The Ross-Littlewood Paradox and Mathematical Explanations

As normal people do when they’re bored and 4+ years out of college, I read a mathematics book for fun.

I’ve always reserved a corner in my heart for mathematics. It was my best subject in high school, and when the business school I hoped ot attend rejected me, I chose to pursue a pure mathematics degree instead.

While in college, I experienced numerous phases where I loved math more than anything. Life was good. But each of those phases was accompanied by a complementary, and often longer, period where I hated math and regretted my major. I even considered switching to an English degree after reading and writing seduced me. However, seeing as I had one semester left of my math degree, I deferred that path.

The same on-again, off-again trend of loving math continued after college. On any given day, I never know whether I’ll wake up adoring or detesting math.

But right now, I love math and hope this phase sticks. Whether it does or does not is unknown. What I do know, however, is that mathematics will always be a passion of mine, and I will frequently write about it here.

The book I’m currently reading is Eight Lessons on Infinity: A Mathematical Adventure by Haim Shapira, PhD.

What I love about it so far is that Shapira doesn’t only tell stories about mathematicians or history, as most non-textbooks do. Instead, he gives readers fun puzzles, riddles, and paradoxes to explore and test their critical thinking skills.

One paradox, in particular, is why I’m writing this post.

The paradox in question is the Ross-Littlewood paradox. Reading about it engaged my mind in ways no other book has this year. The paradox, which has multiple variations but the same core concept, goes like this:

Imagine you have an empty room and an infinite number of balls lined up to enter. The balls are numbered 1, 2, 3, 4, and so on.

Thirty seconds (or half a minute) before midnight, balls 1 and 2 enter the room, with ball 1 being immediately removed. Fifteen seconds (or a quarter of a minute) before midnight, balls 3 and 4 enter, while ball 2 leaves. An eighth of a minute prior to midnight, balls 5 and 6 enter, and ball 3 exits. The trend continues…

The paradox asks, how many balls will be in the room at exactly midnight?

Shapira is kind enough to provide a hint. He reveals the mathematical way (or mathspeak) to describe the balls’ movement at any given time before midnight. It is that at (1/2)^n minutes before midnight, balls 2n-1 and 2n enter the room while ball n is ejected.

Using this, I concluded that there would be an infinite number of balls in the room at exactly midnight.

(1/2)^n is an exponential equation whose limit approaches zero as n increases, but never actually hits zero. So, there is an infinite number of balls entering the room before midnight. I also deduced that at each stage of n, two balls enter while one exits. That leaves us with a net addition of one ball at each stage.

If we add one ball at each stage, of which there are an infinite number, then there must be an infinite number of balls in the room at midnight. That was easy. It seemed as though I debunked the supposedly unanswerable.

Extrapolating this logic made me wonder why this paradox is called a paradox.

Unfortunately, there’s more depth to the paradox. I continued reading and discovered that there is no concrete answer. Many people, including Shapira, agree with my logic. However, there’s another answer that is as popular—that no balls are in the room at exactly midnight.

Interesting. Infinite balls versus no balls. It fascinated me how the two most common answers lie on opposite ends of the spectrum.

Here’s how people who answer “none” think…

Since each ball n leaves the room at (1/2)^n minutes before midnight, you can give the precise time every ball exits the room. Thus, you can say every ball must exit at some point, so none would remain at exactly midnight.

Once you understand the reasoning behind the two answers, they both make sense. Neither explicitly trumps the other. Fascinating, right?

Shapira also introduces two variations of the Ross-Littlewood paradox, and the second interested me.

In this variation, we again begin with an empty room and an infinite number of balls. But instead of sending two balls in at once, we send 10. So, half a minute before midnight, balls 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 enter the room while ball 1 is immediately removed. At a quarter of a minute before midnight, balls 11, 12, 13, 14, 15, 16, 17, 18, 19, and 20 enter while ball 2 is ejected. And so on. The trend continues for an infinite number of stages n.

Once more, we can deduce that an infinite number of balls or no balls will remain in the room at midnight, justifying either with the same logic from the two-ball variation. That’s not what interested me.

Instead, I obsessed over the mathematical way to explain the balls’ movement, which Shapira does not provide. So, I set out on a mission to find it myself.

Since I took notes while reading and planned to write down my thought process, I didn’t want to do so for 10 balls. It seemed like a lot. So, I simplified the variation to use five balls.

This means that half a minute before midnight, balls 1, 2, 3, 4, and 5 enter the room, and ball 1 leaves. Then, at a quarter of a minute before midnight, balls 6, 7, 8, 9, and 10 enter while ball 2 exits. And so on…

The mathspeak should have been obvious. However, since the mathematical side of my brain had been out of practice for months, I struggled. I made poor assumptions and followed red herrings.

So, I assumed that at (1/2)^n minutes prior to midnight, balls 2n-1, 2n, 2n+1, 2n+2, and 2n+3 enter the room. For n=1, this hypothesis holds true. However, it falls apart when n=2, where my variables have balls 3, 4, 5, 6, and 7 entering. But we know balls 3, 4, and 5 enter at n=1, so they cannot enter twice. My idea failed.

I tried another iteration, where balls (2^n)-1, 2^n, (2^n)+1, (2^n)+2, and (2^n)+3 enter at (1/2)^n minutes before midnight. But, once again, this fails for n=2.

I was stumped.

I couldn’t understand why deducing this mathematical explanation troubled me. I decided to move on to the next section of the book and return to this idea later. As I lay in bed that night, on the brink of falling asleep, my mind revisited the five balls.

This time, I approached the problem from a different perspective. Rather than thinking about all five balls entering at each stage, I thought about the final ball.

At half a minute before midnight (n=1), ball five is the last to enter. At n=2, ball 10 enters last. And ball 15 enters last at n=3. The answer became obvious. 5 = 5(1), 10 = 5(2), and 15 = 5(3). For each stage n, the last ball to enter is 5n. I realized I didn’t need 2n or 2^n in my explanation. Instead, I needed to focus on multiples of five.

This realization helped me deduce that the proper explanation for five balls entering the room as once is: at (1/2)^n minutes before midnight, balls 5n-4, 5n-3, 5n-2, 5n-1, and 5n enter while ball n is immediately ejected.

Going one step further, I can generalize the equation for an arbitrary number of balls entering at once, whether it be 2, 5, 10, or 100,000,000. This universal explanation is as follows:

For any b number of balls entering the room at once, balls bn-(b-1), bn-(b-2), bn-(b-3), … , bn enter the room at (1/2)^n minutes prior to midnight while ball n is ejected.

Cool, right?

I recognize that I’m not the best at writing about math concepts, so some of my ideas and explanations may be confusing. However, if you made it this far, I hope my ideas gave you a mental workout.

Although math may not be your favorite subject, there are many ways in which it can be entertaining, stimulating, and, dare I say, fun.

Who knows? Maybe you’ll join me and start reading mathematics books for fun, too.