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): No Subject by Tristan)

I would agree, Tristan, that the limit is infinite, as we approach the one-minute mark.  But, that would be my same answer to part B of the original Marbles Bonanza. (The limit of an arithmetically increasing series is infinity.) And, I would also assert, in that case, that it's impossible to say what the final number of marbles is at the one-minute mark.  But, in any case, how can anyone claim it's ZERO, when the limit is infinite?

Here's a similar problem with no meaningful answer, and I think you'll agree.  Only one marble and one bag.  At the 1/2 minute mark, I put the marble in the bag.  (By the way, it's marble #1, in case that matters, and I promise not to relabel it.) 

At the 3/4 minute mark I take it out.  I keep repeating this an infinite number of times: in at 7/8 minute, out at 15/16 minute, etc.  How many marbles are in the bag at the end of one minute?  One? Zero? Somewhere in between?  There's no meaningful answer to this much simpler case, so how can we assert that there is a meaningful answer to the more complex one?  (Incidentally, in this case, there's no limit either.)

So, why again, does my problem statement not have enough information?  Why does it make a difference how I get from one step to the next, as long as you can see, clearly and unambiguously, the result of each step?  If I told you the bag had one marble (marked #1) at the end of step 1, and two marbles (marked #1 and #2) at the end of step 2, etc., would that be enough information to determine an answer? Or would you still need to know that I'm not sneaking marbles in and out, or relabeling them while you're not looking?

 

Edited on June 26, 2005, 2:23 am

Edited on June 26, 2005, 2:32 am

Posted by Ken Haley on 2005-06-26 01:49:34