## My write-up of the "Blue Eyes" solution (SPOILER A

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Potatoberg
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### Re: My write-up of the "Blue Eyes" solution (SPOILER A

phlip wrote:
Potatoberg wrote:You being able to comprehend this is proof that this is English.

But how do I know I'm comprehending you? How do I know with mathematical certainty that you're not using a word I've never seen before that happens to be spelled the same as "comprehend" but actually means something totally different? How do I know you're not testing out a new conlang that has words that happen to be spelled the same as English words, but all of them mean different things? You could be telling us all about how cute your new pet cat is, and we just think you're talking about standards of proof, because we're making the assumption that you're using English. I just can't rule out that option with mathematical certainty.

You do realize you're arguing that we don't have proof for an axiom, right? I doubt it because you're asking for mathematical proof of an axiom. Can you give me mathematical proof that 1=1? You can't, it's an axiom.
The guru is not color blind is not an axiom.

rmsgrey
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### Re: My write-up of the "Blue Eyes" solution (SPOILER A

Potatoberg wrote:
phlip wrote:
Potatoberg wrote:You being able to comprehend this is proof that this is English.

But how do I know I'm comprehending you? How do I know with mathematical certainty that you're not using a word I've never seen before that happens to be spelled the same as "comprehend" but actually means something totally different? How do I know you're not testing out a new conlang that has words that happen to be spelled the same as English words, but all of them mean different things? You could be telling us all about how cute your new pet cat is, and we just think you're talking about standards of proof, because we're making the assumption that you're using English. I just can't rule out that option with mathematical certainty.

You do realize you're arguing that we don't have proof for an axiom, right? I doubt it because you're asking for mathematical proof of an axiom. Can you give me mathematical proof that 1=1? You can't, it's an axiom.
The guru is not color blind is not an axiom.

If you choose to take "The guru speaks English" as an axiom, but not "the guru's statement is unquestionable" then that's a perfectly valid, and self-consistent position. It just happens to (with appropriate other axioms) generate a contradiction within the problem statement.

It is also true that it is either impossible or, at the very least, incredibly difficult, to come up with a wording for the problem that doesn't allow alternative interpretations due to out-of-the-box thinking (if it were easy to create comprehensive, precise wording, then there would be far fewer professional lawyers). The wording does specify that it is not a trick question - under a reasonable interpretation of "trick question", "Hah! The guru might be colour-blind" qualifies as the answer to a trick question, so actually is ruled out by the wording. It also directly contradicts "The answer is not 'no one leaves.' "

The generally accepted answer doesn't involve treating the question as a trick question, does lead to people leaving, and is an interesting outcome. The wording may not be perfect, but it's pretty good, and it's pretty clear that the accepted answer is the intended answer.

Potatoberg
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### Re: My write-up of the "Blue Eyes" solution (SPOILER A

rmsgrey wrote:If you choose to take "The guru speaks English" as an axiom, but not "the guru's statement is unquestionable" then that's a perfectly valid, and self-consistent position. It just happens to (with appropriate other axioms) generate a contradiction within the problem statement.

It is also true that it is either impossible or, at the very least, incredibly difficult, to come up with a wording for the problem that doesn't allow alternative interpretations due to out-of-the-box thinking (if it were easy to create comprehensive, precise wording, then there would be far fewer professional lawyers). The wording does specify that it is not a trick question - under a reasonable interpretation of "trick question", "Hah! The guru might be colour-blind" qualifies as the answer to a trick question, so actually is ruled out by the wording. It also directly contradicts "The answer is not 'no one leaves.' "

The generally accepted answer doesn't involve treating the question as a trick question, does lead to people leaving, and is an interesting outcome. The wording may not be perfect, but it's pretty good, and it's pretty clear that the accepted answer is the intended answer.

"The guru speaks English" is not an axiom. "I see someone with blue eyes is a statement in English" is an axiom.
"This is not a trick question" is just a cop out for a lack of clarity. But even if I accepted that reasonable assumptions are permissible in a logic puzzle, the puzzle has been reworded several times, which means that the currently accepted solution was reached through unreasonable assumptions.

phlip
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### Re: My write-up of the "Blue Eyes" solution (SPOILER A

Potatoberg wrote:"The guru speaks English" is not an axiom. "I see someone with blue eyes is a statement in English" is an axiom.

But "'I see someone with blue eyes' is only a statement in English" is not an axiom, and is in fact provably false. In order to claim that the puzzle containing those words means it's making that statement in English, you need to assume that (1) it's intending to make a statement at all, and isn't just a random string of letters and spaces, and (2) that the statement is written in English, and not one of the vast state-space of other languages that happen to contain a homographic statement. This is before you get to questions of whether the statement is unambiguous within the language, true, or reliable.

Now, sure, both of those are reasonable assumptions to make, and the possibility that they are false is, intuition says, vanishingly unlikely. But, as you say, "reasonable assumptions" aren't allowed here.

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Gwydion
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### Re: My write-up of the "Blue Eyes" solution (SPOILER A

Potatoberg wrote:"The guru speaks English" is not an axiom. "I see someone with blue eyes is a statement in English" is an axiom.
"This is not a trick question" is just a cop out for a lack of clarity. But even if I accepted that reasonable assumptions are permissible in a logic puzzle, the puzzle has been reworded several times, which means that the currently accepted solution was reached through unreasonable assumptions.

Could you please list the axioms that you're using to solve this puzzle? I don't know how you're differentiating between necessary axioms and "cop outs". More to the point, why is it inappropriate to take the color vision of the Guru as a given, in a logic puzzle explicitly stated to be about people's perceptions of colors and with no mention of the possibility that these perceptions might be false? I would argue that such a claim is even more of a "cop out" because it reduces the whole discussion to "nope, no solution exists."

As for the claim that the solution was reached by unreasonable assumptions because it has been reworded several times, I only see one rewording - by you. Your statement boils down to "even if I accepted that reasonable assumptions were reasonable, I don't accept your definition of reasonable so your assumptions, and therefore your solution, are unreasonable." Along with the above list, you might be able to strengthen your case by explaining what makes an assumption reasonable or unreasonable.

