What do you mean? Are you talking about imprecision in the details surrounding how the puzzle is presented? If so, we could present it with all humans removed entirely:
-There are N perfect AIs. Each one has absolutely perfect intelligence and deduction capability.
-Each AI has a number assigned to it: 1 through 10. No AI knows what its own number is, but each AI knows the number of every other AI
-Once every great while, so that there’s always plenty of time for thinking between questions, every AI is simultaneously and secretly asked the question “do you know what your number is?”. The AI can respond with “no”, or a number. Once all the AIs have responded, then it will be publicly revealed to all AIs which AIs responded with a number, and what that number was
-AIs can not communicate with each other
-AIs have two goals. Most importantly, do NOT ever answer the question wrong. Second most importantly, if possible, answer the question right.
-All AIs are aware of all of the above, are aware that all AIs are aware of all the above, to infinitum
There are 100 AIs. Twelve of them are "3"s. At some point, with plenty of time to think after this statement is made, an external source, which all the AIs trust, tells all the AIs “there is at least one ‘3’”. All the AIs believe this, and all the AIs know that all the AIs were told this and believe this.
What happens?
Or do you mean something else by “interpretation”?
I don’t think it’s possible for a perfect logician to look at 99 green-eyed people who can all see each other and come to the conclusion that there might be someone who thinks there are zero green-eyed people, or any number less than 98. There is no logic that could lead to that conclusion.
I think it works more like this.
I know there are either 99 or 100 green eyed people. I also know every other person on the island knows there are at least 98 green eyed people. Because I can see them looking at each other. I also know that every other person knows that every other person knows that there are at least 98 green-eyed people.
In theory, if there were 1 green-eyed person, they would leave on day 1 when hearing the interloper’s announcement. But we all know that’s not going to happen, it’s impossible given what we can see. If there were 2, they’d leave day 2, etc. So the only question remaining is: is everyone going to leave on day 99? If so, I know I have non-green eyes. If not, we all have green eyes.
I’m not sure the interloper needs to share any information at all, except perhaps “Today is day 1”.
Yes, for highly mixed eye color cases, the solution becomes more difficult. But my point would remain true. Whatever mechanism is necessary to make the distinction will minimize the number of steps to the absolute minimum required. If we all know that there’s at least 99 people with green eyes (in the single-color case), and we all know that the simple solution is to iterate through the options until we count to 99 or 100, then it makes the most sense to shortcut the solution and simply act out the last two days. The first 98 are pointless.
If we can take it that we’re all infinitely smart and infinitely desirous of escape, we can assume that we’ve all arrived at the same solution, and we can assume that we will want to reduce that solution to its fastest variant.
A smart and logical being wouldn’t send uncompressed data over a wire, if he doesn’t need to - because entropy is wasteful - and he wouldn’t go through a needless waiting phase to decide between 2 numbers. Regardless of the number of eye colors, that would remain true. The delay until leaving might be longer, but redundancy will be removed from the system and the steps simplified to the optimal.
However, he would also try to give the most information to his compatriots that he possibly could get away with.
Assuming that the other perfect logician’s will just skip over the first 98 days seems like a leap to me. That’s a good guess, but it involves more than logic. Being desirous of an early escape is not necessarily logical.
:dubious: If we assume that, then we would need to assume that escape itself might not be desirable, in which case none of the strategies (including the “correct” one) would work, as they all rely on the concept that:
We all want to leave.
We’re all completely logical.
For the official solution to work, they all have to assume both of these facts.
If there’s any doubt that a guy wants to leave at any particular date, then when he is still there the next day, you have no new information. He might be there because of his eye color, or he might be there because he really likes the chow on Tuesdays.
If there’s any doubt that a guy might not be completely logical - he just didn’t leave last night because he’s a wimp, or because he might not have realised the significance of the statement - then, again, the fact that he did not leave last night tells me nothing.
