Math Fun

[Carleas]

If you had only 2 on the island and you used that same "counting and deducing" procedure, given that everyone on the island is starting with the same count, would they still deduce their proper color? The obvious answer is "yes", but it seems you are hell bound to never say it.

Without the guru saying anything? Really?
Every blue sees 1 blue, and has no idea what color his own eyes are. He also knows that that blue has no idea what color his own eyes are. Why would counting the days change the this sum total of knowledge on the island? Neither of them should expect anything to happen on the first day: they don't know that the person they're looking at knows anything about blue eyes. What have they gained on the second day? Even if they start with the same number, they number isn't related in a meaningful way to the color of their eyes, or to the color of anyone's eyes.

Can you make a syllogism for this case?

[James S Saint]

Geeezzz...

Carleas, if someone asked you "If the Earth was flat, would the water spill off the edge?", how many pages would you continue to argue and demand to know how the Earth got flat before you actually addressed the question?

If everyone knew there was at least one blue on the island, even if they didn't know that everyone else knew it, and there really was only one, he would deduce that it was him immediately.

[Carleas]

If everyone knew there was at least one blue on the island, even if they didn't know that everyone else knew it, and there really was only one, he would deduce that it was him immediately.

Agreed. But, if everyone on the island knew that there was at least 1 blue when there are 2 blues, they don't deduce anything unless they know that the other person knows it.

For 3 blues, the change is similar: they must know that each other islander knows that each other islander knows.

For 4, they must know that the others know that the others know that the others know that the others know.

With each additional blue eyed islander, they must nest the knowledge an additional degree.

[James S Saint]

Still just can't bring yourself to address a hypothetical question, huh.
I think you have lived in DC too long.

[Carleas]

Your proposed solution doesn't work because you are taking knowledge out of context, and missing the distinction between someone knowing something, and someone knowing that someone knows something.

On an island with 100 blues, everyone knows that 99 have blue eyes, but no one know how many blues the 99 blues they see see. And no one knows how many blues the 98 blues that the 99 blues see see see. And no one knows how many blues the 97 blues the 98 blues the 99 blues see see see see.

Let me repeat that, with parentheses, because it is both grammatically and logically complex: No one knows how many blues (the 97 blues (the 98 blues (the 99 blues see) see) see) see.

[James S Saint]

Your proposed solution doesn't work because you are taking knowledge out of context, and missing the distinction between someone knowing something, and someone knowing that someone knows something.

On an island with 100 blues, everyone knows that 99 have blue eyes, but no one know how many blues the 99 blues they see see. And no one knows how many blues the 98 blues that the 99 blues see see see. And no one knows how many blues the 97 blues the 98 blues the 99 blues see see see see.

Let me repeat that, with parentheses, because it is both grammatically and logically complex: No one knows how many blues (the 97 blues (the 98 blues (the 99 blues see) see) see) see.

I am not "proposing a solution". I am asking a question which seems to be akin to having a discussion that starts with "IF men really were superior to women,..."... with a woman.

I have asked a hypothetical question (about 50 times now it seems). Your only answer has been "no", yet you cannot demonstrate your answer's correctness. Instead you repeat (probably close to 50 times now) the same argument that everyone involved in this thread knew very long ago and hasn't argued against.. with the exception of another possibility, which you totally avoid discussing.

I am not interested in the canonical proposed solution. That one has been beat to death and is "old news". I am proposing a hypothetical "new angle" from which to discuss a possible new solution. How often you repeat the old solution isn't going to change anything nor address the actual question being asked.

How about actually answer the question with a demonstration of any single value (between 0-99) that didn't produce the correct color deduction.

[Carleas]

As I did pages ago, I will take 98. 98 is a number between 0-99, and even if all islanders start counting from 98, they will not deduce the color of their eyes.

Let's start in the position of an islander. He doesn't know how many blue eyes there are. He doesn't know his eye color. He sees 99 blue eyes and he knows that "there are at least 98 blue eyes, and all of the people with blue eyes that I can see know that there are at least 98 blue eyes."
He knows that if there were 98 blue eyed islanders, and they know "there are at least 98 blue eyes", they would leave the island.
But his knowledge that there are at least 98 blue eyes doesn't transfer to this hypothetical group: he knows there are at least 98 blues because he can see at least 98 blues. But if there were only 98, they could not see at least 98 blues. By the very assumptions made to construct the hypothetical, they would not have access to the information that one of the 100 blues on the island has access to. They thus would not leave on day 1, so nothing would be learned when they haven't left on day 2, and on day 3 he would still not know what color his eyes are.

