After processing an infinite number of marbles, not once but
twice in solving the puzzle
Marbles Bonanza, you've grown rather tired of moving all these marbles around. Nevertheless, it is your duty to press on forward and try a third experiment. This time, though, you decide that you'll cut down on the amount of work by not
removing any balls ever.
At the start of the minute, you put marbles 1-9 in the bag, and then add a 0 to the end of marble 1 (so that you now have duplicate marble 10s, one in the bag that you just modified, one out). Therefore you now have marbles numbered 2-10 in the bag, as in scenario B of the previous puzzle, and marble 10 outside the bag as in scenario A. 30 seconds later, you put marbles 11-19 in the bag, and add a 0 to marble 2, so that now you have two marbles numbered 20 - one in, one out. You continuously repeat this process, with each interval half as long as the one before. In general, for the nth operation, you put marbles 10n-9 to 10n-1 in the bag, and add a zero to marble n in the bag, so that it becomes marble 10n in the bag.
- How many marbles are in the bag at the end of the minute?
- What are the numbers on the marbles ?
- Is the situation inside the bag identical to either of the previous two problems after 31 seconds? 50 seconds? at the end of the minute? How about the situation outside the bag?
How clever! At every point in time up to but not including the one-minute mark, there were a finite number of marbles, each with a finite integer marked on it. But at the one-minute mark (and not at instant before), BOOM, we have a BIG BANG, where there are suddenly an infinite number of marbles, each one with an infinite number of digits on it. (We assume, that the person writing the zeros on the marbles makes each zero half the size of the previous one, so they'll all fit.)
The interesting thing is that this is the same as scenario in which you start by placing #1-9 in the bag, then adding #10-19 and taking out #1, etc. At the end of each step, the marbles in the bag have the same count and labeling as we have in this puzzle! No one could reasonably assert that the bag is empty at the end of one minute in this puzzle, so how could anyone assert that the bag would be empty in the previous one? And yet, in the previous puzzle, it can genuinely be argued that every marble that went in came back out at a later time--so the bag must be empty.
This is why I assert that the earlier problem is meaningless--mainly because different valid approaches lead to wildly different conclusions. But why is it meaningless? I think it's because in that analysis we're adding an infinite number of marbles to the bag and removing an infinite number, and trying to assert that that leaves a finite value (namely, 0). That just can't make sense--you can't subtract infinities and end up with anything meaningful, just like you can't divide 0 by 0.
No Subject
