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
 Next remove balls 3 & 4
 Next remove balls 5 & 6
 Next remove balls 7 & 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 nanosecond 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
 Next remove balls 3 & 4
 Next remove balls 5 & 6
 Next remove balls 7 & 8
 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 nanosecond 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.
