A while back we discussed the interesting process where each step consisted of placing consecutively numbered balls into a bucket, two at a time, and then removing the lowest-numbered ball. We postulated an infinite number of balls being processed in this manner. Completion was "ensured" by executing each step after an interval of time that was one-half of the interval of the previous step. An Infinite number of steps could thus be executed in finite time, after which the bucket could be inspected.

The puzzle was to describe the final contents of the bucket. The unintuitive result is that it would be empty. Even though after each step the number of balls it contains increased by one.

The interested reader can enjoy my misguided discussion of the outcome here.

We propose here a different kind of very long task, also involving placing and removing balls into and from a bucket. Completion is again the central issue, and the bucket's final contents if it's the case that the task does in fact complete. Nothing in this task is infinite, but it is unbounded.

You are given a bucket that contains an arbitrarily large number N of balls, numbered 1-N. You are to empty it, by a series of steps described below. To make the task interesting, an adversary named Alex tries to prevent you from emptying the bucket. Alex is well equipped to prevent your success. For each numbered ball initially in your bucket, Alex has another bucket with an infinite number of balls marked with the same number. That is, if your bucket initially contains a ball that is numbered 37, Alex has his own bucket with an infinite number of balls also numbered 37.

Here is how the steps are executed. You remove a ball of your choosing, say its number is p, and discard it. Alex may then add to the bucket an arbitrarily large number of balls numbered p-1, using his infinite bucket of p-1 balls. An individual step can thus add to your bucket a set of new balls of size equal to the number of electrons in the universe. Moreover, your bucket might have started with that many balls initially. But even if your bucket starts with as few as 2 balls, it is not possible to place a bound on the number of steps needed to empty the bucket.

## Question

## bonanova 84

A while back we discussed the interesting process where each step consisted of placing consecutively numbered balls into a bucket, two at a time, and then removing the lowest-numbered ball. We postulated an infinite number of balls being processed in this manner. Completion was "ensured" by executing each step after an interval of time that was one-half of the interval of the previous step. An Infinite number of steps could thus be executed in finite time, after which the bucket could be inspected.

The puzzle was to describe the final contents of the bucket. The unintuitive result is that it would be empty. Even though after each step the number of balls it contains increased by one.

The interested reader can enjoy my misguided discussion of the outcome here.

We propose here a different kind of very long task, also involving placing and removing balls into and from a bucket. Completion is again the central issue, and the bucket's final contents if it's the case that the task does in fact complete. Nothing in this task is infinite, but it is unbounded.

You are given a bucket that contains an arbitrarily large number N of balls, numbered 1-N. You are to empty it, by a series of steps described below. To make the task interesting, an adversary named Alex tries to prevent you from emptying the bucket. Alex is well equipped to prevent your success. For each numbered ball initially in your bucket, Alex has another bucket with an infinite number of balls marked with the same number. That is, if your bucket initially contains a ball that is numbered 37, Alex has his own bucket with an infinite number of balls also numbered 37.

Here is how the steps are executed. You remove a ball of your choosing, say its number is p, and discard it. Alex may then add to the bucket an arbitrarily large number of balls numbered p-1, using his infinite bucket of p-1 balls. An individual step can thus add to your bucket a set of new balls of size equal to the number of electrons in the universe. Moreover, your bucket might have started with that many balls initially. But even if your bucket starts with as few as 2 balls, it is not possible to place a bound on the number of steps needed to empty the bucket.

So here's the question: Can you empty the bucket?

Edited by bonanovaAdd link to previous topic

## Link to post

## Share on other sites

## 7 answers to this question

## Recommended Posts

## Join the conversation

You can post now and register later. If you have an account, sign in now to post with your account.