No, the "timing" by itself is not sufficient, as has been clearly laid out above.
Not only is that not obvious, it is not correct.

Oh, you'll definitely need that! And by the way, I'm still awaiting the answer to my simple query to you.
Correct, that is precisely what will happen if the students are sharp enough, and can trust each other to be sharp enough.

Sam5, this is a very foolish question. Figuring out the color relies on the behavior of the students at each iteration of your exercise, just as it requires observing the behavior of the blues in the puzzle. I thought you understood that. Tell me the behavior of all the students at each iteration, and I will very definitely tell you the color of my card. THe one caveat is-- they must be better logicians than you are, so this experiment may be doomed to fail! Otherwise, it would be an excellent way to get to the heart of the matter. Tell you what, I will carry out this experiment with you on one condition-- we do it several times in succession, starting out where I only see 2 red cards. Then you can pick a card color for me, and tell me what the students do, and I'll figure it out. Then we'll do it again where I see only 3 red cards, and I'll figure out my card again. We'll keep repeating this until we build up to where I see 8 red cards. That way, you will see the error in your thinking, but we have to do it this way or you won't be able to figure out what the students will do correctly.

edited to add the caveat.

...assuming the students are acting with utmost logic.

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0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 ...

Furthermore ...

It's The 8^th Iteration That Is The KEY ...

So, Do All of The Red Cards Go Up Then, or Not?

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If you Ignore YOUR Rights, they Will go away.

So you believe that one must work through what each thinks the others think before it can be built back up through what happens to one blue on day one, two on day two, etc? I suppose I can see the point in that, since it might be necessary to demonstrate first that one blue might think another thinks another thinks another sees no blues in order to eliminate the possibility of any metaknowledge. So I guess I have to give you the benefit of the doubt on that one. We already know that if n blues die on day n, there is no metaknowledge. But my thinking is that this wouldn't necessarily eliminate the possibility of metaknowledge in the first place. We could say, for instance, that 'all dogs are carnivores', but that doesn't necessarily mean that 'all carnivores are dogs'. I can see that these two ways of thinking might be and probably are completely mutually exclusive, but I still have not seen anything yet that conclusively demonstrates that one way of thinking must be chosen over the other, only that which is preferred, the one that will work in order to reach the conclusions provided in this puzzle. I am sure a proof exists one way or the other, as a proof or "disproof", and it will probably be very simple in the end. Until then, though, I am still torn. I am not completely convinced yet that one way of thinking about it is right and the other is wrong, except, as I said, that one way works toward the purposes of this puzzle and the other does not.

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Let's put together the pieces of The Grand Puzzle . (website - now revised)

"Let's define another operator, Sz, which we won't pay any attention to."
"This transformation will automatically make zero equal zero."
"It may be true that zero equals zero -- and that is certainly an equality -- but I don't want to go into the details at this time."

It isn't so hard to understand.

Just pretend you are in the tribe and you can see 1 blue and 18 browns. You don't know what colour you are. When the anthro says there is blue then you know that the blue will commit suicide the next day if you are brown, but won't if you are blue.

So we can say that, if there is 1 blue he will die on the first day and if there are 2 blues they will die on the second day.

Now pretend you are in the tribe and you can see 2 blues and 17 browns. You don't know what colour you are. We have already established that 2 blues will die on the second day. If they don't do that there must be 3 blues, so you are a blue.

So we can say that, if there are 2 blues they will die on the second day (we already knew this) and if there are 3 blues they will die on the third day.

Now pretend you are in the tribe and you can see 3 blues and 16 browns. You don't know what colour you are. We have already established that 3 blues will die on the third day. If they don't do that there must be 4 blues, so you are a blue.

So we can say that, if there are 3 blues they will die on the third day (we already knew this) and if there are 4 blues they will die on the fourth day.

You see, it's not so complicated really.

clop

Indeed so, which is why I must add a caveat to the challenge! It doesn't matter, it'll never happen, it's like Randi challenging an ESP claim.

Quite so, after the eighth iteration I will announce my card color.
That's up to Sam5's choice for my card color.

Your meaning would be precisely the same if you were talking about the statement 2+2=4.

