A Favorite Infinity Puzzle

Okay. There's this guy. And he's standing outside of a very dark room. At his feet are a pile of balls. Like this.

There are a lot of balls. In fact, there are an infinite number of balls. To keep track of this situation, he has labelled each of the balls with a name. Because he needs an infinite number of names and he wants them to have a simple pattern, he simply numbers the balls. The first one is named "1," the second one is named "2," etc. This way he can keep track of all of his balls.

Note:

Since there are an infinite number of balls we have entered the realm of imagination. It is not physically possible to have that many balls. There just aren't that many made, you know. So, as this problem goes on and the real-life speed gets to fast and the poor guy's arm would get tired--you'll see--just remember, this is now, right here, a problem in your mind. So, if you will accept that it's possible to talk about an infinite number of balls, let's pass on the other, equally irrelevant, physical impossiblities that come up in this problem.

Back to the problem.

Now, at 11:30 am, the guy picks up balls one and two, and throws them in the dark room across the way. Virtually instantaneously, some unknown force (another person? a rubber wall?) throws ball one back out. It rolls out of the dark room, but far away from the poor guy. He's not about to go get it.

So, ball two is in the dark room and ball one is outside, far away from the guy and his pile of balls. All the other balls are still in the pile.

Now, at 11:45 am (which you will notice is halfway to 12:00 pm from 11:30 am), the guy picks up balls three and four and throws them into the dark room. Almost instantaneously (ah, let's just say instantaneously), ball two rolls out and lands where ball one already is laying, out of reach.

So, ball three and four are in the dark room and balls one and two are outside but out of reach.

Now, at 11:52:30 am (which you will notice is halfway to 12:00 pm from 11:45 am), the guy throws balls five and six into the dark room, and that instant ball three rolls out and into the pile with balls one and two.

The guy continues this pattern. What pattern? He keeps cutting the time in half. For example, his last throw was at 11:52:30 am. Halfway between that time and 12:00 pm he will throw the next two balls. The lowest numbered ball still in the room will be tossed out. And so on. Since, time keeps on breaking up like this, he should be able to throw halfway between his last throw and 12:00 an infinite number of times. In other words, he should eventually throw all the balls.

When is he finished? At 12:00 pm! By that time, he will have gone through an infinite number of throws.

Break!:

How can he be finished? Doesn't infinity go on "forever?" Well, no. Sometimes it does, e.g., 1,2,3,4,5,..., but sometimes it doesn't, e.g., 1,1.1,1.11,1.111,1.11111,...The second set of numbers goes on forever but never even gets to 2!

Of course, you and I know it does get to 12:00 pm. It does every single day. And not only that, time doesn't stop dividing up, right? I mean there's 1 second, and 1/2 a second, and 1/4 a second, etc. It's not like time has a smallest unit, is it? So, by the time it's 12:00 pm, the guy has thrown an infinite number of times, and therefore all the balls.

This isn't physically possible? Yeah, I know. Pretty soon the guy would be throwing balls faster than the speed of light. Pretty soon the balls would simply vaporize. But guess what? This is why I mentioned before that this whole problem happens only in your mind. It isn't physical. It's logical.

Okay. So what's the question?

How many balls are in the dark room at 12:00 pm?

That's the question.