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 Posted: Fri Aug 18, 2006 11:23 pm
Imagine that at an hour before midnight, there is an infinite collection of numbered balls and an urn of infinite capacity. Balls numbered 1-10 are put in and ball 1 is removed and thrown away. At half an hour before midnight, balls 11-20 are put in and ball 2 is removed. And so on, halving the time until next step: balls 21-30 put in at fifteen minutes before midnight and ball 3 removed, etc. Please ignore the physical limitations with this scenario.
How many balls are in the urn at midnight?
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Posted: Tue Aug 22, 2006 7:44 pm
ok so, ignoring the physical limitations, you have an infinite number of time intervals at which you place exactly 19 balls into the urn and each time you take one away. However, if you do this for an infinite number of intervals then you take away an infinite number of balls. So no matter how many balls you put in the urn you are still subtracting an infinte number of balls, which should take you back to zero.
Even though there are 19 more balls every time, there is only one infinity, and infinity - infinity = 0.
There are 0 balls in the urn by midnight.
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Posted: Tue Aug 22, 2006 8:48 pm
poweroutage However, if you do this for an infinite number of intervals then you take away an infinite number of balls. So no matter how many balls you put in the urn you are still subtracting an infinte number of balls, which should take you back to zero. Even though there are 19 more balls every time, there is only one infinity, and infinity - infinity = 0. This reasoning is incorrect. As for the conclusion, no comment at this time.
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Posted: Tue Aug 22, 2006 9:33 pm
VorpalNeko Imagine that at an hour before midnight, there is an infinite collection of numbered balls and an urn of infinite capacity. Balls numbered 1-10 are put in and ball 1 is removed and thrown away. At half an hour before midnight, balls 11-20 are put in and ball 2 is removed. And so on, halving the time until next step: balls 21-30 put in at fifteen minutes before midnight and ball 3 removed, etc. Please ignore the physical limitations with this scenario. How many balls are in the urn at midnight? Well, building a little bit on poweroutage's initial assumptions, we can see that there are an infinite number of time intervals before midnite. So, at any given instant before midnite, we can check and see which balls are inside the urn, and which have been taken away. Let's say that to start with, we have *all* of the balls outside of the urn. At midnite, each of them will *at least* have entered the urn. But we can also pin down the exact instant at which each ball will have been taken out, just as much as we know which have been put in. But if we have an infinite number of time intervals, each ball will have been taken out at some time. So it seems that while, at each instant of time, there are more balls in the urn than have been taken out, at midnite there will be no balls inside the urn. As far as I can tell, the only possible answers are 0 and all of them, however.
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Posted: Thu Aug 24, 2006 1:16 pm
Swordmaster Dragon But if we have an infinite number of time intervals, each ball will have been taken out at some time. So it seems that while, at each instant of time, there are more balls in the urn than have been taken out, at midnite there will be no balls inside the urn. Right. The most important part is not just that there will be infinitely many removals, but that all balls will be removed because of the way the removing operaton is defined. Swordmaster Dragon As far as I can tell, the only possible answers are 0 and all of them, however. By adjusting your removing function, you can conceivably make any subset to be there at midnight. This is kind of interesting. This question is a favorite of mine; usually, when I ask it, most people think in terms of limits to conclude that there are infinitely many balls in the urn at midnight, which would be incorrect.
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Posted: Thu Aug 24, 2006 6:33 pm
I knew it looked familiar, in theory. This seems a lot like Riemann's process of allowing a conditionally convergent series to have any (extended) real inf, sup, lower limit, or upper limit by defining a clear-cut process.
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Posted: Fri Jan 05, 2007 12:30 pm
VorpalNeko usually, when I ask it, most people think in terms of limits to conclude that there are infinitely many balls in the urn at midnight, which would be incorrect. but, but gonk .... how can so many people get it wrong when the pattern is so intuitive!? (it just goes to show that traditional mathematical education is flawed) I reached the right conclusion within 20 seconds...
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Posted: Sat Jan 13, 2007 5:37 am
BloodlvsTxBvtterflies VorpalNeko usually, when I ask it, most people think in terms of limits to conclude that there are infinitely many balls in the urn at midnight, which would be incorrect. but, but gonk .... how can so many people get it wrong when the pattern is so intuitive!? (it just goes to show that traditional mathematical education is flawed) I reached the right conclusion within 20 seconds... You start off with 9 == 10-1 balls. Next you have 18 == 20-2 == 10*2-1*2 balls. Then you have 27 == 30-3 == 10*3-1*3 ... After n steps you will have 10n-n. As 10n increase faster than n, as n tends towards infinity, so does 10n-n. And we have an infinite number of points in time at which we're removing balls. Or, for an even simpler explanation. Each time we increase the number of balls by 9. We do this an infinite number of times. Therefore there are an infinite number of balls. Personally, I think my version is far more intuitive.