Yakk
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### Re: My write-up of the "Blue Eyes" solution (SPOILER A

There is no foundation to human communication that is guaranteed to be solid. So no, there is no way to reword any such puzzle so that there is no ambiguity. Eliminating all ambiguity is not a possible act.

Please stop pretending as if there is.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.

Xias
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### Re: My write-up of the "Blue Eyes" solution (SPOILER A

ITT: One kid walks up to a bunch of kids and tells them that their tricycles are all broken, even though they've been riding around on them all day.

Seriously guys you just need to admit that you've been riding on broken trikes this whole time.

jestingrabbit
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### Re: My write-up of the "Blue Eyes" solution (SPOILER A

Potatoberg wrote:
jestingrabbit wrote:No, I didn't. I suggested that

Ok then, if you want to be like that then join assholes club united, they'll love you.

From their founding member, this is high praise indeed.

Potatoberg wrote:Something being a lie is not the same as the act of lying.

Lying is "the telling of lies, or false statements; untruthfulness: ". Transparently, this can include not just someone knowingly deceiving, but also someone unknowingly doing so. There are clearly two concepts that lying could refer to, it seems natural here to use the broadest sense.

Potatoberg wrote:
jestingrabbit wrote:Regarding your allusion to the term "ipse dixit", meaning an unadorned dogmatic statement: this is the nature of these puzzles! The setup is dogmatically asserted, and the solutions are arrived at by accepting the setup and making arguments regarding it.

But the islanders do not know that they are in a puzzle of this nature, thus they have no reason to take the words of the Guru as unquestionable truths. They also don't have a sign saying "do not question the words of the guru". Not that it matters if they are all "perfect logicians".

I think that the intent of the puzzle is clear.

I really think you should consider the conlang possibility that phlip is raising more seriously. Its impossible to remove every ambiguity. There are always ambiguities in any finite communication, always assumptions made.

But look, lets consider the island with 100 blue eyed people and 100 brown eyed people. When the guru says that there is at least one blue eyed person, why is it logical for the islanders to disbelieve the guru when the guru's utterance agrees with the evidence of their senses?
ameretrifle wrote:Magic space feudalism is therefore a viable idea.

Xias
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### Re: My write-up of the "Blue Eyes" solution (SPOILER A

jestingrabbit wrote:But look, lets consider the island with 100 blue eyed people and 100 brown eyed people. When the guru says that there is at least one blue eyed person, why is it logical for the islanders to disbelieve the guru when the guru's utterance agrees with the evidence of their senses?

Be careful there. Consider the 2-blue-eyes case.

Potatoberg is right that in order for the islanders to leave, they have to know that the Guru's statement is truthful and accurate and that its truthiness and accuracy is common knowledge. Potatoberg is wrong in thinking that the way the puzzle is written doesn't justify assuming that is the case.

Cauchy
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### Re: My write-up of the "Blue Eyes" solution (SPOILER A

jestingrabbit wrote:But look, lets consider the island with 100 blue eyed people and 100 brown eyed people. When the guru says that there is at least one blue eyed person, why is it logical for the islanders to disbelieve the guru when the guru's utterance agrees with the evidence of their senses?

This is the same as the argument "the islanders don't learn anything new, each of them already can see someone else with blue eyes". While the islanders know that what the guru says is true, and they know the others know, they can't know that the others know that the others know that...[the appropriate number of repeats] ... what the guru says is true. An argument for the truthfulness of the guru has to work in the "one blue eyed person" case, or it can't work at all.

That said, I think the puzzle is fairly clear. The objections Potatoberg is bringing up are really objections against logic puzzles as a medium, not against this specific one.
(∫|p|2)(∫|q|2) ≥ (∫|pq|)2
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jestingrabbit
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### Re: My write-up of the "Blue Eyes" solution (SPOILER A

Cauchy wrote:
jestingrabbit wrote:But look, lets consider the island with 100 blue eyed people and 100 brown eyed people. When the guru says that there is at least one blue eyed person, why is it logical for the islanders to disbelieve the guru when the guru's utterance agrees with the evidence of their senses?

This is the same as the argument "the islanders don't learn anything new, each of them already can see someone else with blue eyes". While the islanders know that what the guru says is true, and they know the others know, they can't know that the others know that the others know that...[the appropriate number of repeats] ... what the guru says is true. An argument for the truthfulness of the guru has to work in the "one blue eyed person" case, or it can't work at all.

Yeah, I thought of this objection. I was interested to see what others would say about it mostly.

Cauchy wrote:That said, I think the puzzle is fairly clear. The objections Potatoberg is bringing up are really objections against logic puzzles as a medium, not against this specific one.

Yes. Especially when you consider how vaguely a lot of the puzzles are worded, and yet how what is intended to be set up is clear, its encumbent on reader to make certain assumptions. An unreliable guru's statement is clearly of no consequence, so they must be reliable and their utterance must add to the facts that are known.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.

Potatoberg
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### Re: My write-up of the "Blue Eyes" solution (SPOILER A

phlip wrote:
Potatoberg wrote:"The guru speaks English" is not an axiom. "I see someone with blue eyes is a statement in English" is an axiom.

But "'I see someone with blue eyes' is only a statement in English" is not an axiom, and is in fact provably false. In order to claim that the puzzle containing those words means it's making that statement in English, you need to assume that (1) it's intending to make a statement at all, and isn't just a random string of letters and spaces, and (2) that the statement is written in English, and not one of the vast state-space of other languages that happen to contain a homographic statement. This is before you get to questions of whether the statement is unambiguous within the language, true, or reliable.

Now, sure, both of those are reasonable assumptions to make, and the possibility that they are false is, intuition says, vanishingly unlikely. But, as you say, "reasonable assumptions" aren't allowed here.

