You have an empty container, and an infinite number of marbles, each numbered with an integer from 1 to infinity.

At the start of the minute, you put marbles 1 - 10 into the container, then remove one of the marbles and throw it away. You do this again after 30 seconds, then again in 15 seconds, and again in 7.5 seconds. You continuosly repeat this process, each time after half as long an interval as the time before, until the minute is over.

Since this means that you repeated the process an infinite number of times, you have "processed" all your marbles.

How many marbles are in the container at the end of the minute if for every repetition (numbered N)

A. You remove the marble numbered (10 * N)

B. You remove the marble numbered (N)

In the math of infinity(ies), in this case the infinity that corresponds to the number of integers that exist (Aleph-0), the answer in both cases is still infinity. It holds true (simplified) that infinity + infinity = infinity. More technically, to show that the number of marbles in the container is infinite, you (must?) show a 1 to 1 mapping between the marbles and the positive integers. The marbles in the container (as long as they are not ALL removed at each interval) can be assigned sequential integer numbers, creating this mapping.
So, putting ANY number of marbles in the container an infinite number of times results in a container with an infinite number of marbles in it.

Posted by Andy on 2003-11-18 17:50:17