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?
Or are you saying that the Colonies didn’t actually win the war because they weren’t standing in proper battle formation?
No. First, they can’t start counting from zero. They all know that there are at least 0 blue on the island, and they cant get anywhere from that.
Second, the number of days has to mean something. They’re counting the days because each day reveals information.
In the 3 case, if the guru says “I see at least 1 blue”, they leave on day 3 because A knows that B knows that C knows that at least 1 person has blue eyes. On day 2, A knows that B knows that there are at least 2 blue. On day 3, A knows that there are at least 3blue.
If the guru says “I see at least 2 blue”, they can leave on day 2, because A knows that B knows that at least 2 people have blue eyes. On day 2, A knows that there are at least 3 blue.
But if in the 100 case, the guru whispers to each separately, “I see 99”, they can count the days forever in unison, the won’t get enough information to deduce their eye color.
Gyahd… your thinking like the Brits during the American revolution, “they can’t fight that way. It isn’t proper.”
I very well know the canonical solution. I was explaining it to others long ago, before you got into this.
And thus you still haven’t answered the question. I am not saying that any arbitrary number would be fine because obviously they must all pick the same number, thus it cannot actually be random or arbitrary.
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?
That just makes no sense at all.
Whenever people avoid demonstrations and insist on whatever they care to insist on to avoid demonstrations, you kind of know what’s going on.
FJ, Sil, now Carleas, all avoid everything I write. Fine, be that way.
James, don’t “gyahd” me if your language is imprecise. You asked if they will learn their eye color “as long as they each use the same number to start counting.” If they start counting from zero, it’s clear that they won’t (even assuming the rest of your argument works . . . which it doesn’t).
Of course, accepting that they won’t leave if they start from zero is quite important, because if the guru does not speak, they can only start from zero (in the relevant sense; see below).
Don’t front, James. Where were you when I was using the canonical solution to point out the deficiencies in the Monk problem that developed into the discussion of this problem. Let it be known that I am the oldest school.
The you must agree that merely “picking” a number doesn’t get them anywhere. If they all start counting the days, and happen to start counting the days on the same day, they will never learn their eye color. So, if by “pick”, you mean that they start counting, then every number, from 0-100 inclusive, will fail to produce additional knowledge. If by “pick”, you mean that some outside force tells them publicly that “there are at least X islanders with blue eyes”, then X can be any number from 1-100, inclusive.
Fixed Cross, when you quote my most recent response to your argument immediately before you accuse me of “avoid[ing] everything [you] write”, you hurt your credibility.
I agree it was poorly phrased. Let me try again:
It is not enough to know that there are 97 people on the island with blue eyes. In the 2 and 3 case, every islander knows that there is someone on the island with blue eyes. In the 3 case, everyone on the island knows that everyone on the island knows that there is someone on the island with blue eyes.
The guru’s statement “there is someone on the island with blue eyes” is not important for the content of the message. That’s why I compared the whispering guru case: a guru who whispers separately to two islanders “there is someone on the island with blue eyes” does not make it possible for the islanders to learn the color of their eyes, because in the whisper case they don’t learn something significant about what the other islander knows. The same is true in the three case or the four case.
The only case in which a whispering guru gives them sufficient information to get off the island is when she whispers “I see X islanders with blue eyes”, where X is the number of islanders with blue eyes. The reason this works is that the deduction doesn’t have to include knowing what someone else knows. As long as what the guru said is true, each blue eyed islander can conclude that they have blue eyes, even though they are uncertain about what the guru said to each other islander.
Let’s not continue repeating ourselves, James. If you want a different answer, ask a different question.
So you are deliberately avoiding that question.
In a court of law, you would be cited for “contempt of court”.
And since you personally don’t want to get to the answer, I’ll move on to present the end result for anyone else actually interested.
James, I answered the question: If you mean they’re just counting days, I can think of a lot of numbers between 0 and 99 that don’t work: all of them. If you mean the number the guru says she sees, then none of them won’t work. If you mean something else, say so. “Pick” is an ambiguous term.
Let me ask you a direct question that you have avoided:
Fixed Cross, I’d be curious to know your answer as well.
A) No, you didn’t answer it and still haven’t, merely more belligerent avoidance.
B) If you can’t demonstrate one, I am really not interested in your responses.
For anyone else;
This is a puzzle that challenges people to consider the mind of God, an absolutely perfect thinker who knows all possible ways to get anything done, without exception.
This puzzle, derived from similar forms from more than 2000 years ago, has a canonical solution proposed to be “THE solution”. It proposes that there could be neither alternate nor better solution, “better” being defined as more efficient. If there is an alternate solution then the canonical solution would not work because it depends upon everyone on the island using the same algorithm. If there is even one alternate, it won’t work unless it can be deduced that it is more efficient and thus will be chosen by all of the perfect logicians.
In this puzzle, “God knows”, that all it takes for everyone on the island, is that they all start from the exact same number and count days until they can see that all of the blues they can see didn’t leave, and thus derive themselves as a blue.
So the question becomes by what means could perfect logicians choose a number from which to start the count, knowing that all of the others would be using that same number. They don’t want to use the least efficient of any choices that might be available, else they can’t know that the others would be using that number also. So they want to find the highest number possible for everyone to use that everyone could know that everyone else was using.