"'I see someone with blue eyes' is only a statement in English" is not an axiom, just like 1 only equals 1 is not an axiom as you could say 1=(3-2). However, the statement "I see someone with blue eyes' is only a statement in English" can be broken down into concepts (which by nature are universal--if two men have the same concept, then the concept is the same independently of name given to the concept). For example, the concept of English could be understood by those who use a scripturally identical language. This means that the underlying concept behind "I see someone with blue eyes' is only a statement in English" is axiomatic, just like 1 only equals 1 is conceptually axiomatic. This means that any logical conclusion reached by a reader of a scripturally identical language would part from incorrect concepts and would thus be wrong, as he or she assumed they understood this scripturally identical language. For example, if a German man spoke to me and told me "mit Bea" and I assumed he meant "meet Bea" instead of "with Bea", I would be wrong to say "nice to meet you" because I assumed he was talking in English. Is the German man being ambiguous? No; I'm assuming.
Furthermore, no other language scripturally identical to English exists on earth. Given that the website with said puzzle cannot be accessed from outside of earth, it's only possible that the language used in the puzzle and the solutions proposed is English.
All of this is besides the point though because I previously pointed out something that is ambiguous even within English speakers: the guru's statement could be wrong because he could be color blind.
Yakk wrote:There is no foundation to human communication that is guaranteed to be solid. So no, there is no way to reword any such puzzle so that there is no ambiguity. Eliminating all ambiguity is not a possible act.

So does your statement "Eliminating all ambiguity is not a possible act" contain ambiguities? Care to list them for me?
Xias wrote:Potatoberg is right that in order for the islanders to leave, they have to know that the Guru's statement is truthful and accurate and that its truthiness and accuracy is common knowledge. Potatoberg is wrong in thinking that the way the puzzle is written doesn't justify assuming that is the case.

Here is the key fact: the islanders cannot read how the puzzle is written, thus they have no evidence that the guru is not lying or saying untruthful statements.

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### Re: My write-up of the "Blue Eyes" solution (SPOILER A

Potatoberg wrote:
Yakk wrote:There is no foundation to human communication that is guaranteed to be solid. So no, there is no way to reword any such puzzle so that there is no ambiguity. Eliminating all ambiguity is not a possible act.
So does your statement "Eliminating all ambiguity is not a possible act" contain ambiguities? Care to list them for me?

Yes, all communication contains ambiguities. You cannot eliminate them.

And no, I won't exhaustively list them. I don't claim to be able to do that.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.

phlip
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### Re: My write-up of the "Blue Eyes" solution (SPOILER A

Potatoberg wrote:However, the statement "I see someone with blue eyes' is only a statement in English" can be broken down into concepts (which by nature are universal--if two men have the same concept, then the concept is the same independently of name given to the concept). For example, the concept of English could be understood by those who use a scripturally identical language. This means that the underlying concept behind "I see someone with blue eyes' is only a statement in English" is axiomatic, just like 1 only equals 1 is conceptually axiomatic. This means that any logical conclusion reached by a reader of a scripturally identical language would part from incorrect concepts and would thus be wrong, as he or she assumed they understood this scripturally identical language.

I honestly have no idea what you're trying to say here. Can you reword?

Potatoberg wrote:For example, if a German man spoke to me and told me "mit Bea" and I assumed he meant "meet Bea" instead of "with Bea", I would be wrong to say "nice to meet you" because I assumed he was talking in English. Is the German man being ambiguous? No; I'm assuming.

But if you took the phrase at its German meaning, without preexisting mathematical proof that that was the language he was speaking, then you would still be assuming. Just an assumption that happened to be correct (and an assumption that was probably more likely to be correct given context... though still not 100%).

Potatoberg wrote:Furthermore, no other language scripturally identical to English exists on earth. Given that the website with said puzzle cannot be accessed from outside of earth, it's only possible that the language used in the puzzle and the solutions proposed is English.

So what you're saying is that the conjecture that the puzzle is written in a language commonly used on Earth is a reasonable assumption to make? The conjecture that no-one on the planet has ever created a conlang that intentionally mimics English structure but has different meaning is a reasonable assumption?

(I would actually argue that the second one is not a reasonable assumption, it's actually quite likely false, but if you modified it from "no other such language exists on Earth" to "no other such language is in common use on Earth" then sure.)

Potatoberg wrote:All of this is besides the point though because I previously pointed out something that is ambiguous even within English speakers: the guru's statement could be wrong because he could be color blind.

It's not beside the point. The point is that all this waffling about the guru maybe being colour-blind or maybe one of the islanders being deaf (or maybe one of the islanders thinks one of the other islanders is deaf) is so much hot air because there are much more fundamental ambiguities inherent in the mere concept of communication, and it is impossible for them to be eliminated entirely.

The only difference between "I can't tell for 100% certain it's written in English" and "it doesn't explicitly say thing X which is a standard assumption in logic puzzles" is one of scale, a difference in the probability that your assumption is false. You have to draw the "reasonable assumption" line somewhere, and it can't be at zero. To disagree with "reasonable assumptions" outright is to disagree that communication is possible.

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Xias
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### Re: My write-up of the "Blue Eyes" solution (SPOILER A

Potatoberg wrote:
Xias wrote:Potatoberg is right that in order for the islanders to leave, they have to know that the Guru's statement is truthful and accurate and that its truthiness and accuracy is common knowledge. Potatoberg is wrong in thinking that the way the puzzle is written doesn't justify assuming that is the case.

Here is the key fact: the islanders cannot read how the puzzle is written, thus they have no evidence that the guru is not lying or saying untruthful statements.

In what way does my statement require the islanders to have some meta-knowledge of the puzzle?

The puzzle is written in such a way that "it is common knowledge that the guru is reliable" is a reasonable and justified assumption for the puzzle solver to make.

