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: No Subject by ajosin)

I don't accept your premisses (a) and (b), as obvious as they appear. But let me come back to that. First, let's look at the paradox of the original problem, that I think people are ignoring:

Suppose you have an observer who is watching a person put 10 marbles in a bag, and taking out 1, over and over, etc., as described in the original problem, but he can't see the labels on the balls. Can he, or can he not say how many balls are left at the end of the minute? If he watches a person following procedure A (that supposedly leaves the bag empty) or procedure B (that supposedly leaves the bag with an infinite number of marbles in it), HE CAN'T SEE ANY DIFFERENCE, at any point. Furthermore, if he yells, "STOP" at any given point in time before the one-minute mark, both procedures will result in the exact same number of balls in the bag at that point. So how can the two suddenly become so wildly different in that final infinitesimal time interval just before the one-minute mark??

Now, let's go back to your case (b): "Start with {1,2,3,4,...} and take away {1,2,3,4,...}. You are left with nothing." And let's put those numbers on marbles again. You start by putting all the marbles in the bag. {1,2,3,4...etc.} I then take the bag and place all the odd-numbered marbles in a secret pouch inside the bag. That leaves all the even-numbered marbles in plain sight (although still inside the bag). {2,4,6,8...etc.} I erase and renumber those marbles with exactly half their values--now they're numbered {1,2,3,4...}. I give the bag back to you, and you take out the marbles numbered {1,2,3,4...}, that are in plain sight. Are there zero marbles left? No. I can now show you that there are still an infinite number of marbles {1,3,5,7...} in the bag, that I hid in the secret pouch. Yet, you put in marbles labeled with all the positive integers and took out a set labeled exactly the same way. And I didn't add any marbles with what I did.

So, no...I don't agree that (b) necessarily leaves you with nothing. And by a similar argument, I don't agree with (a) either. You can't subtract infinite sets and get any consistent, reasonable answer.