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Friday, December 7, 2012

Why Infinity Will Drive You Bonkers…



You have an infinite number of balls labelled from "1" to infinity.
At one minute to midnight you place balls #1 through #10 in a box and simultaneously the #1 ball is removed.

At 1/2 minute to midnight you place the #11 through #20 balls in the same box and simultaneously remove the #2 ball.

At 1/3 minute to midnight you place the #21 through #30 balls into the box, removing the #3 ball.

At 1/4 minute to midnight…....... removing the #4 ball.

...and you continue on with the same pattern. . . . .

How many balls are in the box at midnight???  At first glance it would seem there could be an infinite number, i.e. 9+9+9+9+…
However, for ANY ball "#n" that you might assume remains in the box, it can be deduced that THAT ball was REMOVED from the box at the 1/n-of-a-minute time point... thus all balls get removed; the box is empty!

I adapted this from a version at another site:

http://tonysmaths.blogspot.com/2012/12/my-current-favourite-infinity-paradox.html  

This is known as the Ross-Littlewood Paradox and you can read more about it here:
http://en.wikipedia.org/wiki/Ross%E2%80%93Littlewood_paradox

It is similar to a more involved Raymond Smullyan puzzle I've posted about previously (...if your mind isn't already blown):

http://math-frolic.blogspot.com/2011/01/seemingly-impossible-task-that-isnt.html


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