Actually, believe it or not, it is a bit more complicated than your explanation. Your attempt at simplification is equivalent to the effort for an elegant inductive proof by worzel on post 594. Showing that n blues die on day n does not automatically extend to n+1 blues dying on day n+1, because nobody in the tribe might know that n blues would have to die on day n! You need to stay inside the head of a given blue, and insert another layer of metaknowledge to make that extension. I likened it to a fuse that is burning up to n, but it still has to have the n+1 part of the fuse "treated" by the flammable material to keep the burn going. The flammable material here is metaknowledge about what others know. You could easily set up a situation where 1,2, or 3 blues die on days 1, 2, and 3 respectively, but 4 blues are fine. However, if you know that completely common knowledge exists (unlimited metaknowledge) about the existence of blues, then the fuse is a proper fuse all the way up, and you are right-- knowing that n blues must die on day n implies that if they don't die on day n then you know there must be more than n blues ("you" being in the tribe).

I was intending to have another go at that proof using CK(n, x) to mean that x is common knowledge to meta level n. So CK(1,b>=1) means that everyone knows there's at least 1 blue, while CK(2,b>=3) means that everyone knows that everyone knows that there's at least 3 blues.

I think it is true that:

CK(n,b>=m) => CK(n+1,b>=m-1)

and using that with P(n) to mean that n blues will die on day n I was hoping to show something like:

[ CK(n, b>=1) => P(n) ] => [ CK(n+1, b>=1) => P(n+1) ]

Does that sound promising, Ken?

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There are 10 types of people in the world. Those who understand ternary, those who don't, and those waiting for a bus.
If logic doesn't work, then surely it does.

I'm not sure why you keep saying this. As far as I can tell, that statement is true simply by definition (and substitution).

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Let's put together the pieces of The Grand Puzzle . (website - now revised)

"Let's define another operator, Sz, which we won't pay any attention to."
"This transformation will automatically make zero equal zero."
"It may be true that zero equals zero -- and that is certainly an equality -- but I don't want to go into the details at this time."

Actually, it's a little more complicated than that, even. That n blues die on day day n leads to n+1 blues dying on day n+1 is not in question here since that is the definition of n. What is in question is the logic that leads us there to begin with. In order to begin with one blue dies on day one, we must arrive at the conclusion that either there is the actual possibility of there being only one blue, or that one blue thinks another thinks another thinks another thinks another might see no blues. That is what is required to begin the sequence to begin with. Without this, it is difficult to see why that chain of logic should even begin. If noone thinks anyone else thinks anyone else thinks anyone might see no blues, or that there must be more than one blue, provided enough blues are present and can be seen by all, then it is not automatically seen why one should start with one blue would die on day one and work our way up, when that would then have no real chance of occuring. With enough blues, it would seem, although perhaps erroneously I have to add since I'm still not sure, that noone would be waiting to see if anyone offed themselves the first day, and noone would think anyone else was either, or that anyone thinks anyone thinks anyone else is, etc.

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Let's put together the pieces of The Grand Puzzle . (website - now revised)

"Let's define another operator, Sz, which we won't pay any attention to."
"This transformation will automatically make zero equal zero."
"It may be true that zero equals zero -- and that is certainly an equality -- but I don't want to go into the details at this time."

Very promising indeed.

No, it requires a logical reasoning process. You need the principle of associativity: 2+2 = (1+1)+(1+1) = 1+1+1+1 = 4. So if I can't convince you that I can drop the parentheses, using a logical argument, then I can't convince you that 2+2=4. Of course I know I've proven it because of the associativity of addition, but then, I know I've proven this puzzle also. Anyone can say, "but that associativity business is just a claim, I don't believe it" and there's not a lot more to be said about it. So it is with this puzzle, only on a much deeper level.

No, that is not the definition of n. This sounds a bit like my initial criticism of worzel's proof, which was off the mark. The meaning of n is that it may take on an arbitrary value, but it is not the same as n+1 as a result. It is an arbitrary but particular value, and n+1 is one more. Yes, we know all this.

