Math Fun

My latest analysis of the correct solution is that it relies on the assumption that “deducing based on what is known for sure is most logical, when it is only knowing something (one’s own eye colour) for sure that will make a difference”.

Given only two possibilities (to each blue) that there are either 100 or 99 blues (which is exhaustively equivalent to knowing what their own eye colour is), they would therefore start deducing based on the possibility that contains only that which they can initially know for sure: the 99 blues they definitely can see.

For example:
Each of the 100 blues does not see 100 blues. They DO know for sure that they definitely see 99, so they start their deduction based on this rather than anything that contains a variable that could go either way. Importantly, they begin their deductions from the point of view of any blue of the 99 they see - knowing only what any of the 99 would know.

Any of the 100 blues in turn knows for sure that 99 blue-eyed islanders would know for sure that they would see 98 blue-eyed islanders. Deducing ONLY based on what these 98 would see for sure, knowing the only thing that any of these 98 would only know for sure, the series of deductions continues to the point that 1 blue-eyed islander would know for sure that he sees ZERO blue-eyed islanders, based ONLY on what this 1 blue-eyed islander would know for sure, based ONLY on an examination by any of the 100 blues of what can be known FOR SURE.

At this point, the ONLY other thing that can be known for sure (the Guru’s words) CHANGES the only sure knowledge that can be obtained without the Guru’s words, to “1 blue-eyed islander would know for sure that there is ONE blue-eyed islander (whom they cannot see)”. The only islander whose eyes he would not be able to see is his own, therefore he has blue eyes.

And then the rest:
(He can now know for sure that IF there were no other blue-eyed islanders, he would leave on the ferry the next midnight. He can know for sure that IF there was only 1 blue-eyed islander who he could see that did not leave on the ferry the next midnight, that blue-eyed islander that he could see sees another blue-eyed islander, who could only be himself, so they both leave on the ferry on the 2nd midnight. If there were 2 that he could see, they would act in the previously described way if there were in fact only 2 blue-eyed islanders, and if they do not, that leaves the only possibility that hey each see he himself who therefore has blue-eyes, and they all leave on the 3rd midnight. This continues all the way up to what is ACTUALLY the critical juncture: whether 99 leave on the 99th midnight, and if not then 100 leave on the 100th midnight.)

That is all the blues. The browns and the 1 green cannot do the same as they can only know for sure the same and only definite knowledge that the blues could use aside from the Guru’s words. The Guru’s words do not change what 1 brown or green would definitely and only know for sure about the amount of browns and greens that they would definitely see. Accordingly, they cannot build the deduction back up like the blues needed to be able to do in order to leave. Additionally, the fact that all the blues left tells the browns and the green nothing new about their own eye colour, as it would remain a possibility to each and everyone of them that the could have, e.g. red eyes.

I have intentionally emphasised:
(a) the only assumption that is being relied upon in order for this series of deductions to unfold,
(b) that there are no other assumptions that can be made from certain knowledge,
(c) that each deduction in the series operates ONLY from the knowledge that the number of blues in question would know (no transfer of knowledge out of context, such as “what 100 would know” being transferred to “what any less than 100 would know (critically to what 1 would know)”.

As such, this is the ONLY solution.
All other proposed solutions are abusing at least one of the above.
As far as I know, this clears up every single grievance in the entire thread, though I can illogically deduce that this will not end the discussion - but that is not my fault. Carleas, FJ and I can now all step out and take a breather. Everyone else, study it well and swallow your pride.

That is a list of your errors.

…until you said that.

Well argued.

The counter argument was already made. You seemed to have ignored it to focus (again) on only the one thought, making the assumptions that the counter argument pointed out (which you then listed as your strong points). It seemed kind of Sil-ly.

My reasoning illustrates why there is only one solution to focus on.
The consequence of this is that other attempts are going to be flawed, and the emphases that I have made make explicit exactly where and why they are.

Is all you have to say about that that your counter attempts have already pointed out why they’re wrong?
And as unoriginal your play on my name is, S’ain’t relevant.

James, are any of the premises of the canonical solution inherently contradictory? Because the canonical solution by its nature excludes all others, so if there is a non-canonical solution, it means there is some internal inconsistency in the problem itself.

