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(11): No Subject (Summary) by ajosin)

OK, I've given this a bit more thought.  Let's see if we can come to some agreement, if we change our discussion to simply one of sets of integers, and forget about marbles and bags.  To do this, we need to translate the statements about marbles and bags to simple statements about sets of integers.  In doing this, I'm going to concede some points that I disagreed with earlier...

First, your assertion that
 {1,2,3,4...} - {1,2,3,4...} = null set
or to be more formal, if you start with the set of all integers, and remove the set of all integers, you end up with the empty set.  Agreed!

Second, your last two assertions (A) and (B) in your last post are certainly true.  (I won't reword them--they're pretty obvious, and we agree.)

Third, in set theory, a set can't contain duplicates.  If S is a set of integers, 5 is either in the set or it's not.  You can't assert that 5 is in the set several times.  So, the following isn't a set:
 {1,2,3,3,4}
(see http://www.reduce-algebra.com/docs/sets.pdf, and other references by searching Google for ["set theory" duplicate]).  I mention this because one of your posts mentioned "If I relabled the marbles {2,2,2,2} and then applied the algorith, I would be left with no marbles", which really has no meaning in pure set theory because {2,2,2,2} is not a set.  Not a big deal, because the cases we're interested in don't have this problem.

Fourth, when dealing with pure sets of abstract objects like positive integers, "relabeling" has no meaning.  You can't relabel an integer.  So, we have to decide what relabeling a marble means when we translate that to sets of integers.

Fifth, a set is unordered.  The set {1,2,3} is identical to the set {3,2,1}.  So, when you're talking about infinite sets, it's meaningless to talk about any "first" or "last" member.  An ordered collection of elements is called a "sequence".  We can have infinite sequences, and we might map the marbles problems to sequences (instead of simple sets) but I don't see any advantage of doing that.  Let's just keep it to sets.  By the way, sequences can have duplicate members.

Do you disagree with any of this, so far?  If so, we might as well stop here--I don't think we'll be able to get any further.

So, assuming we have agreement, let's try to translate the marbles and bags problems into statements about integers and sets.  Specifically, I'd like to concentrate on part B of the original Marbles Bonanza and this puzzle (Marbles Bonanza II).

In Marbles Bonanza, part B, we start with an empty bag and... 

• step 1: Put marbles #1-10 in the bag, then take out #1.
• step 2: Put #11-20 in the bag and take out #2.
  etc.

This would translate to: Start with an empty set, and...

• step 1: add the integers 1-10 to the set and remove the integer 1.
• step 2: add the integers 11-20 to the set and remove the integer 2.
  etc.

I think you'd agree there.

Now for this puzzle.  Since we can't "relabel" an integer (it is what it is), we have to decide what "relabeling" a marble means.  I would suggest this:  "Relabeling marble x (already in the bag) to marble y" would translate to "replacing integer x in the set with integer y"...that is removing integer x and adding integer y to our set.  We can do this as long as y is not already in the set (avoiding the duplicate issue in my third point above). 

So in this puzzle, we start with an empty bag and...

• step 1: Put marbles #1-9 in the bag and then relabel marble #1 by appending a 0, making it #10.
• step 2: put marbles #11-19 in the bag and then relabel marble #2 by appending a 0, making it #20.
  etc.

This would translate to: Start with an empty set, and

• step 1: add the integers 1-9 to the set and then replace the integer 1 with the integer 10.
• step 2: add the integers 11-19 to the set and then replace the integer 2 with the integer 20.
  etc.

Since "replace" just means adding and removing, if we collect the add operations together and isolate the remove operation, we can restate these steps as:

• step 1: add the integers 1-10 to the set and remove the integer 1.
• step 2: add the integers 11-20 to the set and remove the integer 2.
  etc.

But this last set of steps is exactly the same as the set of steps in the previous puzzle!  So the results of the two puzzles should be the same!

And, as I've maintained all along, you can draw two logical, but contradictory, conclusions from applying this infinite sequence of operations.  (A) The final set must be infinite in size because, at every step there's a net increase of 9 integers in our set, and (B) the final set must be empty because every integer added to the set is subsequently removed.

Now, I'm no longer asserting that the problem is simply "subtracting infinities." (I was off base there.)  I think you can remove an infinite set of objects from an infinite set of the same type of objects and decided unambiguously what the size of the remaining set is.

But there's something wrong with applying this infinite sequence of steps, and it has to do with a combination of adding and removing elements an infinite number of times--leading to contradictory results depending on how you look at it.  I think it may be the fact that, although set theory allows infinite sets, I don't know if there's anything that allows a function that consists of an infinite number of steps to be defined. 

It's just like the other example I gave.  Starting with the empty set, and applying the infinite sequence of adding the integer 1 to the set, and removing the integer 1 from the set, ad infinitum.  What's the resulting set? It can't be determined, and the question is meaningless. Just because a process is well-defined, doesn't mean you can apply it an infinite number of times with no issues.

Tristan?  SilverKnight?  Anybody else?  Comments?

Posted by Ken Haley on 2005-06-29 05:54:47