Paradox Lost...and found.

Copyright 1996 by Paul Niquette. All rights reserved.

At midnight, how many balls are in the vase?
Zero
Countless times I have posed this thought experiment to friends and associates, to university audiences and management seminars. The answers I invariably hear are of the form: The mathematician places net nine balls into the vase at each step; after an infinite number of steps, there must be an infinite number of balls in the vase. 
    Not so, I chortle. At the stroke of midnight, the vase would be empty.
Name any ball and I will tell you it is not in the vase. Not only that, but I will tell you the exact moment at which it was removed from the vase: 
  • Ball number one is not in the vase; the mathematician removed it at one minute to midnight.
  • Ball number two is not in the vase; the mathematician removed it at one-half of a minute to midnight.
  • Ball number three is not in the vase; the mathematician removed it at one-third of a minute to midnight.
  • Ball number four is not in the vase;...
How can infinity and zero both be the right answer? 



Off on a Tangent

Consider the trigonometric tangent of a 90-degree angle.

As a transcendental function, the tangent does something mighty strange in the vicinity of 90 degrees.

  • At 89.99999o, the tangent has a value of plus 5,729,578, which is not infinity or anything, but it sure is heading in that general direction.
  • At 90.00001o, the tangent has a value of minus 5,729,578, which is heading toward a negative infinity.
Half-way in between -- at exactly 90.000...0o, the tangent is supposed to be infinite; however, it must also be passing through zero at the same exact place.

Mathematicians shrug. "What do you expect from a discontinuous function?" they explain.
 

Where two parallel lines meet, so too
do the infinite and the infinitesimal.
-- Paul Niquette, 1966

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Paradox Found

This puzzle is a ball-in-the-vase version of the famous paradox [seemingly contradictory statement but nevertheless true] attributed to Zeno of Elea, a Greek mathematician and philosopher who lived back in 5th century BC (long before numbered balls).  We have come across this fellow Zeno before (see Hand Over Hand).  Another version of this paradox was discovered in the solution to Rational Roots.


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