Also, it’s worth noting that in your breakdown of the situation where the islanders don’t want to leave, the fact that they didn’t want to leave was not used. If they are uncertain about whether the other islanders heard the guru, they can’t complete the syllogism, whether they want to or not. The canonical solution assumes that there is no doubt about whether the islanders heard and understood the guru.

I have another extension of the suicidal tribe presentation of the puzzle, which I think is interesting (again moving beyond defending the canonical solution):

The suicidal tribe presentation goes like this: there’s a tribe of perfect logicians. Some of them have blue eyes, some have brown eyes, but no one knows their own eye color. In fact their religion requires that if any tribe member learns her eye color, she must kill herself the next day at noon in the middle of the village. As a result, they never talk about eye color. A foreign anthropologist finds the tribe and is welcomed. He spends some times, learns the language, but only too late learns about the religious customs: at a meeting involving the whole village, he makes an off-hand remark that it is strange to see that some members of the tribe have blue eyes.

The solution is the same: all the blue eyed people must kill themselves on the same day, and that day will be X days after the foreigner spoke, where X is the number of blue eyed people.

As I mentioned before, this setup allows the anthropologist to save everyone on the island by pointing to one individual and saying “that’s the guy who has blue eyes”, effectively revoking the information that was used to start the inference. But I think he would have to point to 1 person on the first day; on the second, I think he’d have to name 2; on the third 3 and so on. Does anyone disagree with that?

Also, since in this version, the tribe members can communicate, it’s not their inability communicate, but their actual desire not to know (and their respect for the desires of their fellow tribe mates) that prevents them from telling each other. But does the ability to communicate allow them to short circuit the logic before it completes in such a way that everyone is saved?

My first thought is that two people privately discussing the fact that a third person has blue eyes would save them (i.e., with three people, A and B talk about C, B and C talk about A, and C and A talk about B), but that doesn’t seem to solve it: A and B will still expect C to leave.

The question is essentially asking whether there is a way to limit knowledge by increasing the number of agreed truths (i.e., I’m not interested in answers where they all agree to doubt the foreigner). I will have to think more on this.

Quid pro quo, bro

That isn’t true.

This list of assumptions must be addressed;

• “everyone knows that everyone heard the guru”
• “everyone knows that everyone is a perfect logician”
• “everyone knows that everyone can see everyone”
• “everyone knows that everyone is thinking the same scheme”
  proving that there is no alternative to them. A perfect logician cannot deduce anything until he can prove to himself that there is no alternative, “nothing is possible until something is impossible”.

The first three we have been taking as a given and I suspect that if we don’t, the puzzle isn’t worth addressing. But the fourth is the one I still see as an issue. There is nothing in the canonized version that says that there is no possibility of any other person on the island thinking anything else, such as merely a higher number to begin the count down.

The incentive issue really isn’t an issue because perfect logicians with instant thought, would instantly know what number every perfect logician would be using and thus instantly be using it himself trapping himself into destiny and his fate.

Btw, that is the whole point of this resonating, not personal or social egos.

Until you can prove that there are absolutely no alternatives to “the solution”, you don’t know the solution.

Quite, but therein lies the rub…

The rules exist after the fact by logical extension, some people find it hard to work them out, some people think there are none, some more ignorant people thing there are not just no rules but they have no need of them, but there are, and that is the point of the problem, it’s why I liked it, and hated it at the same tiem.

• “Standing before the islanders, she says the following: “I can see someone who has blue eyes.”” - though nothing is said about their attention/auditory/comprehension capacity. An assumption to which it would be valid to object, given the information explicitly given.
• “They are all perfect logicians… Everyone on the island knows all the rules in this paragraph.” - known.
• “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).” - known.
• “if a conclusion can be logically deduced, they will do it instantly.” - this means that everything that can be deduced certainly from certainty will be known instantly. I have explained what these things are - no assumptions, just fact and deduction from fact. Apart from the unspecified hearing and understanding of the Guru’s words, this is no issue, and it gives rise to the one and only solution, which I have explained and that you have ignored.

