Infinite balls exchanged between buckets problem

**slightlyoddguy** · May 15th 2008, 04:44 AM

I apologize beforehand if this problem isn't particularly clear. It's an extra credit problem for pre-cal, and the correct answer (and my explanation of that answer) will make up for a lot of missed work (I was absent quite a bit). Any help would be greatly appreciated.

There are two buckets, each of which can hold an infinite number of balls. They are arranged something like this (I apologize for the somewhat crude drawing):

One of the buckets is full of an infinite number of balls, each of which is marked with a number (1,2,3, and so on). There is a mechanism able to move the balls from one bucket to the other (as shown in the picture). The mechanism does this instantly.

At 12:00, balls 1 and 2 are moved from bucket A to bucket B, and ball 1 is moved back to bucket A from bucket B (remember that this happens instantly).

At 12:30, balls 3, 4, and 5 are moved from bucket A to bucket B, and ball 2 is moved back to bucket A from bucket B.

At 12:45, balls 6, 7, 8, and 9 are moved from bucket A to bucket B, and ball 3 is moved back to bucket A from bucket B.

At 12:57 and 30 seconds, balls 10, 11, 12, 13, and 14 are moved from bucket A to bucket B, and ball 4 is moved back to bucket A from bucket B.

The interval between movements is always half that of the previous interval, and the balls continue to move in the pattern given above. At 1:00, there are no balls left in bucket B. Why?

Feel free to ask for any clarification if needed.

**topsquark** · May 15th 2008, 04:51 AM

Originally Posted by slightlyoddguy

I apologize beforehand if this problem isn't particularly clear. It's an extra credit problem for pre-cal, and the correct answer (and my explanation of that answer) will make up for a lot of missed work (I was absent quite a bit). Any help would be greatly appreciated.

There are two buckets, each of which can hold an infinite number of balls. They are arranged something like this (I apologize for the somewhat crude drawing):

One of the buckets is full of an infinite number of balls, each of which is marked with a number (1,2,3, and so on). There is a mechanism able to move the balls from one bucket to the other (as shown in the picture). The mechanism does this instantly.

At 12:00, balls 1 and 2 are moved from bucket A to bucket B, and ball 1 is moved back to bucket A from bucket B (remember that this happens instantly).

At 12:30, balls 3, 4, and 5 are moved from bucket A to bucket B, and ball 2 is moved back to bucket A from bucket B.

At 12:45, balls 6, 7, 8, and 9 are moved from bucket A to bucket B, and ball 3 is moved back to bucket A from bucket B.

At 12:57 and 30 seconds, balls 10, 11, 12, 13, and 14 are moved from bucket A to bucket B, and ball 4 is moved back to bucket A from bucket B.

The interval between movements is always half that of the previous interval, and the balls continue to move in the pattern given above. At 1:00, there are no balls left in bucket A. Why?

Feel free to ask for any clarification if needed.

I disagree. There are an infinite number of exchanges before 1:00 so an infinite number of balls has left bucket A. However the balls put back in bucket A are never removed again. So there are an infinite number of balls left in A.

The question in my mind is how many balls are in bucket B at 1:00? I don't know how to answer that one.

-Dan

**slightlyoddguy** · May 15th 2008, 05:19 AM

Originally Posted by topsquark

I disagree. There are an infinite number of exchanges before 1:00 so an infinite number of balls has left bucket A. However the balls put back in bucket A are never removed again. So there are an infinite number of balls left in A.

Why are the balls put back into bucket A never removed again? It seems to me that this has something to do with what happens at 1:00.

The question in my mind is how many balls are in bucket B at 1:00? I don't know how to answer that one.

The answer to that question is 0 -- our teacher told us that much. That isn't to say that it's true because he says it's true (there must be a way to prove it mathematically), but I included it to provide as much information as possible. In any case, whatever process one goes through to answer the question of "how many" will answer the question of "why" as well.

