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?
(In reply to
re: uh oh, here I go again. by Tristan)
Tristan,
Since you can't really have an infinite number of marbles and an infinitely large bag, how would you state the problem in terms of purely numbers and sets? I gave it my best shot (two posts back), and arrived at what certainly appears to be a contradiction. I'd like to hear your argument in those terms. What part of my translation into pure numbers and sets do you disagree with? How would you do it differently?
By the way, nothing in a Google search of Cantor's work and infinite sets talks about applying an infinite number of operations on a set and arriving at any meaningful result--at least not that I could find. Please let me know if you've seen something that I've missed.
re(2): uh oh, here I go again.
