A Small Infinite Puzzle Kenneth S. Friedman (2002) “A Small Infinite Puzzle,” Analysis 62.4: 344-345. ppt file by Selmer Bringsjord for teaching at RPI www.rpi.edu/~brings

Place two balls ‘1’ and ‘2’ on a table. Now discard the ball Numbered ‘1’. balls 1 2 table

2

Add two new balls, ‘3’ and ‘4’, to the table. Interchange numbers ‘2’ and ‘3’. 2 3 4

Discard the ball now numbered ‘2’. 3 2 4

3 4

Continue, following the same pattern. 3 4

3 4 5 6

3 4 5 6

5 4 3 6

5 4 6

4 5 6

4 5 6

4 5 6 7 8

7 5 6 4 8

7 5 6 8

5 6 7 8

5 6 7 8

… 5 6 7 8 9 10

General Instructions Step n: Place balls ‘2n-1’ and ‘2n’ on the table. Next, interchange numbers ‘n’ and ‘2n-1.’ Finally, discard the ball that is now numbered ‘n’.

Question Assume each step takes less time than its predecessor. E.g., the first step can take ½ seconds, the second ¼ seconds, then 1/8 seconds, then 1/16th of a second, and so on, ad infinitum. The entire process thus takes less than a second. At the end of the process, how many balls remain on the table?

The Puzzle Answer 1: None! Any ball remaining on the table must have a number on it. Yet number ‘n’ has been discarded at the nth step and at no point after the nth step is any ball numbered ‘n’ placed on the table. Answer 2: An infinite number! The second ball placed on the table will be renumbered an infinite number of times, but will never be discarded. Initially numbered ‘2’, it is renumbered ‘3’ at the second step, ‘5’ at the third, … The fourth ball placed on the table will also be renumbered an infinite number of times, but will never be discarded. Indeed, every ball that originally had an even number will be renumbered an infinite number of times, but will never be discarded. The number of balls that originally have even numbers is infinite. So an infinite number of balls will remain on the table. At least one of these answers must be wrong. Which is, or are, wrong?