You have in front of you an empty box. You also have on hand an infinite source of billiard balls, numbered 0, 1, 2, 3, 4, and so on forever.
At time zero, you place balls 0 and 1 in the box.
In thirty minutes, you remove ball 0 from the box, and place in two new balls (2 and 3).
Fifteen minutes after that, you remove ball 1 from the box, and place in two new balls (4 and 5).
7.5 minutes after that, you remove ball 2 and place in balls 6 and 7.
And so on.
After an hour, you will have taken an infinite number of steps. How many billiard balls will be in the box?
At time zero, the box contains two balls (0 and 1). After thirty minutes, it contains three (1, 2, and 3). After 45 minutes, it contains four (2, 3, 4, and 5). You can see where this is going…
Naively taking the limit of this process, we arrive at the conclusion that the box will contain an infinity of balls.
But hold on. Ask yourself the following question: If you think that the box contains an infinity of balls, name one ball that’s in there. Go ahead! Give me a single number such that at the end of this process, the ball with that number is sitting in the box.
The problem is that you cannot do this. Every single ball that is put in at some step is removed at some later step. So for any number you tell me, I can point you to the exact time at which that ball was removed from the box, never to be returned to it!
But if any ball that you can name can be proven to not be in the box.. and every ball you put in there was named… then there must be zero balls in the box at the end!
In other words, as time passes and you get closer and closer to the one-hour mark, the number of balls in the box appears to be growing, more and more quickly each moment, until you hit the one-hour mark. At that exact moment, the box suddenly becomes completely empty. Spooky, right??
Let’s make it weirder.
What if at each step, you didn’t just put in two new balls, but one MILLION? So you start out at time zero by putting balls 0, 1, 2, 3, and so on up to 1 million into the empty box. After thirty minutes, you take out ball 1, but replace it with the next 1 million numbered balls. And at the 45-minute mark, you take out ball 2 and add the next 1 million.
What’ll happen now?
Well, the exact same argument we gave initially applies here! Any ball that is put in the box at any point, is also removed at a later point. So you literally cannot name any ball that will still be in the box after the hour is up, because there are no balls left in the box! The magic of infinity doesn’t care about how many more balls you’ve put in than removed at any given time, it still delivers you an empty box at the end!
Now, here’s a final variant. What if, instead of removing the smallest numbered ball each step, you removed the largest numbered ball?
So, for instance, at the beginning you put in balls 0 and 1. Then at thirty minutes you take out ball 1, and put in balls 2 and 3. At 45 minutes, you take out ball 3, and put in balls 4 and 5. And so on, until you hit the one hour mark. Now how many balls are there in the box?
Infinity! Why not zero like before? Well, because now I can name you an infinity of numbers whose billiard balls are still guaranteed to be in the box when the hour’s up. Namely, 0, 2, 4, 6, and all the other even numbered balls are still going to be in there.
Take a moment to reflect on how bizarre this is. We removed the exact same number of balls each step as we did last time. All that changed is the label on the balls we removed! We could even imagine taking off all the labels so that all we have are identical plain billiard balls, and just labeling them purely in our minds. Now apparently the choice of whether to mentally label the balls in increasing or decreasing order will determine whether at the end of the hour the box is empty or packed infinitely full. What?!? It’s stuff like this that makes me sympathize with ultrafinitists.
One final twist: what happens if the ball that we remove each step is determined randomly? Then how many balls will there be once the hour is up? I’ll leave it to you all to puzzle over!
One thought on “A Supertask Puzzle”