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.