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Infinite Ping-pong Ball Machine
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Melissa Neacsu

Suspend your beliefs about ping pong balls. What follows is an adaptation of a thought experiment devised by Edward Burger and Michael Starbird in their excellent textbook, The Heart of Mathematics.



Imagine a machine with a claw arm that can move arbitrarily fast. Attached to the machine are a barrel and a tank, each of arbitrarily large capacity. Calculus students may wish to imagine the tank as conical in shape, but it does not leak water. Instead, it dispenses ping-pong balls, each labeled with a unique number from the set 1, 2, 3, 4, etc..., off to infinity. 

The machine works for only 1 minute. 

In the first 30 seconds, the tank dispenses the balls labeled 1 through 10 into the barrel, then the claw arm reaches in and removes the ball labeled “1.” How many balls are in the barrel?

In the next 15 seconds, the tank dispenses the balls labeled 11 through 20 into the barrel, then the claw arm reaches in and removes the ball labeled “2.” How many balls are in the barrel?

In the following 7.5 seconds, the tank dispenses the balls labeled 21 through 30 into the barrel, which, you may recall, is large enough to hold any number of ping-pong balls, and the claw arm reaches in and removes the ball labeled “3.” Are you keeping track of the number of balls in the barrel?

By now you’ve noticed a pattern. A math teacher might say that at step n, the machine releases balls 10- 9 through 10n into the barrel and removes ball n, all in 60/2^seconds, leaving precisely 9n balls in the barrel. Notice the machine is performing an infinite number of tasks in a finite amount of time. You may need to suspend your beliefs about more than just ping-pong balls.

Question: At the end of one minute, are there balls in the barrel?


While you ponder this question, please enjoy a few memories from this semester.



Removing an axiom or having an unsound proposition may cause the tower to fall!



A joyful group of students. This was Decade Day from Spirit Week. I was thrilled to see the 90’s so well-represented.




Fifth period Algebra II always starts with guitar. Sometimes trumpet as well. And drums. This was Tacky Day, but they wore it well.




The best part of Spirit Week was watching House Lucy dominate at Kajaba Can-can.




I can still hear this picture.



Members of TSCS attended the Houston Symphony last weekend as a collaborative activity between the music and astronomy departments.



Happy Thanksgiving! The Cake Decorating Club is in full swing.



Some Calculus humor posted in my AP Calculus Teachers Facebook group.



There really is always a conical tank. Now, back to the ping-pong ball machine…



At the end of one minute there are no balls in the barrel! 

Suppose we wish to know if ball n remains in the barrel once time runs out. We know ball n was removed at step n, so it is not in the barrel. If ball n is not in the barrel for any n, then the barrel must be empty. This argument seems cut-and-dry, but notice that as the steps progress, more and more balls accumulate in the barrel. The number of balls in the barrel is always increasing, yet not a single ball remains at the end of one minute!


The infinite ping-pong ball paradox demonstrates the sort of counterintuitive conundrum that can arise when contemplating infinity. Mathematics doesn’t mind exploring the “impossible.” At its heart, it is an adventurous game.