Suppose there is an infinite number of small balls, all of which have been uniquely numbered from 1 upwards. Furthermore, you have an infinitely large bucket. That is to say, it can be made as large as necessary. You decide to fill the bucket by throwing in all the balls, in order. Starting from 1, every minute you throw in two balls. But every minute, an impish sprite takes one ball back out, always extracting the lowest-numbered ball in the bucket.(adsbygoogle = window.adsbygoogle || []).push({});

For example,

1st minute: You throw in Ball 1 and Ball 2.

Impish sprite extracts Ball 1.

2nd minute: You throw in Ball 3 and Ball 4.

Impish sprite extracts Ball 2.

3rd minute: You throw in Ball 5 and Ball 6.

Impish sprite extracts Ball 3.

and so on...

QUESTION: After an infinite amount of time has elapsed, how many balls are in the bucket?

Argument 1: There is an infinite number of balls in the bucket. After 1 minute there is 1 ball. After 2 minutes there are 2 balls. After 3 minutes there are 3 balls, etc.

Argument 2: There are no balls in the bucket. If there are some balls in the bucket, what is the number of the lowest-numbered ball? It can't be Ball 1; that was extracted after 1 minute. Similarly, it can't be Ball 2; that was extracted after 2 minutes. It can't be Ball 3; that was extracted after 3 minutes, etc.

(If the phrase 'after an infinite amount of time has elapsed' bothers you, then we can change the problem so that the 1st put-in-and-take-out operation is completed in 1/2 minute, the 2nd operation is completed in 1/4 minute, the 3rd in 1/8 minute, and so on. Now you can ask the question after 60 seconds, and "infinite time" is not longer an issue.)

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# Impish Sprite

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