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(5): uh oh, here I go again. by Tristan)
Figure Eights: Cool puzzle. I would have guessed the opposite result--that they form an uncountable set--but the solution posted is irrefutable. Very clever--I wish I hadn't peeked! Now I'll never know if I'd have gotten it.
By the way, it's been proven that the set of curves in space form a higher order of infinity (aleph-2) than the set of reals (aleph-1). I've never seen the proof, though--have you?
But back to this one. My position has changed since I first started responding to this puzzle. My first objection was that you can't "subtract" infinities and get any meaningful result. And, I still maintain that--but I don't think that's the crux of this problem. Now, I'm saying the issue has to do with performing an infinite sequence of operations on a set, and expecting to be able to declare what the result is. The 1,0,1,0.. case is much simpler, and obviously has no unambigouus answer. So how can we say that this puzzle has one? In fact, I can't think of very many infinite sequences of operations on a set, where you can declare what the result is.
Well...here's a simple one. Start with an empty set. At each step add an integer, starting with 1. At the end, you've got a set with all positive integers. What made that okay? Well--for one thing, we never touched any members in the set once we added them. But, can you think of an infinite sequence of operations that DOES touch members in the set (deleting, replacing, etc.) that has an unambiguous result? One that even you and I wouldn't argue about? <g>
re(6): uh oh, here I go again.
