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

How about this:

The first step takes 30 seconds, and every step takes half the time of the previous.  After a minute, another infinite set of steps is performed with the same half interval method.

Minute 1:
Add marble N on the Nth step.
Minute 2:
Remove marble N on the Nth step.

You see that I have separated the adding and deleting to create a case that, I think, you will agree results in an empty set.  Of course, I can't deny that in the case presented in Marbles Bonanza, the adding and deleting are not separated and are instead done simultaneously.

I'm going to argue that in Marbles Bonanza, the effect is the same as when the addition and deletion parts are separate.  The command to remove each marble is after the command to add each marble.  After deletion, no marble is again re-added.  The two parts do not really affect each other, so it doesn't matter if the deletion part is done a minute later.

In the case of 1,0,1,0,... it's pretty clear that we can't separate the adding and deleting parts of the algorithm.  We can't re-add any element before having removed it.

Posted by Tristan on 2005-07-04 22:10:14