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 uh oh, here I go again. by Cory Taylor)

It does not produce inconsistent results, Cory.  It produces two different results for two different cases.  It is in fact your method with which you "see" the same thing in cases a) and b) that is inconsistent with set theory.

As I said to Ken Haley, first thing, I strongly encourage you to google cantor, infinite sets, and cardinals.  It's actually quite an interesting read if you find a site that explains it properly.  Not to say that everyone would be interested, but I think you would be interested, or you wouldn't keep on come back to this problem.

The two operations are not at all the same.  It may seem that the only difference is the labelling, and labelling does not matter.  But labelling helps us sort out different objects that would otherwise look exactly the same, but are entirely different.  All the marbles may look the same, but you can't give me two marbles and tell me that you only really gave me the same marble twice.

What's interesting about this problem, is that we need to keep track of both the labels, and the true identities of each marble.  In the end, there is a marble labeled 100000..., but it's really the same marble as the one I first put in.

Posted by Tristan on 2005-06-29 21:56:02