Contrast this case with a case in which the guru says "I see at least 98 blues". There, he knows that if there were only 98 blues, they would leave, because they derive their knowledge from what the guru says, and not from what they see.

In short, the reason that people can't deduce their eye color even if they all start with 98 is that if there were 98 blues, they could not see 98 blues. The fact that every blue islander can see 98 blues does not change this.

[James S Saint]

I am blue out of 100 blues;
A) They all know [that they all know] [that there are at least 98 blues].
B) They all know [that they all know] [that they don't know their own color].
C) They all know [that they all know] [that they will leave if they could deduce their color].
D) They all know [that they all know] [that if there were only 98 and (A) were true, they would leave the 1st day].
E) They all know [that they all know] [that if there were only 99 and (A) were true, they would leave the 2nd day].
F) They all know [that they all know] [that if there were only 100 and (A) were true, we would leave the 3rd day].

Looks like it worked to me.
But of course, you are going to say, "but they didn't know [that they all know] [that there are at least 98 blues]", right?
And I am going to reply, "And there you go yet again for the umptenth time avoiding the question".
And then you would repeat your response...again and again and again.

But now compare that quote to this one;

I am blue out of 100 blues;
A) They all start counting with 98 blues. {specified by the question's hypothetical }
B) They all know [that they all know] [that they don't know their own color].
C) They all know [that they all know] [that they will leave if they could deduce their color].
D) They all know [that they all know] [that if there were only 98 and (A) were true, they would leave the 1st day].
E) They all know [that they all know] [that if there were only 99 and (A) were true, they would leave the 2nd day].
F) They all know [that they all know] [that if there were only 100 and (A) were true, we would leave the 3rd day].

Well, that one seemed to have worked too.

So it seems to me that you chose a number, 98, that actually does work considering the question that has been asked, but denied that it worked (because you want to argue an old issue, not the question actually asked).

[Carleas]

You need to rephrase the question if the question you are asking is not answered by "a demonstration of any single value (between 0-99) that [doesn't] produce the correct color deduction."

Also, could you clarify what they are counting? It does not seem true that "if there were only 98 and [they all start counting with 98 blues] they would leave the 1st day," unless they are counting how many blues they know there are.

One way to think of it is this: why can't the 100 islanders start counting at 100, and all leave immediately? Your step D is equivalent to that:
if there were 100, and they all started counting at 100, they would all leave the first day.
But that seems obvious, it's analogous to the case of 1 islander told by the guru "I see blue eyes." It just assumes that they already know their eye color.

[James S Saint]

Still avoiding the question. Do you even know what the question was anymore?
Just state the first half dozen words of it or so. That is the part you seem to want to ignore.

Also, could you clarify what they are counting?

Really??
Well.. this once, stating the more than obvious (again)..
1) they count the number of blues they see above the number they started counting from, the "97" or "1" or whatever.
2) they count the days of no one leaving.
3) if they count less blues above the start number to the count of non-leaving days, they deduce they are the last blue.

It does not seem true that "if there were only 98 and [they all start counting with 98 blues] they would leave the 1st day," unless they are counting how many blues they know there are.

If there were only 98 and you were one of them and starting a count from 98;
1) they count the number of blues they see above the number they started counting from = -1
2) they count the days of no one leaving. = 0
3) if they count less blues above the start number (-1) to the count of non-leaving days (0), they deduce they are the last blue. = Leave

But that seems obvious, it's analogous to the case of 1 islander told by the guru "I see blue eyes."

Hell yeah it's obvious. So why in the hell don't you ever admit it???

It just assumes that they already know their eye color.

No. It assumes that they all STARTED THEIR COUNT WITH THE SAME NUMBER, just as the question's hypothesis required.

[Carleas]

Do you even know what the question was anymore?

Clearly not, which is why I asked you to repeat it, rephrase it, clarify it repeatedly. Please do.

If there were only 98 and you were one of them and starting a count from 98;
1) they count the number of blues they see above the number they started counting from = -1
2) they count the days of no one leaving. = 0
3) if they count less blues above the start number (-1) to the count of non-leaving days (0), they deduce they are the last blue. = Leave

So, again, I'll ask why they can't start counting at 100, and leave the first day. You're happy to have a hypothetical 98 people who know from the start that there are only 98 blues. So why not 100 people who know from the start that there are 100?

