facts. I see two people with green eyes. I’m still wondering if I’m the odd-man out and they’re not leaving because they see me with blue eyes or if they see themselves as potentially the odd-man out. Nobody leaves.
Day 3, same set of facts.
The time passing doesn’t change anyone’s knowledge of the situation.
It tells you what the other people see. If you see two green-eyed people and have brown eyes yourself, they’d be able to deduce as I demonstrated above that they both have green eyes and both leave on the second night. The fact that they are still there on day 3 means they can’t be sure, hence they must both see the same as you.
Not knowing your own eye color – and thinking, gosh, maybe it’s blue – you looked at Guy #2, who (a) didn’t know his own eye color, and (b) was, as far as you know, looking at one green-eyed guy plus your blue-eyed self.
That green-eyed guy, Guy #3, would’ve left if that had been the case; he’d have seen two blue-eyed guys, and known himself to be the one green-eyed guy. But he didn’t leave, which means you or Guy #2 or both have green eyes.
You knew that already, because you could see Guy #2’s eyes. Guy #2 knew it already, because he could see your eyes. But you didn’t know that Guy #2 knew that; until Guy #3 stays put, you had to wonder whether Guy #2 was looking at a blue-eyed guy and wondering whether his own eyes were also blue.
Time doesn’t have anything to do with this. There’s no “start the algorith today” message of importance here. The point is simply this.
Before the guru speaks, everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that there is a green-eyed person on the island,
but unfortunately, not everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that there is a green-eyed person on the island.
However, after the visitor speaks, then everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that everyone knows that there is a green-eyed person on the island.
Got it? No? That’s because it’s impossible to comprehend that sentence. Nevertheless, it’s the answer.
The ninety-nine green-eyed prisoners have never really liked obnoxious Donald, with his ugly blue eyes. By the 99th day, all ninety-nine know they’ve got green eyes, but each decides to wait one more day — what’s an extra 24 hours in jail compared with a chance to perform the perfect murder? On the 100th day, all the prisoners leave … Donald is the only one killed by the guard.
[QUOTE=Grumman]
A perfect logician would not assume that 99 people they’ve never communicated with are perfect logicians.
[/QUOTE]
You changed my mind Grumman, as far as I can see, you are right. The original question locks everyone in prison forever; exactly because they are all perfect logicians who would never stand upon such a faulty premise, or jump the tenuous conclusion that everyone else they see (who they never communicate with in any way) are also perfect logicians. They know that the sorting algorithm only works if everyone is on board and starts counting at the same time.
They know that having any grouping of people at the same level of intellect is vanishingly small, and as such they know to treat everyone else as a wild-card who will inevitably come up with different solutions at different rates, and who might even agree with the “perfectly logical answer” in time but who did not realize the importance on the first day to start counting. Even in this thread, with time to think out and type out our best responses/questions everyone here is still at varying degrees of belief of the “logical” answer that the guy from the XKCD comic came up with.
The only way I can see everyone getting out is if they are all general people who don’t put too much thought into things, and have heard of a similar riddle (thought to be common in that society) before their incarceration. They would smirk to themselves, never really giving it much thought, as they already “knew” the answer, and assume (not very logical!) that everyone else knew the answer as well. Given that the prisoners were brought in piecemeal and not all dumped in on the same day (as that would lead some to jump ahead and start their count on the first day of incarceration to limit their time spent in prison), on the day of announcement everyone would then mindlessly start their sorting algorithm without ever knowing how or why it worked.
It is the difference between assuming that every stranger in a room who you can’t communicate with are equally able to derive advanced Physics formulas from earlier math principles at the same rate as any other in the room, and appreciate how the variables interact to create a new useful operation (like a perfect logician?) vs assuming a room of people who you can’t communicate with have variable skills, and some might not even be able to memorize the necessary final formula and plug in the variables to complete the task in unison with all the others (which is necessary for everyone’s survival).
Or: it is far closer to “perfect logic” to extrapolate from the 99/99 visible green-eyed prisoners that there are probably 100/100 total green-eyed prisoners, than it is to extrapolate from the 1/1 perfect logician prisoner that there are probably 100/100 perfect logician prisoners.
The puzzle does not require that the entire prison/island/whatever be filled with same eyed-people. There could be 100 green eyes out of 740 people, and the answer is the same.
Again, there is no “counting” and timing is not a part of the solution. You could have someone announce in front of everyone “We will start ‘counting’ any green-eyed people tonight!” and the solution would not work. You could say “If you are a perfect logician, start your logical deduction tonight, and the rest of us will do the same” and the solution would not work. You could have someone say to only 99% of the prisoners “I see someone with green eyes” and the solution would not work.
Ask Guy 1 what the least number of green-eyed prisoners could be, and he’d say 99. Ask Guy 2 what the lowest number Guy 1 could’ve said, and he’d answer “98.” Ask Guy 3 what the lowest number Guy 2 couldn’t said is, and he’d respond “97.” Continue down the line, then ask Guy 100 how Guy 99 answered, and he’d say “0. Guy 99 might’ve said 0.”
I’ma write that again so everyone gets it: Ask Guy 100 how Guy 99 answered, and he’d say 0. If you do not understand why this is so, you do not understand the solution.
No strategy of “Let’s coordinate this!” will change this, ever, unless someone speaks of eye color to the entire population of green-eyes.
I was actually going to chime in with asking how the population of the prison came about. Your declaration above doesn’t seem like a given to me, and to be honest it would appear that this isn’t an idle question. If they did come in piecemeal, then there are differing levels of information for each prisoner, depending on when they came in (“OK, a busload of 20 blue guys came in today, first blue eyed guys I’ve seen too…”). If however they all came in on the same day, I see no reason why the algorithm couldn’t commence on that very day, once the rules were announced by say the warden.
Then I must pronounce myself as dense, because EVERYONE SEES AT LEAST ONE GREEN EYED PERSON! :eek::smack: This “I could conceive of one of the other prisoners as actually seeing zero when he must actually be seeing at least one” horsehockey seems to depend on some arcane, paradoxical and twisted use of what it means to possess information. Where mere primitive empiricism doesn’t matter. There seems to be some sort of hidden assumption here that nobody who says they grasp the solution will just come out and say and be 100% clear and nonparadoxical about.
:smack: This “I could conceive of one of the other prisoners as actually seeing zero when he must actually be seeing at least one” horsehockey seems to depend on some arcane, paradoxical and twisted use of what it means to possess information. Where mere primitive empiricism doesn’t matter. There seems to be some sort of hidden assumption here that nobody who says they grasp the solution will just come out and say and be 100% clear and nonparadoxical about.