## The Ross-Littlewood Paradox

Whenever I feel especially frustrated or burned out I will often turn to art or music to rejuvenate myself. Whenever art and music aren’t up to the task, I turn to mathematics. I once spent an entire afternoon working to figure out a formula that could predict prime numbers. This week was one of those frustrating burn out weeks. I started out looking into Hilbert’s 23 problems which were way over my head and then somehow ended up at the Wikipedia page for the Ross-Littlewood Paradox which is really just a variation on Zeno’s Paradoxes or the Thomson’s Lamp Paradox. The Ross-Littlewood Paradox is often referred to as the *balls and vase problem* or the *ping pong ball problem*, but I like to think of it as the *marbles in a jar problem* because that just seems much more elegant to me.

For this paradox you are going to need an empty jar and infinite marbles. You can order marbles online from LandofMarbles.com. I recommend their Big Box o’ Marbles deal but you could splurge on the Fiber Optic Marbles if you really wanted to go high class. Let me know when you have those handy and we’ll continue. Okay fine, you don’t actually need a jar or any marbles. You can just do it in your head. You will have two minutes to complete this paradox at which point I will ask you all to put down your marbles if you haven’t lost them by then. Okay, let’s begin.

**Step One** is to stare at your empty jar for 1 minute. That’s right, 60 seconds of pure empty jar staring bliss. If you are having difficulty with this step you may not want to continue and there may still be time to get a refund on all those marbles you purchased. When the first minute has passed you can move on to Step Two. You have only 1 minute left to finish this paradox.

**Step Two** is to put marbles 1 through 10 into the jar and then remove marble 1. Okay, I’m beginning to think I’ve lost some of you at this point. Let’s pretend that the marbles have numbers on them corresponding to the order in which you add them to the jar. If it helps, you can put the ten marbles in one at a time and then remove the first one you put in. Got it? And don’t forget the order in which you’ve added your marbles. I would have asked you to write numbers on all your marbles but frankly, we don’t have that kind of time. Some of you advanced students may observe that the jar now has 9 marbles in it. This must feel very rewarding for you but don’t get cocky, you only have 30 seconds to complete this step and there are still an infinite number of steps left to go. Once 30 seconds have passed you can move on to Step Three. You now have only 30 seconds left to complete this paradox.

**Step Three** is very similar to Step Two but you have to do it twice as fast because you only have 15 seconds to complete this step. Add marbles 11 through 20 to the jar and then remove marble 2. That’s right, you are adding 10 new marbles to the jar and then removing the second marble you added from Step One. Some of you in the front row are smiling because you’ve figured out there are 18 marbles in the jar right now. However, with only 15 seconds left in this paradox, you are beginning to question the wisdom of the sizable investment you made purchasing those infinite marbles and renting that infinitely large shed to store them in. How the hell are you going to use up all those marbles in the next 15 seconds? And no, we are never going to put an infinite number of marbles into the jar in a single step. That was a good guess though. After Step Three you have only 15 seconds remaining to complete this paradox.

Hopefully some of you will have guessed what **Step Four** might be, but if not don’t worry. I’ll walk you through it because I’m that kind of guy. For this step you need to put marbles 21 through 30 into the jar and then remove marble 3. Hopefully you’ve figured out a quicker way to do this because you have only 7.5 seconds to finish this step. That’s less than one second for each marble you’re adding! I know you may think I’m a bit of a task master but in a few more steps you going to be introduced to the idea of a supertask. Those of you who can see where this is going are probably wondering whether we should have done some stretching exercises or drank a full pot of coffee before we began, but no. I assure you by the time we are done, your fingers are going to be cramped and you won’t have the time to be making trips to the bathroom. And yes teacher’s pet, there are now 27 marbles in the jar but you only have 7.5 seconds to use up the rest of those infinite marbles.

