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(5): uh oh, here I go again. by Tristan)
I don't understand. An "element" is defined to be a member of a set. It can be an integer, or anything else. I'm just asking you to say what the elements are, in the abstract case. They can be anything you like--but pick something. I've tried twice: first, I picked integers, then I picked ordered pairs of integers. Your turn. :-)
BTW, if the elements ARE integers, you certainly can add 1 and 2 together to get 3. That's part of the definition of add for integers. But if they're ordered pairs, I agree...you can't add them--unless we define what it means to add ordered pairs. It doesn't matter though--for the sake of our discussion we don't need to define an operation for "add", but we do need an operation for "relabel". That's what will complete the isomorphism to our present problem.
Looking forward to the rest of your response.
Edited on July 2, 2005, 5:21 am
re(6): uh oh, here I go again.