As has been repeatedly explained to you, any argument that it is not a reasonable and justified assumption could be made against any other arbitrarily chosen assumption, to the point of absurdity.
If you want to say that a statement like "the guru is reliable, and this sentence is common knowledge" is a necessary addition to the puzzle to yield the desired solution, then (a) you are demonstrably wrong given the number of people who came to the desired solution without such a statement and (b) we would then have to add an infinite set of further caveats and explanations, and even then we would not have a solvable puzzle by your standard.

marzis
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### Re: My write-up of the "Blue Eyes" solution (SPOILER A

All these ideas of conlang, color blind, can't ever know the situation perfectly, I feel like these all fall into the categories that XKCD actually eliminates from being necessary in order to solve the riddle, namely:

There are no mirrors or reflecting surfaces, nothing dumb. It is not a trick question, and the answer is logical. It doesn't depend on tricky wording or anyone lying or guessing, and it doesn't involve people doing something silly like creating a sign language or doing genetics. The Guru is not making eye contact with anyone in particular; she's simply saying "I count at least one blue-eyed person on this island who isn't me."

And lastly, the answer is not "no one leaves."

So, Potatoburg, if we take those above statements to be true (I mean, if you want to say that XKCD could be lying, well, then just stop trying to solve logic puzzles or something, they're not for you) then how is 'the Guru could be color blind' any different from 'one of the islanders could find a mirror'. Sure, we can't know that the Guru isn't colorblind, but is that assumption 'logical' and 'not dumb' and 'not an answer assuming the question is a trick' and 'somehow relevant to the Guru not making eye contact with anyone in particular' and also results in the answer to the riddle being not 'no one leaves'?

I appreciate your objections to the phrasing of the puzzle and the ambiguities of English and all those other things you've mentioned, but XKCD provides us not only with the phrasing of the problem, but also with these guidelines I've copied above, to allow us to say 'okay, a) there is an answer and it is not 'no one leaves', and b) the answer isn't tricky or of the sort that he describes above (which falls into a certain category often found with sub-par logic problems, your objections to which are perfectly valid in those situations) so I don't have to worry about silly things and instead know that focusing on a purely logical solution will, if I am good enough, lead me to the solution'

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### Re: My write-up of the "Blue Eyes" solution (SPOILER A

phlip wrote:Can you reword?

Words represent universal concepts. The concept we name "tree" does not change if we call it the German word for tree "Baum" just like the concept we name "1" does not change if we call it "2". So while "a tree can only be a tree" is not an axiom because a tree can be a pine or a tree can be a Baum, if we substitute the words for the concepts they represent, such phrase is an axiom: a [concept of tree] can only be a [concept of tree].
So basically you can substitute "I see someone with blue eyes' is only a statement in English" for: [concept of I][concept of see][concept someone]....[concept of English], which is an axiom. You have no reason to assume I am talking English right now, but you have conclusive scientific evidence that I am. Some reader of a language written exactly the same as English would not have conclusive evidence that this is his language and not English.
phlip wrote:But if you took the phrase at its German meaning, without preexisting mathematical proof that that was the language he was speaking, then you would still be assuming. Just an assumption that happened to be correct (and an assumption that was probably more likely to be correct given context... though still not 100%).

But how would you express that mathematical proof? in roman numerals?
phlip wrote:So what you're saying is that the conjecture that the puzzle is written in a language commonly used on Earth is a reasonable assumption to make? The conjecture that no-one on the planet has ever created a conlang that intentionally mimics English structure but has different meaning is a reasonable assumption?

I've been thinking about it a bit more and I've come to the conclusion that such a language (let's call it ConEnglish) is actually impossible. You cannot change the definitions of words in such a way that any given text retains it's coherence while at the same time having a coherent grammatical set of rules. If ConEnglish interchanged the concepts of yes and no, you would know you're not reading your ConEnglish as soon as you read: yes is an affirmation. If you interchanged the concepts of affirmative and negative too you would know you're not reading ConEnglish as soon as you read: "The result is positive; yes, you are pregnant" affirmed the doctor". If you also replaced dead with alive you would have a problem reading Did he die? because you cannot alive. You would have to replace Did with was, but then you would have problems reading "yes, he was correct" because you cannot "do" correct.
ConEnglish, a language that an English reader would be fooled into thinking is English, is impossible. It would be like creating a coding language called ConJavascript that could read anything coded in Javascript and give you a usable but different program every time.
phlip wrote:It's not beside the point. The point is that all this waffling about the guru maybe being colour-blind or maybe one of the islanders being deaf (or maybe one of the islanders thinks one of the other islanders is deaf) is so much hot air because there are much more fundamental ambiguities inherent in the mere concept of communication, and it is impossible for them to be eliminated entirely.

The only difference between "I can't tell for 100% certain it's written in English" and "it doesn't explicitly say thing X which is a standard assumption in logic puzzles" is one of scale, a difference in the probability that your assumption is false. You have to draw the "reasonable assumption" line somewhere, and it can't be at zero. To disagree with "reasonable assumptions" outright is to disagree that communication is possible.

If a scientist gathers enough evidence that copper conducts electricity, then he has conclusive evidence that copper is a conductor; he is not saying "it's reasonable to assume copper is a conductor" because science makes no assumptions. It's true that science goes against the problem of induction, but we're using computers for a reason: we have conclusive evidence that they work. Similarly, we have conclusive evidence that the English language works--we're using it right now because of this. However, if we were on a desert island and a guru told us that she can count at least 1 person with blue eyes, we would not have conclusive evidence that the guru is not color blind, is not lying or is not high on whatever leached out of that leaf he cooked his fish in. Also, saying that nobody leaves because an islander has no evidence that the other islanders aren't deaf is a valid solution.
marzis wrote:So, Potatoburg, if we take those above statements to be true (I mean, if you want to say that XKCD could be lying, well, then just stop trying to solve logic puzzles or something, they're not for you) then how is 'the Guru could be color blind' any different from 'one of the islanders could find a mirror'. Sure, we can't know that the Guru isn't colorblind, but is that assumption 'logical' and 'not dumb' and 'not an answer assuming the question is a trick' and 'somehow relevant to the Guru not making eye contact with anyone in particular' and also results in the answer to the riddle being not 'no one leaves'?

