Suppose I have an infinite number of pots, labelled p1,p2,p3,... initially empty but each capable of holding one of an infinite number of balls, labelled b1,b2,b3,...
I start by placing ball b1 in pot p1, 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.
Start: Place ball b1 in pot p1
Step 1: Move ball b1 from pot p1 to pot p2, and place ball b2 in pot p3
Step 2: Move ball b1 from pot p2 to pot p4, place ball b3 in pot p5
Step 3: Move ball b2 from pot p3 to pot p6, place ball b4 in pot p7
Step n: Move the ball from pot pn to pot p2n, place ball bn+1 in pot p2n+1
Each pot pn is filled exactly once, at step n/2 (even n) or (n-1)/2 (odd n)
Each pot pn is emptied exactly once, at step n
Each ball bn is placed into a pot at step n-1
After step n, pots p1...pn are all empty, and balls b1...bn+1 are all (somewhere) in pots pn+1...p2n+1
The first few steps look like:
Code: Select all
- 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/(2n) to complete - and examine what has happened at time t=1 when an infinite number of steps have been performed.
Every ball b1,b2,b3,... has been placed in a pot, and no balls have been discarded. But every pot p1,p2,p3,... has been filled once and subsequently emptied, so all the pots must be empty.
So, the question is... where are all the balls?
Ball bk will at some point be found in all pots p(2k-1)(2n) where n=0,1,2,...