The infinite bag of billiards balls

Question

The following trio of puzzles comes from http://skepticsplay.blogspot.com/

My question is: What is the solution to variation 2?

_{The original author of these puzzles described them as being abstract. I understand this to mean that we don’t have to consider any real world physical constraints. For example, we don’t have to worry about the bag being big enough to hold all the balls OR whether it is possible to have an infinite number of balls OR any other considerations.}

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This is a classic puzzle that often provokes lively discussion among puzzling communities.

We start with an empty bag at 11:00. We put in 10 billiards balls, which are numbered 1 through 10. But then we take out the ball labeled #1. At 11:30, we add 10 more billiards balls, which are numbered 11 through 20, and remove the ball labeled #2. At 11:45, we add balls 21 through 30 and remove #3.

This process is repeated infinitely. At each step, the next ten billiards balls are put into the bag, and the lowest number in the bag is removed. There are an infinite number of steps before 12:00, since the time between each step decreases (halved) at each step.

Obviously this puzzle is meant to be abstract, as you cannot actually perform an infinite number of steps within an hour, nor can a bag actually hold that many billiards balls. But we can still ask a genuine mathematical question about the situation: what's left in the bag after 12:00?

Variation 1: Instead of removing the lowest numbered ball in the bag at each step, we instead remove the highest numbered ball. So on step 1, we remove #10, on step 2, we remove #20, and so forth. What's left in the bag after 12:00?

Variation 2: On the first step, we add only billiards balls labeled 1 through 9. Instead of adding #10, we just take #1 and paint an extra zero at the end. On the second step, we add numbers 11 through 19, and add an extra zero to the end of #2, resulting in #20. On the third step, we add numbers 21 through 29, and add an extra zero to the end of #3. This process is repeated infinitely before 12:00. What's left in the bag after 12:00?

Answer 1

The balls which are added to the bag are exactly those balls where the numbers on it have no trailing zeros. During the infinite process the number $X$ on such a ball changes from $X$ to $X*10$, $X*100$, $X*1000$... ad infinitum but nervertheless the ball can be indentified by its original number $X$ on it.

Answer 2

As OP acknowledges the whole question is an abstract one and as such it requires unambiguous definitions to be well posed. A bag that can hold infinitely many balls does not exist. Therefore it is mute to argue what properties such a thing may have.

Unless:

We clarify that our bag is metaphor for a set, more precisely, a family $S_1,S_2,...$ of sets indexed by time steps. The elements of set $S_j$ are the balls in the bag after $j$ steps of adding/removing balls. This allows to rigorously define what we mean by "bag after infinitely many steps". Namely the set $$\bigcup_{k=1}^\infty\bigcap_{j=k}^\infty S_j$$ If you do not read mathematical notation: It says that the set contains and only contains each element that is at some point added and never removed afterwards.

With that in mind the original question and variation 1 are easily answered.

Variation 2 introduces new ambiguities and unlike the "magic bag" one it does not have a generally agreed on resolution. Specifically, it is not clear what the "objects", i.e. things that are allowed to be in a set ("bag") are.

In the original problem and variation 1, balls are helpfully but redundantly labelled with unique numbers. It is an acceptable and reasonable mental economy to not distinguish between balls and labels. So, for example, $S_1$ is the set $\{1,2,...,10\}$ and not the set $\{\text{ball labelled with }1,\text{ball labelled with }2,...\text{ball labelled with }10\}$

Variation 2 introduces and leaves unanswered the question:

Is a relabelled ball 1) the same object as before? Or is it 2) a distinct object. If (2) then variation 2 is identical to the original problem. If (1) then no balls are ever removed.

Note (only applies to case (1)): Irrelevant for the question as asked but maybe interesting in its own right. What are the labels after infinitely many steps?

Answer: undefined. AFAIK, there is no "natural"/consensus/broadly accepted "limit" of the sequence of labels on a given ball.

Answer 3

Solution to Variation 2:

As an example, consider what happens to the ball initially labeled 5. It is put into the bag in the first step. In the 5th step, its label is changed to 50. In the 50th step, its label is changed to 500. In fact the label on this ball is changed an infinite number of times (on the 5th, 50th, 500th, ... steps).

So at 12:00 this ball will be labeled:

5000... where there is an infinite number of zeros.

All balls are relabeled in a similar fashion so at 12:00 the bag contains balls labeled:

1000...
2000...
3000...
$\cdots$
9000...

11000...
12000...
13000...
$\cdots$
19000...

$\cdots$

In summary, at 12:00 for every positive integer $n$ that is not a multiple of 10, the bag will contain a ball whose label is $n$ followed by an infinite number of zeros.