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

Ok, I've gotten around to reading this.

I couldn't have defined relabeling better myself.  The way you have explained your paradox is the clearest yet.  I find myself thinking that only the first component in the ordered pair is important when counting the elements (btw, it doesn't really matter what the elements are, but if you don't like relabeling integers, I guess they can't be integers), but I am less inclined to say you've got it all wrong.  If we only considered the first component of each ordered pair, it's pretty easy to show there are infinite elements at the end.

Another point you bring up, is that at no point is an infinite label added.  This point seems to relate to the infinite induction fallacy, which I always find myself stumbling around.  I don't have as much confidence in this area, is what I mean.  Speaking of which, you might be interested in the "Figure Eights" puzzle where you might see me clumsily trying a proof when owl steps in with a much cleverer and more successful one. :-)

So as I was saying, infinite induction means incorrectly making the jump from all finite numbers to infinite numbers.  For example: 1 is finite.  If k is finite, then k+1 is finite.  Therefore infinity is finite.

I think that we're allowed to say that multiplying an integer by 10 or 13 an infinite number of times leads to an infinite number.  I think what we're not allowed to do is--if each step takes exactly 1 second rather than half of the previous step and each step multiplies a number by 10--say that the number ever reaches infinity.  I feel that I already know what your response to this is--that this means that in case B) of Marbles Bonanza I should also have marbles with infinite labels.  I'm really not clear on how to avoid infinite induction in some cases, so I'm not going to say anymore for now.

I was just thinking... what if instead of adding a zero to the end of each label, we add two zeroes, and erase the first of all the zeroes?  Just a random thought...

Going back and forth between labels 0 and 1 an infinite number of times results in one element, but I can't say what it's label is... either 1 or 0.

Of course I'm having fun!  Set theory, especially infinite set theory, is very interesting (which is why I like to recommend googling it).  The debates often involved are quite interesting and I usually learn lots of new things.

Posted by Tristan on 2005-07-02 07:02:19