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(2): uh oh, here I go again. by Ken Haley)

Ok, in terms of purely numbers, I disagree on one part of your translation that lead to a contradiction.  That part is switching the word "relabel" for "replace."  Replacing a marble/number/element implies changing it for a completely separate marble/number/element.  Relabeling implies that it is the same marble/number/element, but our name for it has changed.

In step 1, we add elements 1-9 and rename element 1 as element 10.  10 is the new name for element 1, but it's true identity is that of element 1.  Element 1 never left the bag/set but instead got renamed 10 as if someone meant to call it 1.0 but forgot the decimal point.
In step 2, we add elements 11-19 and rename element 2 as element 20, but again, element 2 never truly disappears.
And so forth...

Some might ask, "What elements are in the final set?"  It seems that any integer N you choose, there exists no element called N in the set at the end of the minute.  This does not exactly mean that there are no elements in the set, only that there are no elements with finite numbers as names.  Each has been renamed an infinite number of times.

One of the points you have made over and over again is that infinite-infinite, depending how you define it, can be anything.  However, infinite-0 is always infinite.

Posted by Tristan on 2005-06-30 22:30:36