I appreciate your objections to the phrasing of the puzzle and the ambiguities of English and all those other things you've mentioned, but XKCD provides us not only with the phrasing of the problem, but also with these guidelines I've copied above, to allow us to say 'okay, a) there is an answer and it is not 'no one leaves', and b) the answer isn't tricky or of the sort that he describes above (which falls into a certain category often found with sub-par logic problems, your objections to which are perfectly valid in those situations) so I don't have to worry about silly things and instead know that focusing on a purely logical solution will, if I am good enough, lead me to the solution'

I don't know if this whole problem arises from a stunted ability to put oneself in the position of others, so let me explain. First, ignore the fact that this is a puzzle. You are a perfect logician and arrive at the island with some people who are also perfect logicians (which implies they do not make assumptions of any kind--maybe that is why they don't talk to anyone, huh?). One of them, the guru, says "I count at least one blue-eyed person on this island who isn't me". Why would you trust her? Even if someone told you "the guru cannot say untruthful statements", why would you trust them? A perfect logician would require evidence that the guru does not lie. Of course, you could say that it is stupid to require evidence because what is the evidence for axioms? we must only know we know nothing (includes eye color). In either case, there is no solution to this puzzle other than "nobody leaves".
I understand you want a logic puzzle, but if the puzzle only gives one possible solution then that's the solution. Real life is like this. Einstein's theory of relativity could be considered an exercise in lateral thinking and a tricky solution, but it's the correct solution. Nobody said "you're wrong because it's reasonable to assume space-time is constant". Another example was how the Copernicus heliocentric-model was not accepted for a long time because it was reasonable to assume that the earth was the center of the universe; so they came up with these relatively complex calculations of the orbits of the planets and called that the solution to the "wanderers".
You can't make assumptions in science. Once you have conclusive evidence then you have facts or theories. Saying "ooh but problem of induction" is not a reasonable refutation, you are using a product of science: a computer. With me you're doing the same, when I say "there islanders have no evidence that the guru is not color blind" you tell me "ooh but you have no evidence that the language that the puzzle is written in is only English" (which begs the question: why are you using it then?). It's childish. To debate in terms of "I only know I know nothing" is itself a performative contradiction given the implicit rules and truths you accept by participating in a debate. For example, you cannot debate that the truth is unknowable because even if you're right, you're wrong, it's unknowable.
Xias wrote: If you want to say that a statement like "the guru is reliable, and this sentence is common knowledge" is a necessary addition to the puzzle to yield the desired solution, then (a) you are demonstrably wrong given the number of people who came to the desired solution without such a statement and (b) we would then have to add an infinite set of further caveats and explanations, and even then we would not have a solvable puzzle by your standard.

You can have a solvable puzzle without adding anything simply by accepting that the solution is "nobody leaves". Why the effort to force the puzzle into an erroneous solution?

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### Re: My write-up of the "Blue Eyes" solution (SPOILER A

Potatoberg wrote:.
Xias wrote: If you want to say that a statement like "the guru is reliable, and this sentence is common knowledge" is a necessary addition to the puzzle to yield the desired solution, then (a) you are demonstrably wrong given the number of people who came to the desired solution without such a statement and (b) we would then have to add an infinite set of further caveats and explanations, and even then we would not have a solvable puzzle by your standard.

You can have a solvable puzzle without adding anything simply by accepting that the solution is "nobody leaves". Why the effort to force the puzzle into an erroneous solution?

But the solution to the puzzle is not "nobody leaves," as defined by the puzzle. So what is the solution to the puzzle as written?

I'm also quite certain from reading your reply that you don't actually understand the conlang rebuttal. The conlang wouldn't have to be useful for communication. It would just have to translate words that sound like "I see at least one person with blue eyes" into a sentence with a different meaning entirely.

Also, your insistence that a game involving fictional characters on a fictional island must perfectly reflect how we approach real world problems demonstrates that you don't actually understand the purpose of a puzzle in the first place. We aren't trying to get real people off of an island, here. We are trying to explore common knowledge as an abstract concept.

Lunch Meat
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### Re: My write-up of the "Blue Eyes" solution (SPOILER A

Potatoberg wrote:You can have a solvable puzzle without adding anything simply by accepting that the solution is "nobody leaves". Why the effort to force the puzzle into an erroneous solution?

Because that's not a puzzle, that's a boring story. By these standards, the solution to all those hats puzzles and prisoners in a jail with two switches would be "The prisoners, being perfect logicians, realize that the jailer is sadistic and insane and they have no reason to trust he will let them go, so they either give up and don't bother counting hats or they stage a revolt." And the Traveler's Dilemma would be "The passengers write the manager's supervisor and complain about not getting their refund in a proper manner and the manager is fired."

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### Re: My write-up of the "Blue Eyes" solution (SPOILER A

In the context of the puzzle, a "Guru" is someone whose statements are believed by everyone, and everyone knows that the Guru is a Guru so everyone knows that every knows that... the Guru is believed by everyone".

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### Re: My write-up of the "Blue Eyes" solution (SPOILER A

Potatoberg wrote:What you call playing with tricky edge cases I call being rigorous with logic.

What you call "being rigorous with logic" is very unlike what mathematicians, philosophers, or logicians would understand by that phrase.

Potatoberg wrote:Here is the key fact: the islanders cannot read how the puzzle is written, thus they have no evidence that the guru is not lying or saying untruthful statements.

So would you be satisfied if it were added, to the statement of the puzzle, that it is common knowledge among the islanders that the Guru's statements are all true?
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### Re: My write-up of the "Blue Eyes" solution (SPOILER A

Xias wrote:I'm also quite certain from reading your reply that you don't actually understand the conlang rebuttal. The conlang wouldn't have to be useful for communication. It would just have to translate words that sound like "I see at least one person with blue eyes" into a sentence with a different meaning entirely.
It even could be useful for communication. It could be a fully fleshed-out language spoken by millions of people on ConEarth. It's just that eventually someone would say something strange enough for someone else to realize English and ConEnglish aren't the same language.