It's not a matter of the number of blues, it's a matter of the degree of metaknowledge. All that happens with more blues is, it's harder to get your head around. That's why worzel is working on a more concise inductive proof, so no one will have to get their "head around" 5 or more blues.

That's true that anyone could say that. The problem is in defining which way we are required to think, not just in order to reach some end we are aiming for, but also to show that no other way of thinking about it is possible, or at least logical, so that no other result can be had. I would think that to be an extremely difficult task, as this thread demonstrates.

The substitution part of 2+2=4 may have closer similarity to this thread if stated as a sequence like 1+(1+(1+(1)))=4, where the logic would follow as #1 thinks (#2 thinks (#3 thinks (#4 might see no blues))). But this would mean that #4 might see no blues, #3 thinks #4 might see no blues, and #2 thinks #3 thinks #4 might see no blues could stand on their own, which they can't, so maybe it would fare better the other way around, where (((1)+1)+1)+1=4, so (((#1) thinks #2) thinks #3) thinks #4 might see no blues. I'm not even sure how to think about that one, though, so it sounds about right, then.

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Let's put together the pieces of The Grand Puzzle . (website - now revised)

"Let's define another operator, Sz, which we won't pay any attention to."
"This transformation will automatically make zero equal zero."
"It may be true that zero equals zero -- and that is certainly an equality -- but I don't want to go into the details at this time."

Indeed. 1+(1+(1+(1))) only equals 4 because 1+(1)=2, but at no time is the answer to 1+(1+(1+(1))) actually 2, it's always 4. So that should seem illogical to you as well, grav

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There are 10 types of people in the world. Those who understand ternary, those who don't, and those waiting for a bus.
If logic doesn't work, then surely it does.

That's true.

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Let's put together the pieces of The Grand Puzzle . (website - now revised)

"Let's define another operator, Sz, which we won't pay any attention to."
"This transformation will automatically make zero equal zero."
"It may be true that zero equals zero -- and that is certainly an equality -- but I don't want to go into the details at this time."

I think you are beginning to understand. All that's left is to realize that there is no "alternative" logic here, there's only what is right and what isn't.

is this thread closed?

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"potential is not accomplisment"

Closed threads have a little padlock image next to them in the thread list, and the Reply buttons inside the threads also have padlocks in the images. And you wouldn't be able to post in them. So, no, the lack of padlocks and the fact that you could post in it at all mean it's not closed. It's just an open thread that nobody's responded to for a while.

thank you delvo

i have spent some hours pouring over proofs offered as solutions in this thread and found that i could not explain an objection that was raised by a friend when i sent him this puzzle.

i am in desperate need of help.

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"potential is not accomplisment"

What was the objection?

We are on pins and needles! Perhaps the objection will relate to the need for this puzzle to cross the line from the "fuzzy" logic of real people into the formal logic of puzzles like this. There is no resolution for that objection, as this puzzle expressly involves the thinking of minds. We say the tribe are all well known to be "perfect logicians", but that's a cheat to set up the puzzle. The puzzle cannot be taken too seriously-- a real tribe of this type might do any number of things, it's a bit like game theory!

Though I don't pretend to understand (or accept) either the puzzle or the logic behind this article (Honey, I doomed the universe), I couldn't help but see a similarity.

http://www.news24.com/News24/Technol...225036,00.html

Unfortunately the article gives no support to this contention, one can only guess what Krauss may be thinking. In the absence of any argument, the stance certainly seems quite unconvincing on the surface. There certainly are things that we can learn about the universe that cause us to lower its life expectancy, just as any visit we make to a doctor might lower our own. The article seems to suggest Krauss thinks we could live longer by never going to that doctor, and that is of course a foolish idea so he must mean something different. Perhaps his meaning is indeed something analogous to this puzzle-- that knowledge itself is the danger. But again, there's no argument cited that could make such a connection.

sorry for the delay in responding but my shop has demanded much of my time.

when i gathered up my notes and reviewed the main objections raised to the various "proofs" and presented them again to my attorney he could no longer remember his quibble but was sure it was both as profound and penetrating an insight as you are likely to find.

with that i must thank you for a great mental exercise.

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"potential is not accomplisment"