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This is an 'infinite task' kind of problem, and it takes place given one minute of time. Let's say you have an urn with infinite capacity and an infinite number of balls, marked 1,2,3.... At the start of the clock there are no balls in the urn. After 30 seconds you place balls 1 through 10 in the urn and then remove the lowest numbered ball, which is 1. At 45 seconds (half the remaining time) you place balls 11 through 20 in the urn and remove the lowest numbered ball, which is 2. So at every step you are adding a net amount of 9 balls. Keep going like this, adding and removing balls every time the clock counts down to half the remaining time. At the completion of one minute, how many balls are in the urn?
(Note: It might be objected that because there are an infinite number of balls, the task cannot be completed. But hypothetically it can [Zeno's dichotomy paradox].)
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* Doesn't it depend on the speed the monkey (knowledgeable and cultured, reading the thread) is operating at?
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rmberwin wrote:
(Note: It might be objected that because there are an infinite number of balls, the task cannot be completed. But hypothetically it can [Zeno's dichotomy paradox].)
I reckon practically the task can be completed, but hypothetically it can not be. The first action starts at 30 sec, and hypothetically it is assumed that the time lapsed in putting in the ten balls and taking out one ball is nothing, so the next action begins at 31st second. Practically, there will be some time lapse which after some iterations will cross the 60 second mark. So, hypothetically there is always some time left in completing the set of actions, in billionth part of a second or trillionth part of a second or............... We are responsible for what we are, and whatever we wish ourselves to be, we have the power to make ourselves. ~ Swami Vivekanand


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srirr wrote:rmberwin wrote:
(Note: It might be objected that because there are an infinite number of balls, the task cannot be completed. But hypothetically it can [Zeno's dichotomy paradox].)
I reckon practically the task can be completed, but hypothetically it can not be. The first action starts at 30 sec, and hypothetically it is assumed that the time lapsed in putting in the ten balls and taking out one ball is nothing, so the next action begins at 31st second. Practically, there will be some time lapse which after some iterations will cross the 60 second mark. So, hypothetically there is always some time left in completing the set of actions, in billionth part of a second or trillionth part of a second or............... I should have added that the time to complete each action should be counted as zero. I won't get into an explanation of Zeno's dichotomy because: 1) I'm not an expert, and 2) There is still some controversy about it. But the paradox states that you cannot walk across the room, because first you have to cross half the distance, then half the remaining, half the remaining, etc., and you can't complete an infinite sequence of events. But in calculus such infinite sums are computed all the time, not by actually calculating them, but by taking the limit of the series of partial sums. In this case, 1/2 + 1/4 + 1/8 + ... = 1. And needless to say, we can cross the room, so the paradox must be confronted, not only in the physical realm, but in the mathematical. But the question remains, How many ball are left in the urn at the completion of the task? A thing of beauty is a joy for ever.


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srirr wrote:[quote=rmberwin] hypothetically there is always some time left in completing the set of actions, in billionth part of a second or trillionth part of a second or............... Quite. So, at an infinitely small fraction of a second before the oneminute mark is reached, there will be an infinite number of balls in the urn. Of course, since that infinitely small fraction of a second can be halved (and another 9 balls added), this can go on infinitely.


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tunaafi wrote:Quite. So, at an infinitely small fraction of a second before the oneminute mark is reached, there will be an infinite number of balls in the urn. Of course, since that infinitely small fraction of a second can be halved (and another 9 balls added), this can go on infinitely. The irony of the situation is that whatever ball you take there's a precise moment at which it is removed. So at the end there are no balls in the urn.
აბა ყვავებს ვინ დაიჭერს, კარგო? გალიაში ბულბულები ზიან.


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