If we have to assume that some of the people might like being there and might not be completely logical, then the initial statement and the passage of time are both insufficient to tell us anything.
It’s possible I’m missing something here, but I think the problem is that while person A might know that the number of green eyed people is either 99 or 100, he doesn’t know that everyone else knows that. As far as he knows, everyone else knows either 99/100 or 98/99.
So… let’s take smaller sample. There are 7 total people. 4 of them have green eyes. So 3 people think that it’s either 4/5 and 4 people think that it’s 3/4. What precisely happens in your solution?
And is there a triggering piece of information at all? Because I don’t see how any solution could ever work without a triggering piece of information.
No one in this thread ever alleged that that was possible. Not a single islander thinks anyone else sees 97 green eyes. They do, however, think it’s possible that their friends are mistakenly thinking it a possibility.
It works like this: Jim has the hots for Suzie. I know this because he told me so. Thing is, I don’t know if Jim told our mutual friend Steve about his love interest. And Jim swore me to secrecy, so I can’t even talk to Steve about it!
Thus, I think it’s a possibility that Steve is oblivious to our friend’s swooning. Steve could quite possibly think that Jim hates Suzie and finds her annoying! I know that Steve is wrong about this.
But I’m actually a horrible secret-keeper. I went to Steve and was like “DID YOU HEAR ABOUT JIM?!?!” and he told me that Jim told him yesterday, but swore him to secrecy as well! So Steve and I spent the whole bus ride gossiping about it.
This all came to a head one day when blabber-mouth Betsy stood up in the cafeteria and shouted “Jim and Suzie, sittin’ in a tree! K-I-S-S-I-N-G!”
So the cool thing is this:
I knew the secret.
Jim knew, cuz it’s his secret.
Steve knew cuz Jim told him.
Thus, everyone knew.
Jim knew that Steve knew, because he told Steve.
Jim knew that I knew because he told me.
I knew that Jim knew, cuz it’s his secret.
I knew that Steve knew because I spilled the beans.
Steve knew that Jim knew, cuz it’s his secret.
Steve knew that I knew because I spilled the beans.
Thus, everyone knew that everyone knew.
But Jim didn’t know I betrayed his secret. It’s cool, cuz I didn’t actually spread any new information. Steve already knew before I told him. But still, I broke my promise to keep it between Jim and I. So did Steve.
Since Jim didn’t know that I knew that Steve knew, it is FALSE that everyone knew that everyone knew that everyone knew.
But you know what? Betsy opened her big, fat mouth. She told everyone something everyone already knew, that Jim and Suzie were hookin’ up under the bleachers. But still, Jim knew the jig was up. His love life was now on public display. After that day… everyone knew that everyone knew that everyone knew.
All true.
False. I know that every person knows there are at a minimum 98 green-eyed people. But I can’t prove that they all think the same thing. For all I know, some people think that some people think that some people think there are 97, minimum.
Remember how Jim thought I’d kept his secret? He thought I was ignorant of Steve’s brain, even though I wasn’t? That Jim thought I thought Steve was ignorant? Same thing here. Jim was wrong to think that, and any given islander is wrong to think another islander is that ignorant. Jim had no way to know I was more knowledgeable about Steve than Jim assessed I was. Alice has no way to know Bob assessed Charlie to have a higher count. Alice thought Bob could assign Charlie a lower “minimum” than he did, but Alice had no way to know that.
That doesn’t have the same effect as Betsy announcing Jim’s business to the whole school. Only because of this did Jim go “Well great, now Chessic Sense knows Steve is in the secret-keeping-club, and now they’ll gossip about it together.”
Alice is mistaken about others’ knowledge. Only by saying “someone has green eyes” can this be corrected at the “deepest” level. It forces Alice to say “now it is no longer possible for someone to be under erroneous assumptions.”
We don’t, though. I, a prisoner, acknowledge that this may be the case, but I cannot prove it. I see 99 greens, but I have no way of knowing if anyone else sees 99 greens. They might see 98, since they might not be counting me.
