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(8): No Subject by Ken Haley)

I agree that the in-and-out problem has no solution it efectively asks for the last element of the infinite set {1,0,1,0,1,0,...}, which is unknown.

I think that in your 9n problem there is not enough information because you don't describe how to act on the complete set. Operatoins on infinite sets have the curious property that you can describe the finite steps however you want. However, they remain meaningfull and logical because the end result is allways the same. For the sake of the argument, let's accept for the moment that,

{1,2,3,4,.....} - {1,2,3,4,...} = {}

The above operation could be though of as adding one marble and then removing it from the bag over and over again. It can also be though of as adding 100000 marbles, and removing #1 in the first step, adding another 100000 and removing #2, etc. Under the second description it "seems" like you are going to end up with infinite marbles, but that is just an illusion hidden by the fact that you are in the finite part of an infinite operation.

Posted by ajosin on 2005-06-26 04:08:24