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(7): uh oh, here I go again. by Tristan)
Well that's two, separate distinct infinite operations. One has to finish before the other one starts. Yes, I agree the result is empty, but I can't really agree that it's a single infinite sequence of operations, where the later steps affect members added in the earlier steps. ...which is what I meant.
When you look at the 1,0,1,0... case, it could be argued as follows. "The problem is equivalent to the following: step 1: add the integer 1, then remove it. Step 2: same thing. And so on. The set is empty at the end of each step, so it must be empty at the end of an infinite number of steps." But we both know that argument won't fly--the result of the 1,0,1,0... example is indeterminite. So, I can't buy the equivalence argument that you presented ("the effect is the same when the addition and deletion parts are separate"), for the same reason.
There's something just not right about an infinity of operations that consist of adding and removing elements from a set.
re(8): uh oh, here I go again.
