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In order to properly answer this problem, we must first develop a theory of infinite marble operations capable of analyzing the situation after the end of the minute, for of course this problem cannot practically be carried out.
The question of what happens at any finite time after we begin operations on the marbles is intuitively determinable by practice, and we wish that there should be some continuity between the finite cases and the infinite (post-minute) case. We will take the following approach: if a marble is at rest either inside the bag or out of it after a certain finite time and is untouched by whatever algorithm we are following, then after the minute is up, the marble will continue to be there. So for each marble, to determine what state it is in at the infinite mark, we must determine at what finite step it is last "touched" and see how it is left. In the case that there is no finite step at which we cease to operate on a marble, we can consider its number and its position seperately: if at any finite step we stop moving the position on the marble, it is considered to rest there. If there is no finite step on which we cease to renumber the marble, then we consider the number on the marble at the minute mark to be indeterminate.
So now we analyze the problem as follows:
For scenario A, every marble that is not divisible by 10 is left untouched after the step that puts it in the bag, so we know there are an infinite number of marbles inside the bag. Also, every marble divisible by 10 is left untouched after the step that takes it out of the bag, so there are infinitely many marbles outside of the bag.
For scenario B, for every marble, there is a finite step when it is put into the bag as well as a finite step when it is taken out at which point it is no longer touched, so we can ascertain that at the one minute mark, all marbles are outside the bag.
For the new scenario, every marble that is originally divisible by 10 will never be touched. So these we know will end outside the bag. Every other marble however will be added into the bag at a finite step and never be removed, so they will all be in the bag at the 1 minute mark. Regarding the number on each ball, though, we can see that by our marble theory, they are undefined, because ball N is renumbered on step N, then step 10N, then step 100N, and so on. (Theoretically, each ball might end up with their original number followed by infinitely many zeroes.)
So the result is after 31 seconds, 50 seconds, or indeed after any finite step, scenario C is identical to scenario A outside the bag (having multiples of 10 outside up to 10N, and then all remaining numbers), but identical to scenario B inside the bag (having all balls between N+1 and 10N). After the 1 minute mark, however, the situation is still identical to scenario A outside the bag (all multiples of 10), but inside the bag, unlike in scenario B, there are infinitely many marbles of indeterminate number! |
