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(3): uh oh, here I go again. by Tristan)

Tristan:

We're talking about a set of *integers*.  What does it mean to relabel an integer? It can't be done--a 4 is a 4 is a 4. It can't be relabeled as a 40--that's a different integer.  The only way to go from this set of integers {1,2,3} to this one {1,2,4} is to remove the 3 and insert the 4.  You can't relabel the 3 as a 4.  If you do, then the elements you're thinking of aren't integers to begin with, but something that we're "labeling" with integers.

So, maybe that means my suggested correspondence isn't complete, because it won't allow me to model "relabeling" in a way that preserves the original identity of the element.  So, let's try to set up a correspondence that does that:

Let's say each marble corresponds to an ordered pair of integers.  The first number corresponds to the original marble label.  We'll call it the ID.  No two of the pairs that we're working with will ever have the same ID (the first number in the pair).  Now let's define U, the set of all marbles we're dealing with whether in the set B or not.  Oh, and set B corresponds to the bag.  Initially, the all the marbles with their initial labels corresponds to the set of ordered pairs of the form (a,a) where a is any positive integer.  Now let's look at relabeling.  Assume you want to relabel a marble from 10 to 100, and the marble's original label was 1.  That would correspond to replacing the ordered pair (1,10) with (1,100).  This operation can't violate our rule that no two ordered pairs have the same first component, so the operation is safe. And now, we have a correspondence for "relabeling" that doesn't forget the "original identity" of the marbles, which is what your objection to my previous model was.

Now, let's look at what this puzzle asks us to do.

Step one: put marbles 1-9 in the bag and relabel marble 1 as marble 10.
That translates to:
Step one: Put the ordered pairs (1,1), (2,2)...(9,9) in the set and then remove (1,1) and replace it with (1,10).
Step two would translate to put the ordered pairs (10,10) .... (19, 19) in the set, and remove (2,2) and put (2,20) in the set.
Step n would be: Add 9 more pairs (a,a)...(a+9, a+9) to the set, where a is one higher than the highest ID already in the set.  Then remove the ordered pair whose SECOND component has the smallest value, and replace it with an ordered pair with the same first component as the one removed with the second component multiplied by 10.

Now, at any time, we can look at the set of second components of all the ordered pairs in Set B and see that they form a contiguous set of integers after each step in the process. As we apply successive steps in our operation, the smallest integer in this set increases without limit, as does the largest integer (which is growing 10 times faster).  But, never in the process is there an ordered pair whose second element is anything but an integer.  We don't have ordered pairs with second components that are infinitely long series of digits as was asserted in this problem's solution.

And, once again, I see the same paradox.  Each step results in 9 more elements in Set B, so the set's count should be infinite after an infinite number of steps.  But every ordered pair with any given second component is eventually removed from the set, and no pair with that 2nd component or smaller is ever added back. So, the set must end up empty, since any subset of the positive integers has to have a smallest member, and Set B doesn't.

That may have been hard to follow, but read it carefully, and find a flaw if you can.

If you still disagree, show me (a) where I've made an analytical error, or (b) why my correspondence is still wrong.  In the latter case, can you define a correspondence with pure numbers that you believe works, and supports your side of this argument?  In other words, exactly what are the elements in our set, if not integers, ordered pairs, or something similar? 

------

Reply to some of your other remarks:

"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." Yes, but never did a relabeling ever result in anything but an integer.  If I relabel a marble an infinite number of times back and forth between 1 and 0, would you assert that the final number on the marble is an infinite series of digits? If not, what is it? Hey--that's a good question...what is it? 

Here's another question: if instead of "adding a 0" to the label on the marble, what would your answer be if each step was "erasing the label and replacing it with 13 times its value", (and we add 12 balls at each step instead of 9).  Here, you have to write the entire label each time you replace it.  Would you still assert that the final result would be an infinitely long series of digits on every ball?

"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."  True. I have no problem with that.  Where does that lead?

P.S.  Are you having fun with this?  I am.

Posted by Ken Haley on 2005-07-01 06:06:16