Balls In Barrels

   
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On the page about infinity we mention that there are things that go wrong if you rely on your intuition. This page describes one of them. You may also want to see the page where Cantor Visits Hilberts Hotel.

I wonder if he indulges in the sport of Stacking Blocks?

Introduction

images/inf_times.png
Here's a small example of the sorts of things that can go wrong. It comes in three stages, but common to them is the timing. We start by defining a sequence of points in time. The first is one hour to midnight, 23:00. The next is 1/2 hour to midnight, 23:30. Then 1/4 to midnight, 1/8 hour to midnight, 1/16 hour to midnight, and so on. You get the idea.

This gives us lots of specific times, all of them before midnight. If you think of any large number, there are more times than that. Loosely speaking, we would say that there are infinitely many times.

Stage 1

Suppose we have a barrel with lots of balls in it, and they're labelled 1, 2, 3, 4, etc. You'll notice that for any number you think of, there are more balls than that in the barrel.

We work to the following timetable. These actions take place at successive times as defined above.

  • First, remove balls 1 & 2
    • Replace ball 2
  • Next remove balls 3 & 4
    • Replace ball 4
  • Next remove balls 5 & 6
    • Replace ball 6
  • Next remove balls 7 & 8
    • Replace ball 8
  • etc.

Here's the idea laid out as a table of timings and actions:

Step number Time Action Balls outside




1 23:00:00 Remove 1 & 2 1, 2
2 23:30:00 Return ball 2 1
3 23:45:00 Remove 3 & 4 1, 3, 4
4 23:52:30 Return ball 4 1, 3
5 23:56:15 Remove 5 & 6 1, 3, 5, 6
6 23:58:07.5 Return ball 6 1, 3, 5
7 23:59:03.75 Remove 7 & 8 1, 3, 5, 7, 8
7 23:59:03.75 Remove 7 & 8 1, 3, 5, 7, 8
8 23:59:31.875 Return ball 8 1, 3, 5, 7

At each odd time step we take out two balls, and at each even time step we replace the highest numbered ball.

At any given moment we know exactly which balls are in the barrel and which balls are outside. At 1 minute to midnight balls 1 through 6 have been removed, and balls 2, 4 and 6 have been replaced. At 1 second to midnight balls 1 through 12 have been removed and balls 2,4,6,8,10 and 12 have been replaced. At 1 nano-second to midnight balls 1 through 42 have been removed and the even balls up to 42 have been replaced. And so on.

So what is the situation at one minute past midnight?

  • If you name any odd number, I can tell you exactly when it was taken out.
  • If you name any even number, I can tell you both when it was taken out and when it was put back.

So we can see that after midnight the odd numbered balls are out, the even numbered balls are in.

Stage 2

Once again we take balls out and put them back. This time:

  • First, remove balls 1 & 2
    • Replace ball 1
  • Next remove balls 3 & 4
    • Replace ball 2
  • Next remove balls 5 & 6
    • Replace ball 3
  • Next remove balls 7 & 8
    • Replace ball 4
  • etc.

This time at each odd time step we take out two balls, but now at the even time steps we replace the smallest numbered ball. Here it is in a table ...

Step number Time Action Balls outside




1 23:00:00 Remove 1 & 2 1, 2
2 23:30:00 Return ball 1 2
3 23:45:00 Remove 3 & 4 2, 3, 4
4 23:52:30 Return ball 2 3, 4
5 23:56:15 Remove 5 & 6 3, 4, 5, 6
6 23:58:07.5 Return ball 3 4, 5, 6
7 23:59:03.75 Remove 7 & 8 4, 5, 6, 7, 8
8 etc. Return ball 4 5, 6, 7, 8

Again, at any given moment we know exactly which balls are in the barrel and which balls are outside. At 1 minute to midnight balls 1 through 6 have again been removed, but now balls 1 to 3 have been replaced. At 1 second to midnight balls 1 through 12 have been removed and balls 1 through 6 have been replaced. At 1 nano-second to midnight balls 1 through 42 have been removed and balls 1 through 21 have been replaced. And so on.

So what is the situation at one minute past midnight? You name any number and I can tell you both when it was taken out and when it was put back.

By the time midnight comes around every ball has been put back in the barrel.

This doesn't sound right. Leading up to midnight the number of balls outside the barrel is increasing, and yet after midnight they're all back in the barrel? Yes, it's very definitely unexpected, definitely not what your intuition tells you, but it's true.

Stage 3

So now let's rub off the numbers from the balls, and at each of our points in time we remove two balls and then put one back in. At 1 minute to midnight there are 6 balls outside. At 1 second to midnight there are 12 balls outside. At 1 nano-second to midnight there are 21 balls outside.

How many are outside of the barrel after midnight? We can't tell !

Without being explicit, without actually naming the balls and saying which ones are being removed and replaced we simply cannot tell. Case 1 says it might be infinitely many, case two says it might be none. We simply don't know because we don't have enough information.

Weird - but that's the strange world of infinity.


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