Balls and vase problem

The balls and vase problem (also known as the Ross-Littlewood paradox or the ping pong ball problem) is a hypothetical problem in abstract mathematics and logic designed to illustrate the seemingly paradoxical, or at least non-intuitive, nature of infinity.

The problem starts with an empty vase and an infinite supply of balls at a starting time before noon. At each step in the procedure, balls are added and removed from the vase. The question is then posed: "How many balls are in the vase at noon?"

At each step, balls are inserted into and removed from the vase in a particular order:
* In the first step, ten balls (numbered 1 through 10) are added to the vase, and then the first ball (numbered 1) is removed from the vase.
* At each subsequent step, ten more balls are added to the vase (numbered 10("n"−1)+1 through 10("n"−1)+10 at step "n"), and then the lowest numbered ball ("n") is removed from the vase.

As part of the problem statement, it is assumed that an infinite number of steps is performed. This is allowed by the following conditions:
* The first step is performed at one minute before noon.
* The second step is performed at 30 seconds before noon.
* The third step is performed at 15 seconds before noon.
* Each subsequent step "n" is performed at 2su|p=−"n"+1 minutes before noon.This guarantees that a countably infinite number of steps is performed by noon.

Solutions

Answers to the puzzle fall into several categories.

Vase is empty

Since by noon every ball "n" that is inserted into the vase (at step frac|"n"|10) is eventually removed in a subsequent step (step "n"), the vase is empty at noon.

Vase contains infinite balls

Since at each step ten balls are inserted but only one is removed, a net nine balls are added at every step before noon. Clearly, the number of balls (as a function of the step) equals 9 times the step: "B" = 9"n" for all "n". So as "n" tends to infinity, "B" likewise diverges towards infinity. Therefore, the vase is filled with an infinite number of balls by noon. Alternatively, since one ball out of ten is removed every step, the number of balls will be 1/10 of infinity, which is infinite.

Answer to problem depends

Some people say that the number of balls that one ends up with depends on the order in which the balls are removed from the vase.

The following procedure outlines exactly how to get n balls remaining in the vase.

Let "i" denote the number of the operation currently taking place.Let "n" denote the desired final number of balls in the vase.Then:for "i" = 1 to infinity,::if "i" ≤ "n", then remove the ball at position 2"i"::if "i" > "n", then remove the ball at position "n"+"i"

Clearly, the first "n" odd balls are not removed, while all balls greater than or equal to 2"n" are. Therefore, exactly "n" balls remain in the vase.

Problem is underspecified

Some point out that although the state of the balls and the vase is well-defined at every moment in time "prior" to noon, no conclusion can be made about any moment in time "at" or "after" noon. Thus, for all we know, at noon, the vase just magically disappears, or something else happens to it. But we don't know, as the problem statement says nothing about this. Hence, like the previous solution, this solution states that the problem is underspecified, but in a different way than the previous solution.

Problem is ill-formed

Finally, some people contend that the problem is ill-posed. To be precise, according to the problem statement, an infinite number of operations will be performed before noon, and then asks us about the state of affairs at noon. But, as in Zeno's paradox, if infinitely many operations have to take place (sequentially) before noon, then noon is a point in time that can never be reached. On the other hand, to ask us how many balls will be left at noon, is to assume that noon will be reached. Hence there is a contradiction implicit in the very statement of the problem.

References

* "On Some Paradoxes of the Infinite", Victor Allis and Teunis Koetsier, "The British Journal for the Philosophy of Science", v.42 n.2, Jun 1991, pp. 187-194
* "Ross' Paradox Is an Impossible Super-Task", Jean Paul Van Bendegem, "The British Journal for the Philosophy of Science", v.45 n.2, Jun 1994, pp. 743-748

See also

* Supertask
* Thomson's lamp
* Zeno paradox

External links

This problem and its variations appear frequently on the "sci.math" newsgroup. A sampling of some of these discussions:
* [http://groups.google.com/group/sci.math/browse_frm/thread/5276b15032c5d79f?scoring=d sci.math] , "infinity", 2005-08-03
* [http://groups.google.com/group/sci.math/browse_frm/thread/b3e5df6da07a29f4?scoring=d sci.math] , "INFINITY Revisited", 2005-08-26
* [http://groups.google.com/group/sci.math/browse_frm/thread/6bd413656b43ee5d?scoring=d sci.math] , "Marble problem - put in 10 marbles, then remove 1", 2005-10-21
* [http://groups.google.com/group/sci.math/browse_frm/thread/c5e8522696fb2b97?scoring=d sci.math] , "Calculus XOR Probability", 2006-03-13
* [http://groups.google.com/group/sci.math/browse_frm/thread/ce6b1a9c57e8c799?scoring=d sci.math] , "Infinity: An interesting variant", 2006-03-18