But since none of the islanders ever communicates anything to each other after the problem starts, and we don't know about anything they may have communicated with each other before it started, we only have the one sentence to go on, which could either be English or one of the infinite number of possible conlangs that aren't English but in which that set of characters is nonetheless a valid grammatical proposition.
Unless stated otherwise, I do not care whether a statement, by itself, constitutes a persuasive political argument. I care whether it's true.
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### Re: My write-up of the "Blue Eyes" solution (SPOILER A

Lunch Meat wrote:Because that's not a puzzle, that's a boring story. By these standards, the solution to all those hats puzzles and prisoners in a jail with two switches would be "The prisoners, being perfect logicians, realize that the jailer is sadistic and insane and they have no reason to trust he will let them go, so they either give up and don't bother counting hats or they stage a revolt." And the Traveler's Dilemma would be "The passengers write the manager's supervisor and complain about not getting their refund in a proper manner and the manager is fired."

Perfect logicians are boring to most people. An interesting solution from a perfect logician is wishful thinking because there are so many assumptions made in even the simplest of arguments.
gmalivuk wrote:It even could be useful for communication. It could be a fully fleshed-out language spoken by millions of people on ConEarth. It's just that eventually someone would say something strange enough for someone else to realize English and ConEnglish aren't the same language.

But since none of the islanders ever communicates anything to each other after the problem starts, and we don't know about anything they may have communicated with each other before it started, we only have the one sentence to go on, which could either be English or one of the infinite number of possible conlangs that aren't English but in which that set of characters is nonetheless a valid grammatical proposition.

How would the Guru know that she is saying something truthful if she has no evidence that the islanders talk English and not ConEnglish? Why would the islanders assume the Guru is talking in English if they have no evidence that she is?
I get why most people think it is a reasonable assumption so assume she is talking in English and get on with the puzzle, but a perfect logician would not accept any assumptions as valid.
mward wrote:In the context of the puzzle, a "Guru" is someone whose statements are believed by everyone, and everyone knows that the Guru is a Guru so everyone knows that every knows that... the Guru is believed by everyone".

Belief is not logical. These are perfect logicians we are talking about.
Xias wrote:But the solution to the puzzle is not "nobody leaves," as defined by the puzzle. So what is the solution to the puzzle as written?

That's not part of the puzzle, that's a hint; it's completely unnecessary in a puzzle. The hint is wrong.
Xias wrote:I'm also quite certain from reading your reply that you don't actually understand the conlang rebuttal. The conlang wouldn't have to be useful for communication. It would just have to translate words that sound like "I see at least one person with blue eyes" into a sentence with a different meaning entirely.

Sorry, I understood ConLang as "con language", what you're saying now is what is called a "coded message".

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### Re: My write-up of the "Blue Eyes" solution (SPOILER A

TheGrammarBolshevik wrote:So would you be satisfied if it were added, to the statement of the puzzle, that it is common knowledge among the islanders that the Guru's statements are all true?
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### Re: My write-up of the "Blue Eyes" solution (SPOILER A

Potatoberg, if you don't like the sorts of premises you need to accept for every instance of human communication ever, then you should maybe try to develop your own system for transmitting ideas and information from one person to another.

And if you don't like the conventions of logic puzzles, this is perhaps the wrong subforum for you?
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### Re: My write-up of the "Blue Eyes" solution (SPOILER A

Potatoberg wrote:How would the Guru know [...] ? Why would the islanders assume [...] ?
I get why most people think it is a reasonable assumption so assume she is talking in English and get on with the puzzle, but a perfect logician would not accept any assumptions as valid.

That's exactly the point of the conlang rebuttal to your original issue. If we hold hard and fast to ridding the puzzle of any assumptions made on the part of the puzzle solver then we are left with it being impossible to write a puzzle with the desired solution.

Potatoberg wrote:
Xias wrote:But the solution to the puzzle is not "nobody leaves," as defined by the puzzle. So what is the solution to the puzzle as written?

That's not part of the puzzle, that's a hint; it's completely unnecessary in a puzzle. The hint is wrong.

I disagree. It is a continuation of the puzzle in that it excludes undesired solutions such as "they look in a mirror" or "nobody leaves."

Potatoberg wrote:
Xias wrote:I'm also quite certain from reading your reply that you don't actually understand the conlang rebuttal. The conlang wouldn't have to be useful for communication. It would just have to translate words that sound like "I see at least one person with blue eyes" into a sentence with a different meaning entirely.

Sorry, I understood ConLang as "con language", what you're saying now is what is called a "coded message".

Yes, a "constructed language" does not need to be useful to still be a "constructed language," and in this case, arguing that there might be some way to distinguish between some conlang and English is just a red herring. So long as a conlang could exist in which "I see at least one person with blue eyes," means anything other than that sentence does in English, then I have to make an assumption about that sentence before I can do anything with it.

And it's not about what the islanders have to assume. It's about what I, as a player in this game, have to assume about the islanders.

I have a question for you. Knowing what the intended solution is, can you provide a wording for the puzzle that does have that solution, by your criteria? Can you provide one that Potatoberg #2 won't come along and criticize in the future?

If the answer is "you can't" (which is the answer, spoiler alert) then my response is that we as players are going to make conventional, reasonable assumptions in order to actually play the game we want to play. I'm sorry that you don't fancy that.

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### Re: My write-up of the "Blue Eyes" solution (SPOILER A

Potatoberg wrote:
mward wrote:In the context of the puzzle, a "Guru" is someone whose statements are believed by everyone, and everyone knows that the Guru is a Guru so everyone knows that every knows that... the Guru is believed by everyone".

Belief is not logical. These are perfect logicians we are talking about.

Perfect logicians believe in axioms, rules of inference and valid deductions from the axioms. In the context of the puzzle, these statements about the Guru are believed by perfect logicians: therefore they are either axioms or valid deductions from axioms (it makes no difference which). As G.K.Chesterton put it:

Logic, then, is not necessarily an instrument for finding truth; on the contrary, truth is necessarily an instrument for using logic--for using it, that is, for the discovery of further truth and for the profit of humanity. Briefly, you can only find truth with logic if you have already found truth without it.