We have finally reached **Step Five**! Some of you look relieved and are thinking with four steps down and only 7.5 seconds remaining, we must be close to the end. You seem to have forgotten that I mentioned in Step Two that there are an infinite number of steps so unfortunately, you won’t be going anywhere for a while – or at least not for another 7.5 seconds. I can tell by the band-aids that some of you are not using a jar with a wide enough brim to allow your hand to enter and exit easily. I blame myself for not being specific about the size of jar you will be needing before we started, but considering you needed an infinite number of marbles you should have been able to figure this out. OKAY QUICKLY! You have only 3.75 seconds to add marbles 31 to 40 to the jar and remove marble 4! No time for questions! Only 3.75 seconds left to finish the paradox!

Whoah! I can tell by the terrified looks on your faces that some of you are beginning to panic so I am just going to describe all the rest of the steps for you right here and now. Yes, all infinite of them. For this step and all the other steps, you just need to add the next ten marbles and then remove the marble that has been in the jar the longest. Also, you will need to perform each step twice as fast as the previous step. So, for **Step Six** you have 1.875 seconds to add marbles 41 to 50 and remove marble 5. Remember, you removed marbles 1 through 4 in steps two through five above, so marble 5 is now the marble that has been in the jar the longest. **Step Seven** gives you .9375 seconds to add marbles 51 to 60 and remove marble 6. **Step Eight** gives you .46875 seconds to add marbles 61 to 70 and remove marble 7. You can figure out the rest of the steps on your own. What is developing here is known as a *supertask*: a quantifiably infinite number of operations that occur sequentially within a finite interval of time.

Okay! Your two minutes are up. Put down your marbles! Now we get to the paradox part. **Who can tell me how many marbles are in the jar right now?**

Well don’t all raise your hands at once. Okay, the kid who used the imaginary jar and marbles has figured it all out in his head and wants to answer. He knew his answer before I even explained steps four through infinite. So how many marbles are in the jar after 2 minutes? “There is an infinite number of marbles in the jar” he says. Give that kid a marble! We can now all see why he is the president of the chess club. He’s smart and he knows it. He’s the master of anything having to do with numbers and using an *infinite* amount of anything doesn’t scare him. “How did you come up with that answer?” I ask him. “Easy” he says. “With each step after step one, the jar has a net gain of nine marbles and there are an infinite number of steps. Nine times infinite equals infinite!” And he’s right of course, so we pat him on the head for being so smart and tell him to sit back down.

Oh wait, someone else wants to answer? Yes, the shy girl who always sits by herself appears deep in thought and wants to challenge our genius. Well, let’s hear what she has to say. “And how many marbles do you think are in the jar after 2 minutes?” I ask her. “There are no marbles left in the jar” she says. This makes a lot of people in the room laugh including our chess club president. “And how did you come up with that answer?” I ask. “Well, at each step after step one, we removed one marble from the jar and if there are an infinite number of steps, then we must have removed all the marbles.” It’s a little more quiet in the room now but we can still hear some chuckling. She’s feeling a little uncomfortable now so she blurts out “I can prove it mathematically!” I say “Please Do” and she continues. “For every marble, 1 through infinite, I can tell you at which step that marble should have been removed from the jar”. Someone shouts out “How about marble gazillion! When did we remove that marble?” It was probably the kid who gave us our first answer. “Easy” she says, “We removed the gazillionth marble at step gazillion and one.” And she’s right of course. We congratulate her on her creative thinking and she sits back down.

Okay class is dismissed, but wait! We’ve heard from the smart kid and the creative kid, how about the rest of the class? The class bully raises his hand. Oh, this is going to be good. “This is bullshit!” he says. “My dad gave me a whole hour to do three chores last night and I didn’t get them done in time. There is no way you can finish an infinite number of chores in just two minutes!” We all stop and think about that for a minute and then realize that he’s right of course. If you have an infinite number of tasks to perform, regardless of how long they each take, you will never be finished. So we congratulate him for not giving a shit and everyone leaves class happy.

And that ladies and gentlemen, is why it’s called a paradox. Whether you’re smart, creative, or just don’t give a shit, you’re still going to lose all your marbles.

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