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**)

Consider a system of infinite numbers. Call the base unit of infinity w. Unlike other definitions of infinity, w is different from w+1 and 2*w. In particular 2*w>w+1>w. Also w is defined to be a multiple of every finite integer, but some infinities like w-1 and w+1 are relatively prime to every finite integer.

The process is said to be reapeated an infinite number of times. We can define this instance of infinity as w. Since 10 marbles are introduced in each iteration, there are a total of 10*w marbles. The set of marbles can then be numbered {1,2,3,4,...,w-2,w-1,w,w+1,...,2*w-1,2*w,2*w+1,......,10*w-1,10*w}.

In case A, every multiple of 10 is removed, one in each iteration. In particular the set of marbles {10,20,30,40,...,w-20,w-10,w,w+10,...,2*w-10,2*w,2*w+10,......,10*w-10,10*w} is removed. This leaves 9*w marbles. All the finite numbered marbles which are not multiples of 10 and all the infinite numbered marbles which are not multiples of 10 remain.

In case B, every marble 1 through w are removed, leaving 9*w marbles the smallest of which is w+1. Every finite marble and some infinite marbles were removed, but even larger infinite marbles are left.