Find out about an old paradox dreamed up by mathematicians in the roaring twenties. It takes a simple (and hypothetical) hotel and uses it to peer into the eyes of the infinite.

After a long, tedious journey, you head to a hotel and try to check in. It's a twenty-room hotel, and all the rooms are full. You're informed that you can't stay there. You walk to the next hotel. It's a hundred-room hotel. All the rooms are full, so you can't stay there. The next hotel has 2000 rooms, and they're all full. You can't stay there.

The last hotel in town is Hilbert's Grand Hotel. All the rooms are full. The clerk at the desk informs you that indeed, you can stay there. Why? Because it's an infinite-room hotel. But really, you argue, it doesn't matter how many rooms are in the hotel. What matters is that all the rooms are full. An infinite number of rooms, all of which are occupied still can't accommodate you.

It's true that the rooms are all occupied, but infinity has a strange characteristic. In most tallies, the amount of a certain group of numbers (say, the odd numbers) is smaller than the overall amount of numbers. For every group of odd-numbered rooms (1, 3, 5, 7...), there have to be a larger number of overall rooms (unless the total number of rooms is one). For every group of even numbered rooms, there have to be a larger number of total rooms (even if there are only two rooms, only one of them will have an even number). But infinity is different. Take away one, or a hundred, or a million, and it's still infinity. Add one, or a hundred, or a million, and it's still infinity. Divide it by anything you want. It's still infinity.