[James S Saint]

Still avoiding the question.

Do you even know what the question was anymore?

Clearly not, which is why I asked you to repeat it, rephrase it, clarify it repeatedly. Please do.
.
.
So, again, I'll ask why they can't start counting at 100, and leave the first day. You're happy to have a hypothetical 98 people who know from the start that there are only 98 blues. So why not 100 people who know from the start that there are 100?

If there were only 100 and you were one of them and starting a count from 100;
1) they count the number of blues they see above the number they started counting from = -1
2) they count the days of no one leaving. = 0
3) if they count less blues above the start number (-1) to the count of non-leaving days (0), they deduce they are the last blue. = Leave

But that seems obvious, it's analogous to the case of 1 islander told by the guru "I see blue eyes."

Hell yeah it's obvious. So why in the hell don't you ever admit it???

It just assumes that they already know their eye color.

No. It assumes that they all STARTED THEIR COUNT WITH THE SAME NUMBER, just as the question's hypothesis required.

The question that you seem to fear to the bone;
"IF everyone chooses the exact same number to use in counting days before deducing that they are the remaining blue (or not), is there any number between 0-99 that would cause them to deduce improperly?"

But the canonical solution says...

But the question, still incorrectly answered was simply;
"IF everyone chose the SAME number to start counting the days before deducing their color, would it always lead to an accurate deduction of their color?"

But the canonical solution says...

Please tell us one number between 0 and 99 such that "even if everyone picked that same number", it would still not work to allow them to discover their actual eye color.

But the canonical solution says...

This is a clowns show.

"If everyone on the island happens, for whatever reason, to start counting days with the exact same number would they deduce their proper eye color?" And since you are going to say, "no", please give an example of a number between 0-99 where their deductions would fail?

But the canonical solution says...

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."

But the canonical solution says...

Do you understand that as long as everyone starts with the same number, it will always work?

But the canonical solution says...

Okay, let me reword it,
Do you understand that the islanders will always be able to discover their eye color as long as they each use the same number to start counting the days?

But the canonical solution says...

But can you show me any case where they ALL pick the same number, for whatever reason, yet they would end up being wrong?
If they all pick 5, I know they would all discover their eye color.
If they all pick 39, I know they would all discover their eye color.
If they all pick 69, I know they would all discover their eye color.

Can you think of any number between 0-99 that would lead to error even if ALL of them pick that same number?
What would that number be?

But the canonical solution says...

Again, because everyone is starting with the same number, after a certain amount of days, they all leave.

As long as everyone is starting with the same number, it will always work.

But the canonical solution says...

If they all merely take the total number on the island divided by 4 and start counting days from there with the same, "if 50 blues didn't leave..", they solve the problem. They can deduce that. But that is only one of many.

Do you believe that isn't deducible or wouldn't work?

Go ahead, say it again, "But the canonical solution says..."

[Flannel Jesus]

I don't even know why there's still an argument about the 'counting from 97' thing. It was already demonstrated that it doesn't work.

The solution basically works like this:
You see 99 blues, you see that they don't leave after 3 days, and you conclude that there must be 100 and you're blue-eyed.

But James already agreed that if there were in fact only 99, they wouldn't leave after 3 days anyway, so a brown-eyed guy would, as above, see that there are 99 and see that they don't leave after 3 days. So clearly seeing 99 blue-eyed people and seeing that they don't leave after 3 days is not enough to conclude that there are 100 and you're blue-eyed, since...that's exactly what a brown-eyed person would see if there were 99.

I explained it once, and James agreed once. It's pretty clear to see why it doesn't work.

[James S Saint]

I don't even know why there's still an argument about the 'counting from 97' thing. It was already demonstrated that it doesn't work.

The solution basically works like this:
You see 99 blues, you see that they don't leave after 3 days, and you conclude that there must be 100 and you're blue-eyed.

But James already agreed that if there were in fact only 99, they wouldn't leave after 3 days anyway, so a brown-eyed guy would, as above, see that there are 99 and see that they don't leave after 3 days. So clearly seeing 99 blue-eyed people and seeing that they don't leave after 3 days is not enough to conclude that there are 100 and you're blue-eyed, since...that's exactly what a brown-eyed person would see if there were 99.