If Bob made a list of green-eyes, did he put me on it? I don’t know. If he guessed what Charlie’s list looks like, is Bob on it? He couldn’t know. He knows if I’m on it, but I haven’t a clue. Maybe Bob knows I’m on Charlie’s list. Maybe Bob knows I’m definitely not. I know Bob has the answer, but I can’t know what it is.
Meanwhile, I know Bob’s on Charlie’s list, but Bob doesn’t know that. So it’s possible that I’m not on Charlie’s list, and it’s definite that Bob doesn’t know for sure that he’s on Charlie’s list.
In addition, I think it’s obvious Charlie’s not going to be on his own list. So Bob is definitely going to not count Charlie, even though we both know Charlie’s a green eye.
So Bob might consider it a fact that there are only 97 greens on Charlie’s list. Of the 99 greens I see, Bob and Charlie are not on the list, at least according to Bob’s thought.
So how many days should we skip ahead? 97? 98? 99?! My God, what if there are 99 greens, I’m brown, and they’re all plotting to leave without me! I’ll be stuck here!!
Your day-skipping idea only works if he all agree on how many days to skip, but that’s not possible without already knowing the “green count.”
For what it’s worth, this is an excellent example for applying the technique of mathematical induction (which, despite the name, is a rigorous deductive technique). The statement we wish to prove is “For all values of n, if there are n green-eyed people, they will all leave on night n”.
First, we prove that the statement is true for n = 1. That’s easy. If there’s only one green-eyed prisoner, then he didn’t know that there was any green-eyed prisoner, and so the statement is direct new information for him, and he leaves immediately.
Next, we prove that if the statement is true for n = m, then it’s also true for n = m+1. This is also fairly straightforward: If there are m + 1 green-eyed prisoners, then each of them knows that the number is either m or m + 1 (because each sees m others, and doesn’t know whether he is or not). But the statement for n = m states that if n = m, then a bunch of prisoners will leave on night m. If that statement is true, but nobody leaves on night m, then everyone knows that n ≠ m, and hence knows that n = m+1. And since everyone now knows that, they all leave on night m+1, just like the statement says.
So the statement is true for 1, and if it’s true for m, it’s true for m + 1. And thus, it’s true for all m.
It’s impossible to come up with a number that everyone knows that everyone can agree on, though (unless you count zero). I’d be curious to read an attempt to come up with such a number, though. Assume there are 100 green-eyed people. What number of days can they skip and how do you prove that everyone can safely pick that number?
To pre-rebut a bit: If you were one of the islanders, the number of days you’d be able to skip would have to be independent of whether you had green eyes or not. But for another islander (who knows your eye color but not their own), the color of your eyes would move their count up or down. How can you factor that in while also keeping in mind that this other islander doesn’t know their own eye color and must factor that in to his own attempt to come up with a maximum number of days to skip?
Also, what’s the difference between what you can say about the puzzle if you imagine yourself as being one of the islanders when there are 100 green-eyed people on the island and you can see 99 other people with green eyes and what you can say if you imagine yourself as one of the islanders when there are only 99 green-eyed people on the island and you can see all 99 other green-eyed people?
If I can see 99 green eyed people, then the worst case scenario is that I’m brown eyed and the guy who is looking at me doesn’t know his own eye color. From his vantage, he will think that it is plausible for there to be 98 non-green eyed people.
There is no case to be made for anyone thinking that there are only 97 green eyed persons, nor any level below that.
Everyone else knows this, and thus 97 days can be skipped. We need to know whether there are 98, 99, or 100 green eyed people.
So all the brown eyed people skip 97 days and all the greens skip 96. The greens wait 3 more days to leave, but the browns think it’s day 100, so they must be the 100th green, so they all try to leave at the same time. All the browns are killed.
If you’re just trying to get the greens to escape, you could’ve done that any prior day, with or without a sage/guru/soothsayer/whatever.
