Newspaper article
*The Christian Science Monitor*

# Math Chat: Ping-Pong Balls and Presidents' Names

## Article excerpt

Old challenge (Jonathan Kravis and Dave McMath)

A process involving infinitely many ping-pong balls is carried out in one hour as follows. In the first half hour, two balls are placed into a huge empty barrel and then one is removed and discarded. In the next 15 minutes, two more balls are put in and then one of the three inside is removed. At each step, in half of the remaining time, two more balls are added and one is removed. How many balls are in the barrel at the end of the hour after infinitely many steps?

Answer

After n steps, there are n balls in the barrel. After infinitely many steps, there might be infinitely many in the barrel or there might be none at all or any number in between!

Suppose for example that at the nth step you remove the nth ball. Then every ball gets removed at some time, and none are left. Of course if at the nth step you remove ball 2n, then balls 1, 3, 5, ... will never be removed. Infinite processes often defy intuition.

If you remove balls at random, it turns out that with 100 percent probability there will be no balls left. For example, the probability that ball 1 remains after one step is 1/2, after two steps is (1/2)(2/3) = 1/3, after three steps is (1/2)(2/3)(3/4) = 1/4, after n steps is 1/(n+1), and after infinitely many steps is 0.

According to Jim Henry, for an engineer, there will be only finitely many balls in the barrel, because after the 19th step after 59 minutes and 59.99 seconds, the engineer will say the hour is up.

Likewise for a quantum-relativist physicist, there will be only finitely many balls in the barrel, because the later ones would have to surpass the speed of light to get inside.

More on arithmetic modulo 7 (Nick Drozdoff)

"I am a high school physics teacher, ex-electrical engineer, and a current physics graduate student. {In the June 27 discussion of arithmetic modulo 7} were many equations such as 52 = 25 = 4. This seems a peculiar form of mathematics. I don't personally see how 25 = 4. What I can see is that 25 divided by 7 is three with a REMAINDER of four, but that is not what was written. What was written was 25 = 4.

"High school kids do this in calculations all the time. They string their work together with equal signs skipping steps in favor of some sort of shorthand. It drives me nuts because they have difficulty checking their work later. I encourage them to write each step of a calculation out. I discourage the tendency to string things together with equal signs that lead to visual/mathematical absurdities.

"Is this a common practice with math instruction? If so, I am appalled and shocked. I ask because this problem is not limited to any particular school. It's everywhere. I am concerned when I see this sort of thing in print in a place where people might regard such techniques as being accepted practice with experts, such as in The Christian Science Monitor, a respected newspaper not known for egregious mistakes. …