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7 replies to this topic

### #1 bonanova

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Posted 10 August 2012 - 07:01 AM

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.
http://brainden.com/...p/topic/3805--/

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 bonanova, 10 August 2012 - 03:53 PM.

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The greatest challenge to any thinker is stating the problem in a way that will allow a solution.
- Bertrand Russell

### #2 CaptainEd

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Posted 10 August 2012 - 03:50 PM

Spoiler for I fear I haven't the time, I'm not retired yet...

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### #3 bushindo

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Posted 10 August 2012 - 05:56 PM

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.
http://brainden.com/...p/topic/3805--/

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?

Some clarification please. What happens when the player removes a ball labeled with the number 1?
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### #4 bonanova

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Posted 11 August 2012 - 03:35 AM

Some clarification please. What happens when the player removes a ball labeled with the number 1?

?

The balls in the bucket decrease in number by one.
There initially were no 0-numbered balls, so Alex wasn't given an infinite bucket of 0 balls.
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The greatest challenge to any thinker is stating the problem in a way that will allow a solution.
- Bertrand Russell

### #5 bushindo

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Posted 11 August 2012 - 11:36 PM

Spoiler for I fear I haven't the time, I'm not retired yet...

Aye, Captain,
Spoiler for In agreement with the captain

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### #6 bonanova

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Posted 12 August 2012 - 01:22 AM

Exactly right, both.

Captain, I held off with congratulations on the off chance someone would succumb to the possible enormity of the ball count and make a case for the other side. When i sketched an array of balls it began with a long line and morphed, successively, into an enormous square, a massive cube, humongous hypercube, and so on. The number of dimensions of the unbounded hyper cube itself is unbounded. It might have overwhelmed one's reasoning.

But not here. Not with these puzzle solvers. Bravo.

Now for a hydrophyllically encrypted summary of the result; with extra points for explanation:
China bests Japan.

Clue: not a reference to the Olympic Games.
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The greatest challenge to any thinker is stating the problem in a way that will allow a solution.
- Bertrand Russell

### #7 CaptainEd

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Posted 13 August 2012 - 03:40 PM

Spoiler for Not a clue...

Thanks for the puzzle(s) as always!

Edited by CaptainEd, 13 August 2012 - 03:45 PM.

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### #8 bonanova

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Posted 13 August 2012 - 05:38 PM

My comment is probably too far fetched to spend much time on:

Spoiler for What occurred to me

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The greatest challenge to any thinker is stating the problem in a way that will allow a solution.
- Bertrand Russell

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