… and if you do think the answer to that last Q is 98, then which person (out of you, Colin and Bill) is Colin counting as an assuredly green eyed person?
I was going to just skip out, since this ^%*^%*&$#& thing kept me up 3 1/2 hours this morning (thankfully, my work schedule today is light). But after some more of the more recent posts, I can’t.
I think you are all wrong, and all this nonsense such as in post #89 (where the premises are subtly changed) are the equivalent of arguing about the number of angels doing the fandango on top of a mushroom. I think you all have been played as fools, and whoever originally designed the puzzle undoubtedly Googles his puzzle periodically to see what new discussions of it have cropped up, and he sits there chuckling to himself. Yes, at the weiners like me who KNOW there is a flaw in there somewhere, but who cannot quite identify it for the life of us. But most of all at those who swallowed his explanation (bait) hook line and sinker and continue arguing it unto the end of time.
The issue to me is directly analogous to privileged reference frames in physics. The 99/100 day thing I totally buy, note. It’s the “Hypothetical Countdown to Zero” thing that I have an issue with.
You say, “Well, think of what Bob thinks, then what of Bob thinks of what Suzie thinks”, ad nauseum.
My issue with that is, what logic in the world compels you to focus on just ONE person at a time like that? Why should I privilege Bob, or anyone else, any single individual, by focusing on him (and his thoughts) first, then on the next person, and the next, etc. etc. etc.? Why would speculating on their minds matter, at all, vs. the simple physical brute facts?
Wouldn’t you instead just focus on them all (the blues) as a group, simultaneously, and simply say, “OK, I see 99, but they might see 99, but perhaps only 98, full stop?”
Every time someone tries to point this out, we get yet another round of the angels boogieing, and nobody actually comes out and says why we have to do the 99-98-97-96 chaa chaa. Recall we are talking about “perfect logicians” here-so why wouldn’t they all approach it in the simple manner I outlined in the previous sentence? I think this speculating about other people’s minds all the way down to zero in this way is utterly beside the point, and makes a simple timewasting children’s puzzle into this preposterous monstrosity.
I know I will just get yet another round of the “Oh no you’re wrongs” and lectures and additional pointless, flawed, irrelevant and too-subtle analogies that will likely go whooshing right over my head again. Fine. I simply think you all have been butts of some sly joke.
I was originally going to just post this originally, decided to keep it in.
Is there a difference between a "perfect logician" and a "perfect thinker"? That was my original motivation with reffing the liar's paradox.
If our Perfect Logicians are told the Liar's Paradox, and then asked to resolve it, I think they would simply sit there going back and forth ("He lied, he didn't lie...") until unconsciousness or death claims them. A Perfect Thinker however would be able to jump out of the deadlocked system, to a higher perspective, as it were, and declare it unsolvable. I suspect something similar is going on here.
Because if someone sees n people with green eyes, each of those n people can see either n or n-1 people with green eyes. You relied on this fact to go from 99 to 98. So if it were the case that the group saw only 98 people with green eyes, they would think that it’s possible for each of those 98 people to only see 97 people with green eyes.
Repeating what I brought up before, there’s no difference between the case where there are 100 people with green eyes, of which I am one, and the case where there are 99 green eyed people and I have brown eyes.
Can someone please walk me through the outcome? Note that I am not trying to fight the hypothetical and I do understand the math tree as stated above but I’m having trouble applying it to this hypothetical prison and prisoners.
I understand Chessic Sense’s interviewer solution but aside from stepping out of the hypothetical, it doesn’t help the first interviewee once you’ve moved on (and again, the original puzzle makes no provision for a third party interviewer).
So given the stated rules:
“[T]here are no reflective surfaces or cameras etc on the island. Communication between prisoners is strictly forbidden, hence, and importantly, no one has ever learned their own eye color.”
What happens next in Max’s above example? Hearing the news that (at least) one of them has green eyes, they both wordlessly stand and walk out? The logic doesn’t hold hold for me.