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?

1) There are an infinite amount of marbles in the bag.

2) After the minute there are no natural numbers on the marbles. Instead, there are some remarcable makings on the marbles; 100..., 200...., 300...., etc. Each marking consists on a natural numer followed by an infinite number of 0s. Moreover, there are an infinite number of marbles with each marking.

3) The situation inside the bag is is identical to B after 31 and 50 seconds. After the minute is over, the situation inside the bag is differente from both A and B because of the unique ball markings. The situation outside the bag is identical to A after 31 and 50 seconds. After the minute is over the situation outside the bag is identical to A;  {10,20,30,...}.

Posted by ajosin on 2005-06-22 22:12:46