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(3): No Subject by ajosin)

Let's review, bypassing all the notational jargon: You put a set of marbles in the bag numbered with all the positive integers. I relabeled some of the marbles (all the even numbered ones), but I didn't add or remove any--the number of marbles in the bag was left unchanged. You then removed a set of marbles from the bag numbered with all the positive integers. When you were done, the bag still had an infinite number of marbles. Isn't that true?

So, it would appear that taking the set of positive integers and removing from it the set of positive integers does not necessarily leave the null set.

Now, actually that isn't what I showed, because of the relabeling. Technically, what I showed is that you can take a set that is in one-to-one correspondence with the set of positive integers, and remove from it a set that is also in one-to-one corresepondence with the set of positive integers, and leave behind an infinite number of elements in the original set. That's not quite the same as removing the set of positive integers from the set of positive integers. And, maybe this is the point you're trying to make, in different words.

But this puzzle is about **counting** marbles. And for that purpose, the labeling should be irrelevant. If I ask you how many people are in a room, do you have to know their names before you can give me an answer?

*Edited on ***June 25, 2005, 5:38 am**