Out of 100 blues, there can be no question in anyone’s mind that there is at least 1. So that could be a number to choose. And there could be no doubt in anyone’s mind that there are at least 2, so that too could be a choice.
As the numbers increase toward the 100 mark, the doubt begins to rise as to whether everyone really knows that everyone else knows that there is at least that many. If you are number 100, not knowing for certain that you are blue, you can’t be certain that there are more than 99 and can’t be certain that everyone knows that there are at least 99. So the puzzle becomes one of seeing if there is a point wherein the questionable becomes the unquestionable so that everyone could knowingly choose the same number.
The doubt in the number to be certain of and thus use, starts dimensioning quickly from the number that you can actually see because you know that everyone can see at least one less that you can. So if you can see 99, it becomes pretty certain that everyone already knows that everyone else knows there are at least 90. But the risk cannot be taken that they would choose 91 or 89 instead of 90, else everyone errors in their deducing of their eye color.
So what is needed is a means to normalize any choice within a range close to what someone would see. In that way, even though they might see a different amount and be slightly more dubious than others as to what number to choose, by a normalization procedure, they all choose the same number.
An example of this is the following;
Since every perfect thinker realizes at least what has been outlined above, they can see that an easy normalization procedure would be for everyone to use the total number on the island as a common number. Everyone sees the same total and everyone knows that every does. If that number is anywhere near twice what you see, then take ¼ of it. In this case, that would be 50.
At that point, you have a common number for everyone to use. If everyone used 50 as the number from which to start counting, everyone would accurately discover their own eye color. And 50 days is twice as efficient as 100 days.
So it being a valid option that is more efficient than the 100 day canonized version, how does anyone know that no one else isn’t using it? And if they are using it then the 100 days canonized version no longer works for anyone.
So it should be unquestionable that the 100 day solution, although sound in itself, cannot be used simply because no one can depend on everyone else using it.
Just as a matter of strict accuracy, you keep saying that it would work if they all start on the same day: that’s not the case. Your solution might work if they all KNEW that they were all starting on the same day. Not just if they happen to start on the same day, but don’t know if the others are starting on the same day as well.
You have that a little backwards.
If they all start on the same day, it wouldn’t matter if they knew it.
Reality is what actually works, end results, not what someone theorizes is supposed to work.
No, it would matter. If I started on day 50 but I don’t know that everyone else started on day 50, and they actually started on day 49, then their plan might have them leaving a day later than my plan would have me leaving, so I’m leaving on day 50 if I don’t see anybody leaving on day 49, but they might not be leaving on day 49 anyway even if there are only 99 because they might have started a different day from me.
So if I don’t know that they’re starting on the same day, then I know that they might be starting on a different day, which means that them not leaving doesn’t say anything to me about my own eye color. I have to know. It’s required.
If not, you end up with brown-eyed people knowing they’re blue eyed. So much for ‘perfect logicians’.
Guy???
What’s wrong with you.
I said IF THEY ALL USED THE SAME NUMBER.
When did 50 become the same as 49??
Mhm, but we’re talking about a guy who doesn’t KNOW that they’re all using the same number, and HE’S considering the possibility that they started on a different day.
You keep making this mistake, James. You keep using information YOU know as if the person on the island would know it as well. He’s on the island, not outside. YOU are talking about them all happening to use the same day, but HE doesn’t know that, so HE’S considering what would happen if they used a different day, and HE, being a perfect logician, would be able to realize that IF they were using a different day than him, which is totally possible from HIS perspective, then them not leaving wouldn’t necessarily tell him anything about his own eye color.
So stop confusing your knowledge with his.
FJ, this whole thing is FILLED with hypotheticals.
Why is it that you only recognize the ones you want to see as hypothetical and the others as actualities?
I said “[size=150]IF[/size] they ALL USE the SAME number”.
IF he doesn’t know, he WILL consider the possibility that they’re starting on a different day. Considering that possibility from his perspective, he will realize that IF they start on a different day from him, them not leaving will not tell him anything about his own eye color.
So, for example, he starts on day 50 but they might be starting on day 49. If he knew everyone was starting on day 50 (which he doesn’t), he would wait 50 days, and on the 51st day see that the 99 he sees haven’t left, and deduce that he has blue eyes.
But since he doesn’t know, everyone else might be starting on day 49 for all he knows, right? And IF they’re starting on day 49, the 99 he sees wouldn’t have left on day 50 anyway, even if there were only 99 of them.
So he sees 99, he sees that they didn’t leave on day 50, and he can’t tell if they didn’t leave because he’s #100 or because they started on a different number but there’s only 99.
And if he can’t tell if there’s 100 or 99, he can’t leave.
It doesn’t matter a single bit what anyone knows, believes, or even wants “[size=150]IF[/size] they ALL USE the SAME number”.
With or without the guru, they have a choice of numbers guaranteed to reveal their eye color.
Those perfect logicians would all know the most efficient number and be able to logically prove it to be so and thus know that all of the others would also use it.
Our game is to discover what that number would be, to “read the mind of the all knowing God”.