Intuitively, I am inclined to think that the clock simply never reaches 1:00, as there are an infinite number of intervals between 12:00 and 1:00. This is similar to the Achilles and the tortoise paradox. The problem comes, however, when I realize that the clock must reach 1:00, since time continues to pass regardless.

**Soroban** · May 15th 2008, 05:22 AM

Hello, slightlyoddguy!

I agree with Dan . . .

At 1:00, all the balls are in bucket A and bucket B is empty.

After $t = 0$ hours, ball #1 is returned to bucket A.

After $t\:=\:1-\frac{1}{2}\:=\:\frac{1}{2}$ hours, ball #2 is returned to bucket A.

After $t\:=\:1 - \frac{1}{2^2} \:=\: \frac{3}{4}$ hours, ball #3 is returned to bucket A.

After $t\:=\:1 - \frac{1}{2^3} \:=\:\frac{7}{8}$ hours, ball #4 is returned to bucket A.

. . and so on . . .

After $t\:=\:1 - \frac{1}{2^n} \:=\:\frac{2^n-1}{2^n}$ hours, ball $n+1$ is returned to bucket A.

Since $n\to\infty$ , all balls are returned to bucket A.

**slightlyoddguy** · May 15th 2008, 05:30 AM

Originally Posted by Soroban

Hello, slightlyoddguy!

I agree with Dan . . .

At 1:00, all the balls are in bucket A and bucket B is empty.

After $t = 0$ hours, ball #1 is returned to bucket A.

After $t\:=\:1-\frac{1}{2}\:=\:\frac{1}{2}$ hours, ball #2 is returned to bucket A.

After $t\:=\:1 - \frac{1}{2^2} \:=\: \frac{3}{4}$ hours, ball #3 is returned to bucket A.

After $t\:=\:1 - \frac{1}{2^3} \:=\:\frac{7}{8}$ hours, ball #4 is returned to bucket A.

. . and so on . . .

After $t\:=\:1 - \frac{1}{2^n} \:=\:\frac{2^n-1}{2^n}$ hours, ball $n+1$ is returned to bucket A.

Since $n\to\infty$ , all balls are returned to bucket A.

Yes! Thanks so much. I'm sorry, I was mistaken about the original problem -- there aren't supposed to be any balls left in bucket B. And, of course, you've shown why this is the case. Again, sorry for the mistake, but this is exactly the solution I was looking for.

**slightlyoddguy** · May 20th 2008, 06:07 AM

Sorry for bumping this (I know it's been a bit since I posted it), but I need clarification on one point:

Since $n\to\infty$ , all balls are returned to bucket A.

I can grasp it in a sort of intuitive sense, but I'm not quite sure how to put it mathematically.

**slightlyoddguy** · May 23rd 2008, 04:49 AM

If anyone could help within the next few hours, that'd be great.

**mr fantastic** · May 23rd 2008, 05:04 AM

Originally Posted by Soroban

Hello, slightlyoddguy!

I agree with Dan . . .

At 1:00, all the balls are in bucket A and bucket B is empty.

After $t = 0$ hours, ball #1 is returned to bucket A.

After $t\:=\:1-\frac{1}{2}\:=\:\frac{1}{2}$ hours, ball #2 is returned to bucket A.

After $t\:=\:1 - \frac{1}{2^2} \:=\: \frac{3}{4}$ hours, ball #3 is returned to bucket A.

After $t\:=\:1 - \frac{1}{2^3} \:=\:\frac{7}{8}$ hours, ball #4 is returned to bucket A.

. . and so on . . .

After $t\:=\:1 - \frac{1}{2^n} \:=\:\frac{2^n-1}{2^n}$ hours, ball $n+1$ is returned to bucket A.

Since $n\to\infty$ , all balls are returned to bucket A.

I'm inclined to think that at 1:00 there is an infinite number of balls in both buckets ........ I might have time for a proof later (either that or take my foot out of my mouth).

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