# Urns of Infinity

by Kevin J. Lin

This puzzle exposes some of the problems that arise when we try to apply traditional mathematics and logic to supertasks (tasks involving infinite operations).

What is in the first Urn at midnight? The obvious answer is an infinite number of balls, but none ending in "0".

What should be in the second Urn at midnight? The second Urn should be empty- every ball that goes in eventually comes out. This seems counterintuitive- at any time prior to midnight, there are nine balls added for each one removed. When does the urn suddenly lose all its balls?

The paradox of this is brought into focus by the Demon #2's "cheat". If the sorcerer had checked on the Demon at any time prior to midnight, unless he was caught in the act of painting, the Urn would look right. On any finite subset, the Demon's operation is identical to the Sorcerer's requested task.

Does this puzzle have an answer? Yes and no; depending on what mathematical model you use for infinite series, either answer is defensible. Here are some more examples to illustrate the paradox:

A Thompson lamp is a lamp that is switched on then switched off after 1 second, then switched on again after a half second, then switched off after a quarter second, then switched on after an eighth of a second... and so on. Two seconds after it is first turned on, is it on or off?

What is the sum of the Grandi series: 1 - 1 + 1 - 1 + 1 - 1 .... ? Consider than this series could also have been written as: (1 - 1) + (1 - 1) ... which would seem to equal 0. Or it could have been 1 + (-1 + 1) + (-1 + 1) which would equal 1.

It seems that our everyday logic breaks down when applied to the infinite. We must either abandon discussions of infinity, or accept that some questions which seem meaningful for all questions of the finite are meaningless when applied to the the infinite.