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The Logic Puzzle

I have found a catch in the solution of this logic puzzle. Provided that you are not allowed to give the prisoners any new information, the statement "at least 99 of you have green eyes" does give them new information. To give 'new information' can be defined as when the statement you say makes the prisoners immediately know about something which they previously did not know about. To prove my point let's shrink the puzzle down to just 2 prisoners Prisoner A and Prisoner B. Imagine the point of view of prisoner A, he would have made a list of what he knows and what he doesn't know.

Point that A knows and is completely certain of- Prisoner B he sees have green eyes (i.e. at least one of them have green eyes)

Point that A doesn't know and is completely uncertain of - Whether prisoner B sees green eyes (i.e. uncertain about B knows that at least one of them have or not) this is because if in case A have non green eyes B would not know that at least one of them green eyes.

Now when you tell them at least one of them have green eyes A will be immediately sure that B knows at least one of them have green eyes as you said the same. Hence A will gain new information immediately after you gave your statement as before your statement he had no information about whether B knows at least one of them gave green eyes. This disproves the point at 3:35 (Timestamp in the YouTube video) that the new information was not contained in your statement itself as it indeed did contain new information. Looking forward for someone to support/oppose my argument.

Edit-I just realised I have been opposing the solution by supporting the very thing the solution is trying to suggest. If you listen carefully the video from 3:35 to 3:40, they mention that "The new information was not contained in your statement itself but in telling it to everyone". If you say atleast one of you have green eyes to everyone privately there's no new information gained by anyone as the prisoner is still not sure of whether the other prisoner knows atleast one of them have green eyes. He will be sure about it only when he knows the same statment was made to the fellow prisoner. Which is the same thing they are mentioning in the solution. Also one of the answer saying that when they were publically told some statement adds new information to them is correct, as the fact comprehended by every prisoner that 'the statment was told to everyone in the same room/auditorium and is know known by everyone' forms the basis of the inductive reasoning performed here.

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  • $\begingroup$ The "one strange rule" - "Any prisoner can approach the guard at night and ask to leave; if they have green eyes, they'll be released" must be public and known to be public (even though this is not mentioned in the puzzle) - because, if it is NOT public, then the proposed solution fails. As all the prisoners already have access to this knowledge (and know it is public) and, as they know there are at least 1 prisoner with green eyes, is the trip to the island a complete waste of time? $\endgroup$
    – Konchog
    Commented Jul 9, 2022 at 11:39
  • $\begingroup$ Actually no, there needs somone to repeat the information that atleast 1(or any ≤99) prisoners has green eyes in the presence of every prisoner such that every prisoner knows that statement is known to every other prisoner. Hence your visit to the island is a must $\endgroup$ Commented Jul 10, 2022 at 12:16
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    $\begingroup$ sorry - can you explain / expand your rationale? Everyone knows that everyone else has green eyes, right? This is self-evidential. In fact all prisoners already know that all of the other 99 prisoners have green eyes. They also all know that the one strange rule is public. Therefore everyone know that everyone else can see either 98 or 99 pairs of green eyes. The visitor doesn't add anything to what they already know: The information is already public. How does it not follow that the prisoners are not trapped for longer than (count of prisoner) days. $\endgroup$
    – Konchog
    Commented Jul 10, 2022 at 13:18
  • $\begingroup$ A textual summary of the original puzzle needs to be included in this question. As it stands now, anyone interested in investigating this is first forced to spend 5 minutes watching a video and recording the relevant information, and many of them (e.g. myself) aren't going to bother. Even worse, a year from now that video might no longer exist, and this question will then have become useless for everyone. $\endgroup$ Commented Jul 12, 2022 at 13:53

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Nice catch, +1!

There is one even more basic catch I found while thinking about your interpretation.

Even saying "at least ZERO of you have green eyes" gives new information!

In your interpretation, you said every one has a list of known facts.
The moment the statement is made, this list also will be updated with "We were all publicly told some statement", which is new information!

What this means is these types of puzzles have lots of implicit assumptions, about, e.g., what a "statement" is, what "knowledge" is, when that knowledge is "new", etc. These will not be well-defined! When you analyze it, you will find contradictions. The solution to the puzzle would involve some trick, but when you analyze it, even there you will find contradictions.

In this particular case, the trick is to give new information to the prisoners but not let the dictator know about that. Essentially, cheating the dictator, but not cheating "logic"!

When you say "1+1=2" to all the prisoners, nobody is going to escape. It is only when you give some new information about the eyes to the prisoners that escape is possible. Other wise it is a logical contradiction: give new information without giving new information! Not possible!

In conclusion:

  1. No matter what you say, you are giving new information, that "the public announcement was made", and all prisoners will get this new information &
  2. You have to give "new information about the eyes", but dictator must be stupid not to know that "new information about the eyes" has been given, and prisoners are smart to know that and then escape.
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    $\begingroup$ Yes you are correct that new information is gained by the prisoners by the event 'All prisoners were publically announced a statement' as it forms the sole of the solution. Kindly refer my edit to my question as I opposed the solution without properly listening to it, they also mean the same thing lol. $\endgroup$ Commented Jul 9, 2022 at 8:50
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Doesn't saying "at least 99 of you have green eyes" give new information?

It depends on how you define "new".

If you define "new" as something you can privately tell every person and at least one person didn't already know it, then No. This is how the dictator is tricked. What each person doesn't know, is that everyone else also knows this. This is precisely why declaring it publicly makes it "common knowledge". Once everyone knows that everyone else also knows what they know, the clock starts for them all to be able to deduce when they can safely leave.

To your point, if you define "new" as literally any new information, then Yes, they were now able to leave because they learned that everyone can start keeping track now. It's much easier to see that with "at least 1 of 2 people" or "at least 99 of 100 people", but since they all knew the information privately, it's not "new" information in the way the dictator was thinking. This is why it was suggested to say "At least 1 of you has green eyes." when there are many people, since, obviously everyone knows that already. I suppose the dictator was not a perfect logician himself.

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