Suppose I have an infinite number of pots, labelled p

_{1},p

_{2},p

_{3},... initially empty but each capable of holding one of an infinite number of balls, labelled b

_{1},b

_{2},b

_{3},...

I start by placing ball b

_{1}in pot p

_{1}, and then perform a series of steps: at each step I move the ball from the first occupied pot to the (even numbered) first pot that has never contained a ball, and add a new ball to the (odd numbered) pot after that one.

So...

Start: Place ball b

_{1}in pot p

_{1}

Step 1: Move ball b

_{1}from pot p

_{1}to pot p

_{2}, and place ball b

_{2}in pot p

_{3}

Step 2: Move ball b

_{1}from pot p

_{2}to pot p

_{4}, place ball b

_{3}in pot p

_{5}

Step 3: Move ball b

_{2}from pot p

_{3}to pot p

_{6}, place ball b

_{4}in pot p

_{7}

...

Step n: Move the ball from pot p

_{n}to pot p

_{2n}, place ball b

_{n+1}in pot p

_{2n+1}

Then:

Each pot p

_{n}is filled exactly once, at step n/2 (even n) or (n-1)/2 (odd n)

Each pot p

_{n}is emptied exactly once, at step n

Each ball b

_{n}is placed into a pot at step n-1

After step n, pots p

_{1}...p

_{n}are all empty, and balls b

_{1}...b

_{n+1}are all (somewhere) in pots p

_{n+1}...p

_{2n+1}

The first few steps look like:

Code: Select all

`1`

- 1 2

- - 2 1 3

- - - 1 3 2 4

- - - - 3 2 4 1 5

- - - - - 2 4 1 5 3 6

Now let's allow each step n to take time 1/(2

^{n}) to complete - and examine what has happened at time t=1 when an infinite number of steps have been performed.

Every ball b

_{1},b

_{2},b

_{3},... has been placed in a pot, and no balls have been discarded. But every pot p

_{1},p

_{2},p

_{3},... has been filled once and subsequently emptied, so all the pots must be empty.

So, the question is... where are all the balls?

Observations:

**Spoiler:**