Balls and vase problem

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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, 10 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, 10 more balls are added to the vase (numbered 10n+1 through 10n+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 2⁻ⁿ minutes before noon.

This guarantees that a countably infinite number of steps is performed prior to noon.

1 Solutions
2 Discussion
3 See also
4 External links

[edit] Solutions

Answers to the puzzle generally fall into three categories:

Since by noon every ball n that is inserted into the vase (at step ⁿ/₁₀) is eventually removed in a subsequent step (step n), the vase is empty at noon.
The problem is illogical, since noon is a point in time that can never be reached. Thus the question of how many balls remain at noon is ill-formed.
Since at each step 10 balls are inserted but only 1 is removed, a net 9 balls is added at every step before noon. Therefore, the vase is filled with an infinite number of balls by noon.

Mathematicians generally agree that solution #1 is the correct answer, given the particular conditions of the question. Set theorists in particular usually characterize the puzzle as a problem of ordinal numbers and infinite sets. Crucial to this view is the order of the insertions and removals, in particular that every ball is added to the vase and then later removed. Other variants establish alternate orderings or have different numbers of balls added or removed at each step, which produce answers that differ from the original solution.

Some logicians prefer solution #2 as the correct answer, pointing 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. Mathematicians usually respond to this criticism by pointing out that the problem is abstract, having no bearing on reality, and that noon is as meaningful a point in time as any other time prior to noon within the conceptual framework of the problem.

Those who accept solution #3 as correct are generally regarded either as mathematically naïve or even as cranks.

[edit] Discussion

The key to understanding the problem is to see the distinction between all points in time prior to noon and the time at noon.

All the steps of the problem are performed at some distinct time prior to noon. Since each step results in the addition of nine more balls into the vase than the previous step, the number of balls in the vase grows without bound as the time approaches noon.

However, all of the steps (an infinite number of them) are completed by noon. So before noon, there is a distinct point in time when each ball is inserted into the vase, and another distinct but later point in time when that ball is removed from the vase. Thus by noon, every ball has been inserted into, and then removed from, the vase.

This distinction between all times before noon and the time at (or even after) noon is key. This is sometimes characterized as a discontinuity in an otherwise smooth function, i.e., the number of balls in the vase at any given time.