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### Re: My write-up of the "Blue Eyes" solution (SPOILER A

Potatoberg wrote:
mward wrote:In the context of the puzzle, a "Guru" is someone whose statements are believed by everyone, and everyone knows that the Guru is a Guru so everyone knows that every knows that... the Guru is believed by everyone".

Belief is not logical. These are perfect logicians we are talking about.

Perfect logicians recognise that logic requires assumptions to be made in order to produce anything.

If you require perfect logic operating on null data, then the best you can possibly achieve is the Cogito - a recognition that logic is being performed, which necessarily implies the existence of an agent to perform the logic.

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### Re: My write-up of the "Blue Eyes" solution (SPOILER A

Potatoberg wrote:I understand you want a logic puzzle, but if the puzzle only gives one possible solution then that's the solution. Real life is like this.

and

Potatoberg wrote:That's not part of the puzzle, that's a hint; it's completely unnecessary in a puzzle. The hint is wrong.

Dude, stop trolling this forum, of course the hint is right, the puzzle creator can give whatever hint he or she wants. Also, you can't tell me you don't see the irony in your 'if the puzzle only gives one solution then that's the solution' comment, right? You must see it. Think about it for a little bit, and lay off the insults to your fellow posters.

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### Re: My write-up of the "Blue Eyes" solution (SPOILER A

Xias wrote:
Potatoberg wrote:That's not part of the puzzle, that's a hint; it's completely unnecessary in a puzzle. The hint is wrong.

I disagree. It is a continuation of the puzzle in that it excludes undesired solutions such as "they look in a mirror" or "nobody leaves."

Yes, it's undersired, but it's the only valid solution that does not require perfect logicians to assume stuff (aka a contradiction).
Xias wrote:I have a question for you. Knowing what the intended solution is, can you provide a wording for the puzzle that does have that solution, by your criteria? Can you provide one that Potatoberg #2 won't come along and criticize in the future?

If the answer is "you can't" (which is the answer, spoiler alert) then my response is that we as players are going to make conventional, reasonable assumptions in order to actually play the game we want to play. I'm sorry that you don't fancy that.

You're correct, you can't reword the puzzle in such a way that the only valid solution is the currently accepted solution (and doing so would mean the current solution was reached through unjustifiable assumptions).
I'm ok with you guys doing puzzles for fun and games, just don't call them logic puzzles unless you're going to be rigorous with logic.
marzis wrote:
Potatoberg wrote:I understand you want a logic puzzle, but if the puzzle only gives one possible solution then that's the solution. Real life is like this.

and

Potatoberg wrote:That's not part of the puzzle, that's a hint; it's completely unnecessary in a puzzle. The hint is wrong.

Dude, stop trolling this forum, of course the hint is right, the puzzle creator can give whatever hint he or she wants. Also, you can't tell me you don't see the irony in your 'if the puzzle only gives one solution then that's the solution' comment, right? You must see it. Think about it for a little bit, and lay off the insults to your fellow posters.

I don't see the irony; the only solution is "nobody leaves", the "hint" says this is not the solution, but by definition a hint cannot be part of the puzzle because it's a hint to the puzzle.
PS: I have not insulted my fellow posters, unlike you calling me a troll.

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### Re: My write-up of the "Blue Eyes" solution (SPOILER A

Potatoberg wrote:it's the only valid solution that does not require perfect logicians to assume stuff (aka a contradiction).

I would love to see your perfectly logical proof of your assertion that it is a contradiction for a perfect logician to make assumptions. Do not make any assumptions!

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### Re: My write-up of the "Blue Eyes" solution (SPOILER A

Importantly, remember that some people deny the law of the excluded middle.

http://en.wikipedia.org/wiki/Law_of_excluded_middle

So you can't use that.

And you still haven't answered this.

TheGrammarBolshevik wrote:
TheGrammarBolshevik wrote:So would you be satisfied if it were added, to the statement of the puzzle, that it is common knowledge among the islanders that the Guru's statements are all true?
ameretrifle wrote:Magic space feudalism is therefore a viable idea.

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### Re: My write-up of the "Blue Eyes" solution (SPOILER A

Potatoberg -
You keep asserting that the desired solution requires unjustifiable, unreasonable, or otherwise "wrong" assumptions. You've previously stated that a perfect logician should not make any assumptions at all, a statement that invalidates many of the popular puzzles on this forum. For example, in the "prisoners with colored hats" puzzle, there is no mention of the possibility that the prisoner behind you might be depressed and suicidal, so you have no reason to assume he is answering correctly. I am not aware of any puzzles that can be solved without any assumptions - can you show me one, or do they not exist? If they don't, then what is the point of your argument?

We call these things "logic puzzles" to distinguish the contents here from math homework, lateral thinking puzzles, jokes, and other content that doesn't fit in with what the rest of us want to find. We enjoy finding a logical path to a solution, given a setup and what we consider reasonable assumptions. Much of the ongoing debate with several of these puzzles comes from the fact that different solutions can arise with different starting assumptions. As an example, take the ongoing thread about how to define a door guard who "always lies" and what that means.

Coming to a pages-long discussion and insisting not only that everyone here is wrong, but that the entire subforum is invalid, is trolling. Please don't take personal offense - this claim comes from your behavior and not your character. You may not have called any of us names, but saying things like "I'm ok with you guys doing puzzles for fun and games, just don't call them logic puzzles" is very insulting to me as someone who has spent more time in this forum than any other, and I suspect to others as well.

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### Re: My write-up of the "Blue Eyes" solution (SPOILER A

An important distinction that we need to make is that it is the player, not the islander, making such assumptions. A solution that required the islanders to make non-axiomatic assumptions would not be a correct solution. A solution requiring the player to make assumptions about the islanders is perfectly fine.

So the "100 days" solution does not require us to reason that the islanders are assuming that "the guru is reliable" is common knowledge.