As just shown, there is no alternative to any of the above assumptions, other than the first one, where they may merely not be paying attention, perhaps distracted by the bewitching green eyes of the only female on the island (known by paying attention to the pronouns). They may be deaf or they may not be - this is not specified. It appears, though, that they are within hearing range, assuming they are both able and willing to listen. There is also nothing to say they did not hear and understand the Guru’s words. I would prefer it you came up with one of your theories based on this ambiguity, rather than from missing certainty and miscontruing from there, as you have done so far.

Everyone has only shown that there is at least one conclusion series that can be known (and everyone here has known that for a week).
“At least 1” =/= “no possible alternatives”.

And until there are no possible alternatives, the perfect logicians cannot use that one.

…and until everyone here knows that everyone knows that everyone here knows that there is no alternative, the arguing continues and no one “leaves this island” (assuming that any two care).

This is another example of your mistaking what we from the outside know with what they from the inside know.
The correct statement would be, “And until there are no possible alternatives, we don’t know that the perfect logicians will use that one.”
What we do and don’t know doesn’t affect what they can and can’t use.

I wasn’t talking about what WE know, but what THEY know.
THEY have to know that there is no alternative concerning any method they are using to predict what others on the island are doing, else they cannot depend on that scheme.

Imagine that you find 200 logicians who have never failed a logic test of any kind. You stick them on an island with those rules. There happens to be 100 blue-eyed. But on the 4th day, one of them leaves. The others are thinking, “what the hell?? How did he do that? Now what? Is someone else going to leave tomorrow?”

James, isn’t it clear that given a syllogism,
A - > B, ~B |- ~A
we don’t have to separately prove that A isn’t also an alternative? The syllogism itself proves that A is not an alternative

So, if the canonical solution relies only on knowns and syllogisms built from them, the proof proves that there are no other alternatives. The logic makes the conclusion necessary; any other alternative would produce a contradiction, which must be inherent in the problem statement (thus my previous question; I didn’t understand your answer).

Carleas, anyone on the island or off can easily see that they need only start counting days from a common number. There are two easy to see common numbers; 1 and 200. If by 1, it takes 100 days to know. If by 200, it takes 51 days to know. But using either method, it must be known as to which method everyone else is using. Both work if chosen by everyone. Neither work if not. Why would they necessarily all choose to use 1 rather than the 200?

This is a problem only for your solutions that rely on everyone somehow choosing the same assumptions without explicit communication.
This really isn’t a problem for the correct solution. As I have pointed out, all they are doing is deductively expanding on what they definitely know for sure, and this just so happens to solve the puzzle. Nothing more is needed for it, it just works from logic and certainty. Nothing is left out or gone to waste, nothing is missed, everything that comes from logic and certainty goes towards the one solution and no other solutions - as shown.

Sure, maybe a breach of the conditions of the puzzle might allow for the otherwise illegal assumptions necessary for your solutions, and at least the pier one would work if the problem was altered in order to allow it.

I think I’ve even said already: if you would only realise/admit that your solutions were only appropriate for an altered version of the puzzle, then that would be fine. It’s the fact that you seem to insist that they’re appropriate for the puzzle as it is that’s incurring the criticism of myself and others. You’ve only been exploring alternative solutions to the puzzle if it were altered - and if you do alter it, there ARE alternative solutions.

Nothing is a problem is you ignore the problem in it.

Yes, I can believe you live by that one.

We’re waiting for a proof of that. If you feel you’ve provided one, link to it. Or, correct my version of it (which I truly meant to be a good faith restatement of my understanding of your proposed solution).

Carleas, before I go make some more formal proof for a hypothetical, I need to ensure that you actually understand the hypothetical. If you understand it, it seems blatantly obvious that it would work, so either you do not understand it (which I only give any credit to at all because FJ couldn’t grasp it), or you are doing your political rhetoric thing again where you divert the conversion upon seeing that you might have a flaw in your reasoning. If it is the former, the proof isn’t going to help because you wouldn’t understand what is being proven. If it is the later, it wouldn’t do any good to present a proof anyway because you would just divert from it again.

So before I go put together some kind of unnecessary proof, how about you show me that you actually understand what it is that I am saying with “IF THEY ACTUALLY START WITH THE SAME NUMBER (between 0-99), IT WILL ALWAYS WORK.” What do you think that statement means?

This whole thing is about clarity and verification.