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(12): No Subject (Summary) by Ken Haley)
I've resisted so long. I'm so proud of myself. But now back into the abyss :(
While I agree with all the "mathematical" evidence supporting an empty bag under the described operations, I still assert (and I can almost hear Ken agreeing with me) that math is a tool and when it produces inconsistent results then we've mis-applied that tool. While set theory indicates an empty bag or an infinite bag under original conditions 1 and 2, someone without the ability to read would see the same thing happening in two cases with different results. The problem is that, our system of numbering and/or labelling (i.e., how humans have chosen to define "set theory") is what causes the discrepancy - adding 10 marbles and removing one, regardless of what the method used to result in this is, should always result in the same ~number~ of marbles in the bag, however, this magical branch of mathematics allows the illiterate (such as myself, set theory wise) to get very confused by seeing the ~same operation~ (with different labellings) produce vastly different results.
So, my conclusion is that, neither Ken nor ajosin, nor any of the host of previous posters are wrong - it is set theory itself! Stupid set theory :(
uh oh, here I go again.