The "100 days" solution does require the player to assume that " 'the guru is reliable' is common knowledge among the islanders." Along with infinitely many other trivial assumptions.

This does not lose step with the definition that the islanders are perfect logicians.

Also, I don't believe that the islanders being perfect logicians is a characteristic of the solution. It's simply a part of the framing device, including that the islanders have colored eyes and that the guru speaks and all of the other arbitrary ways that the narrative creates an inductive process. The islanders could have been framed as robots; the Guru's statement could have been framed as any sort point of mutual contact; the entire puzzle could have been written explicitly as a series of material implications followed by the statement "P." The "puzzle" here is translating the narrative into such a mathematical induction, and a side effect of the narrative is the question of common knowledge. It's still a logic puzzle.

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### Re: My write-up of the "Blue Eyes" solution (SPOILER A

jestingrabbit wrote:And you still haven't answered this.

TheGrammarBolshevik wrote:
TheGrammarBolshevik wrote:So would you be satisfied if it were added, to the statement of the puzzle, that it is common knowledge among the islanders that the Guru's statements are all true?

I guess, at this point, we can extract an answer: if we add that stipulation, we are stipulating that the islanders are assuming something. But perfect logicians cannot assume anything, so the statement of the puzzle would be a contradiction.
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elwood
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### Re: My write-up of the "Blue Eyes" solution (SPOILER A

Xias wrote:Along with infinitely many other trivial assumptions.

And now we're going to have a trolling sequence about how "the number of trivial assumptions is, in fact, finite but unbounded and you're not that clever after all".

Potatoberg: to be fair, you started dragging the thread to that "logic rigor" nonsense after people corrected your misunderstanding of "Theorem 1" exposed in your first post.

Reithan
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### Re: My write-up of the "Blue Eyes" solution (SPOILER A

​I think the solution is wrong. The solution hinges on the fact that the 'new' piece of information you've added is that now the fact that SOMEONE has blue eyes (or green, depending on the telling of the riddle) is common knowledge. We've been told these are all perfectly logical, perfectly computational beings, so, unless over the course of their stay they cannot have all seen each other seeing each other at least once each, then they all have this knowledge already. Thus, the day that each person say the last of the other persons see the other last of the other other persons was THAT individual's "day 0". Thus, THAT person would have left on HIS day 99. So, for each person who saw another person hit day 99, he must also leave. So, I may have fubar'd the math a bit, but in no longer than 99! days after the first person hits critical knowledge, all persons should have left anyway. UNLESS of course, we are to assume that the island is large enough and the movement of persons is disperse enough that this hasn't happened yet...which we are told to make no assumptions. Also, the announcement is unnecessary anyway. As soon as the persons are called together and can see each other seeing each other, it is now each person's "day 0" anyway. In short, unless we assume that the persons have or have not seen each other see each other other previously, then the initial state of the puzzle is untenable.

From the linked "usual objections" thread, this is listed, but the rebuttal states that for each "knows" one adds (i.e: I know that he knows) you must turn n to n-1. However, this is only adds an additional qualifier to my objection. Instead of waiting until they see ONE person see each other person, they must wait until they see EVERY other person see each other person. For a NORMAL person, keeping track of this would be impossible, but these are PERFECT computation beings. So, totally doable, the only limit is how long it takes for this to happen, which if I had the time or the inclination to do so, you could probably create a statistical limit on how long that sort of thing would take given the size, shape etc of the island and the habits of the population. So, again, we must assume that the population is taking pains to ensure this doesn't happen, which we're told they do not.

Additionally, as soon as the guru calls the meeting, even BEFORE he speaks, it is now everyone's day 0 regardless, as they all see each other seeing each other, so they must know that each other person sees each other person seeing at minimum 98 other people with blue eyes, even before the guru says anything.

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### Re: My write-up of the "Blue Eyes" solution (SPOILER A

Additionally, each person can't be certain who's left on which day unless each person also keeps track of the total number of people on the island, or who's leaving it when. So, to avoid having to leave, each person could simply sit in a cave for 99+some random number days and not look, after which point the number of people matched with the timeline of common knowledge is disrupted and they're all safe again. Any time they have all the pieces to the solution, they simply go back to the cave and wait again, thereby resetting the clock.

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### Re: My write-up of the "Blue Eyes" solution (SPOILER A

What is the minimum number of blue eyed islanders required for them to create a "day 0" before the guru speaks?

Re: your second comment, and some things you said in your first comment:

Everyone can see everyone else at all times and keeps a count of the number of people they see with each eye color (excluding themselves), but they cannot otherwise communicate.

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### Re: My write-up of the "Blue Eyes" solution (SPOILER A

Reithan, the thought experiment to demonstrate the solution follows. (Assume the islanders know everyone has brown or blue eyes).

Pretend there are 201 islands, each with a number. The residents of each island do not know what number the island they are on has.

(It is easy to show that the residents of island #X with blue eyes know they are on island #X-1 or #X, and the residents with non-blue eyes know they are on island #X or #X+1, but they do not know what island exactly.)

On island 0, there are zero blue-eyed people. On island 1, there is one. On island N, there are N.

A guru does a lightning tour of the islands. On each island 1 through 200, the guru says "I see at least 1 blue eyed person". On island 0, the guru says "I see no blue eyed people".

Examine what happens the first night. On island 0 everyone leaves. On island 1, the blue eyed person clearly leaves.

On the 2nd night, the people on island 2 with blue eyes notice nobody left. They can work out that if they where on island 1, the one blue eyed person they can see would have left the first night. So they know they aren't on island 1. So they are on island 2! And they have blue eyes! So they leave the 2nd night.

It isn't hard to show that on night X, the people with blue eyes on island X leave, and they do not leave any earlier.

On night X+1, the people with brown eyes on island X now know they are on island X. So they leave, as they now know they have brown eyes.

All of this occurs without any communication occurring between the islands.

Now suppose after the guru does the flight, every island except one is destroyed. We do not tell those islanders. Do the islanders on that island suddenly behave differently?
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