I explained it once, and James agreed once. It's pretty clear to see why it doesn't work.

Gyahd, after all of those explanations, you STILL can't see YOUR mistake in that... Geeezz!!!

[Flannel Jesus]

I certainly can't see how you call it a mistake after explicitly agreeing with it...

[Flannel Jesus]

You agreed that if there were only 99, they wouldn't leave after 3 days.
If there were 100, they also wouldn't leave after 3 days.

It doesn't take much brain power, then, to realize, "Hey, them not leaving after 3 days is completely unrelated to the quantity of blue-eyeds there are, and therefore, I can't use that information to deduce how many there are."

You can't use a fact that would be true regardless of the number of blue eyes, to determine the number of blue eyes.
You can't use a fact that would be true regardless of your eye color to determine your own eye color.

[Carleas]

If there were only 100 and you were one of them and starting a count from 100;
1) they count the number of blues they see above the number they started counting from = -1
2) they count the days of no one leaving. = 0
3) if they count less blues above the start number (-1) to the count of non-leaving days (0), they deduce they are the last blue. = Leave

Only if "starting a count from X" means "know it to be common knowledge that there are X blues". So, for 100, if the blues know there are 100 blues from the get-go (e.g. the guru tells them "there are 100 blues"), they will all leave. But for 99 blues, it's not enough that they know that there are 99 blues, they need to know that (the 99 blues know that there are 99 blues). And they don't know that: they can see 99, they don't know their eye color, so they don't know if the 99 see 98 or 99. They can only leave on the 2nd day if they "start counting from 99", i.e. "know it to be common knowledge that there are 99 blues."

You are using a vague phrase, "start counting from", which you've refused to define precisely. But the way that you're reasoning from it is to treat it as them having common knowledge, i.e. for any hypothetical set of islanders, they know that there are X blue eyes. If they get the knowledge by looking at the other islanders, it doesn't translate the same way: even if we see 99 blues, it's clear that we can say that if there were only 3 blues, they couldn't know that there are at least 99 blues.

[James S Saint]

If there were only 100 and you were one of them and starting a count from 100;
1) they count the number of blues they see above the number they started counting from = -1
2) they count the days of no one leaving. = 0
3) if they count less blues above the start number (-1) to the count of non-leaving days (0), they deduce they are the last blue. = Leave

Only if "starting a count from X" means "know it to be common knowledge that there are X blues".

NO!!
It does NOT!!

It means that they start counting from the same number REGARDLESS OF WHY!!!

You keep denying the hypothesis.
"IF they ALL start with the SAME number,.."

That is all they have to do.
WHY they do it or HOW is NOT the question being asked.

Just answer the question that is asked.

[Carleas]

It means that they start counting from the same number REGARDLESS OF WHY!!!

Can we agree that the following chain of reasoning doesn't work:

I see 99 blues.
If there are 100 blues, I must be one of them.
100.
Therefore, my eyes are blue.

Is it not clear that this doesn't work?

[James S Saint]

It means that they start counting from the same number REGARDLESS OF WHY!!!

Can we agree that the following chain of reasoning doesn't work:

I see 99 blues.
If there are 100 blues, I must be one of them.
100.
Therefore, my eyes are blue.

Is it not clear that this doesn't work?

Using your same reasoning (if we can call it that), "yes, it is not clear. It does work".

[Carleas]

So this must also work:

I am the only person on this island. I don't see anyone.
If there were 1 person on the island with blue eyes, it would be me.
1
Therefore, I have blue eyes.

...

[James S Saint]

So this must also work:

I am the only person on this island. I don't see anyone.
If there were 1 person on the island with blue eyes, it would be me.
1
Therefore, I have blue eyes.

...

Sounds valid to me.

[Carleas]

OK. I am satisfied that you are forced to accept that those are valid proofs in order to maintain that your solution is also valid.

[James S Saint]

So I take it that you didn't get the whole, "I AM NOT TALKING ABOUT A SOLUTION, BUT MERELY ASKING A QUESTION" bit, huh.

[James S Saint]

Carleas, let me ask a different question;

In the canonized solution, it is proposed that a blue (you) is thinking that another blue is thinking that he might be brown, "if I am brown, then...". Why would a blue be thinking that he is brown? He has no way to know that he is brown, especially since he really isn't.