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Posted: Sun Jan 14, 2007 12:05 pm
Dave the lost BloodlvsTxBvtterflies VorpalNeko usually, when I ask it, most people think in terms of limits to conclude that there are infinitely many balls in the urn at midnight, which would be incorrect. but, but gonk .... how can so many people get it wrong when the pattern is so intuitive!? (it just goes to show that traditional mathematical education is flawed) I reached the right conclusion within 20 seconds... You start off with 9 == 10-1 balls. Next you have 18 == 20-2 == 10*2-1*2 balls. Then you have 27 == 30-3 == 10*3-1*3 ... After n steps you will have 10n-n. As 10n increase faster than n, as n tends towards infinity, so does 10n-n. And we have an infinite number of points in time at which we're removing balls. Or, for an even simpler explanation. Each time we increase the number of balls by 9. We do this an infinite number of times. Therefore there are an infinite number of balls. Personally, I think my version is far more intuitive. But your version, however intuitive, must be incorrect. Every ball is eventually removed. As VorpalNeko said, limits don't give you the correct answer.
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Posted: Sun Jan 14, 2007 10:47 pm
As far as I know, the answer to this riddle is "undefined."
That is because both the number of balls taken out and the number of balls put in are divergating rows and we can't work with 2 divergating rows: we know absolutely nothing about the sum, difference or even term-term product (how do you call that in English) of 2 such rows.
Examples
(A_n)_n := 1, 2, 3, 4, 5, ...
(B_n)_n := -1, -2, -3, -4, -5, ...
(C_n)_n := 0, 1, 0, 2, 0, 3, 0, 4, 0, 5, ...
(D_n)_n := 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, ...
A, B, C, D all divergate
A+A divergates, A+B convergates
A-B divergates, A-A convergates
A*C divergates, D*C convergates
Correct me off course if I'm wrong.
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Posted: Wed Jan 17, 2007 12:53 am
The answer is we never reach midnight. It is possible to reach 11:59.99999999999999999999999999, et cetera, but midnight itself will never happen.
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Posted: Wed Jan 17, 2007 10:19 pm
Layra-chan Dave the lost BloodlvsTxBvtterflies VorpalNeko usually, when I ask it, most people think in terms of limits to conclude that there are infinitely many balls in the urn at midnight, which would be incorrect. but, but gonk .... how can so many people get it wrong when the pattern is so intuitive!? (it just goes to show that traditional mathematical education is flawed) I reached the right conclusion within 20 seconds... You start off with 9 == 10-1 balls. Next you have 18 == 20-2 == 10*2-1*2 balls. Then you have 27 == 30-3 == 10*3-1*3 ... After n steps you will have 10n-n. As 10n increase faster than n, as n tends towards infinity, so does 10n-n. And we have an infinite number of points in time at which we're removing balls. Or, for an even simpler explanation. Each time we increase the number of balls by 9. We do this an infinite number of times. Therefore there are an infinite number of balls. Personally, I think my version is far more intuitive. But your version, however intuitive, must be incorrect. Every ball is eventually removed. As VorpalNeko said, limits don't give you the correct answer. Well, it was in response to BloodlvsTxBvtterflies when he said that the answer (referring to all removed therefore none in) was so intuitive, thus I wished to point out that a limits approach is far more intuitive to most people. Cougar Draven The answer is we never reach midnight. It is possible to reach 11:59.99999999999999999999999999, et cetera, but midnight itself will never happen. http://en.wikipedia.org/wiki/Xeno's_paradox#The_dichotomy_paradox
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Posted: Sat Jan 20, 2007 6:17 am
Dave the lost Cougar Draven The answer is we never reach midnight. It is possible to reach 11:59.99999999999999999999999999, et cetera, but midnight itself will never happen. http://en.wikipedia.org/wiki/Xeno's_paradox#The_dichotomy_paradox Indeed. I'd forgotten that was the title.
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Posted: Tue Jan 23, 2007 11:21 pm
it said to ignore all limits (admittedly this approach is woefully impractical in reality; but this is a thought experiment, not a physical one, so the limits of reality simply do not apply)
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Posted: Sun Feb 18, 2007 3:24 pm
The crux of this problem is that when dealing with operations on infinities, the order or way in which you do things is important.
Note that the original problem specified removing the lowest number ball in the urn, i.e., the balls removed were 1, 2, 3, etc..... Now suppose that after putting 10 balls in at each timestep, we took out instead the highest number, so we remove 10, then 20, then 30, etc. What is interesting is that even though at each time step we are putting in the same number of balls as before, now the urn will have an infinite number of balls at midnight!
What this means is that Infinity - infinity = infinity - infinity does not always hold! Crazy, huhn?
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