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)

With all due respect I think we are giving comparative infinity problems a bad name here. While it is valuable and significant to understand that some sequences approach infinity at different rates or ways (e.g. the sum of positive powers of two approaches infinity rapidly, while say the sum of natural reciprocals approches at a snail's pace). This problem leads us to reject hypothetical infinity scenarios altogether.

Since this problem has no real basis with which to work from it is possible to abstract it in any way we wish. I would choose to disregard the numbering of the marbles altogether and choose to imagine that for each iteration of the process I have put 9 marbles in the container and thrown one away. At the end of the minute - even if I have somehow managed to "process" all the marbles - I still have nine times as many in the container as I do in the trash can.

I remember a problem which involved equal amounts of water into two glasses and pouring the water in one of the glasses equally into two other glasses etc. etc. ad infinitum. Until someone brought up the point that eventually you are left with one water molecule which you cannot split (or if you could you wouldn't have water anymore). The point is that our mathematical models must still reflect (at least to some degree) the bounds of reality. And if we think that we can put ten marbles in a container and then go digging around for the one numbered 21 and throw that one away all within a two-millionth of a second - then someone should go check on the dilithilum crystals.

Posted by Eric on 2003-09-08 23:34:58