 Once upon a time there was a traveler, who had been walking for a very long time, and was tired and hungry. Suddenly, out of the mist there appeared a very tall castle, at the top of the castle, a window with light, and a delicious smell coming out of it. But there was no door, and no way to get up, so the traveler shouted, Hello there, do you have any food and shelter for a weary traveler? A castle guard looked out the window and shouted, Hello, we have food and shelter to spare, but you must come up by our magical stairwell. Without a very long stairwell emerged from the castle and led all the way up to the window. Each step after the first one was blocked by a magical gnome. The traveler stepped down the first step, but when he went to step on the next one, the gnome said, Halt, to go to the next step you must give me a gold coin. But I have no gold coins, he said. The traveler said with sadness, I am just poor, very, very traveler and eating a place to lay my head. And the gnome said, well, then I guess it stinks to be you, doesn't it? And then a whole staircase vanished. The traveler was mad. Now look here, he shouted at the castle guard. That's not fair. If you have food and shelter to spare, where do you get off charging me for the rate of one gold coin per step? You should let me up, the guard looked down. I suppose you're right, but we can't change the thing with the gnomes. So instead I'll drop down one gold coin whereupon a gold coin fell to the ground next to the traveler. And I'll tell the gnomes, if you give them a gold coin, they'll give you a new one in return. So you can give the coin to one gnome, you'll get a new one that you can give them in the next gnome, and then you'll get another you can give to the next gnome. And so until you're up here with us, it seemed a little weird, but the traveler was getting hungry. So he said, OK, the magical gnome stairwell appeared again. The traveler picked up his gold coin and started thinking about dinner. But alas, the stairwell didn't reach all the way to the ground. The first step stood hovering 10 feet over his head. Now hang on, man, said the traveler. You've given me a way to get from one stair step to the next. But now I can't even get on in the first place. Stop messing with me, the guard said. Quite all right. Hang on to your coin for a sec. Second, then the stairwell disappeared for a moment and then reappeared on the ground. The traveler stepped on to the first step, took a deep breath, and then gave his gold coin to the gnome. The gnome smiled, and the gnome smiled, and said, thank you. Here's your new coin. He handed the traveler a new and different coin. He stepped on to the second step and handed the new coin to the gnome on the third step. Thank you, he said, and here's your new coin. And he stepped on to the third step. And he only went. Every time he went a new step, he had a way of getting to the next step. And so after a lot of climbing, which was kind of hard because he was tired and all, last made all the way up, and he ate and slept, and everyone lived happily ever after. If you want to know what this has to do with mathematics, stay tuned for the next screencast.