Exponential Growth for Kids: How 42 Paper Folds Reach the Moon

0:00 Welcome to Mass Is Not Boring by Kidopoly.com. I'm Mira. And I'm Finn. If you haven't already, tap follow wherever you're listening so you never miss an episode. So, what impossible thing are we looking at today, Mira? Finn, I want you to imagine a single ordinary piece of paper. Just the standard kind you use to draw or print things out.

0:23 Okay, I'm picturing it. It's white, it's flat, and it's extremely thin. Exactly. Now, imagine you could fold it in half perfectly, and then fold that in half again, and just keep folding it, over and over, stacking it higher. Okay, it's getting thicker. If you could keep doing that, how many folds do you think it would take for that stack of paper to reach all the way to the moon?

0:47 The moon? But paper is basically microscopic when you look at it from the side. You'd have to fold it a billion times. Maybe a trillion. Nope, just 42 times. Wait, what? 42 folds of a tiny piece of paper reaches the moon? That cannot be right. It is entirely true, and it is one of the wildest facts in the universe.

1:11 Exactly 42 folds! Today we are talking about an idea called exponential growth. Exponential growth. Okay, I've heard the name, but how in the world does folding a piece of paper 42 times get you all the way into outer space? Well, let's look at the paper itself. A normal sheet of paper is only about a tenth of a millimeter thick.

1:36 It's so thin you can barely feel the edge of it. Yeah, if you stack a hundred of them, it's still only as thick as my finger. Exactly. But when you fold a sheet in half, it becomes twice as thick. Fold it again, it's four times as thick. But it's still just paper. And honestly, I folded paper a bunch of times making paper aeroplanes.

1:59 It gets incredibly stiff and impossible to fold pretty fast. You are right about that. In fact, for a very long time, people actually believed it was physically impossible to fold any piece of paper in half more than seven times. Just seven times? That sounds like a challenge. I bet I could beat that.

2:20 That is exactly what a high school student in California thought back in January 2002. Her name was Brittany Galavan, and she was trying to get extra credit in her maths class. So did she try to beat the seven-fold rule? She didn't just try, Finn, she completely shattered it. People told her it was impossible, but Britney refused to listen.

2:44 She actually figured out the exact maths of how paper behaves when you fold it. Whoa! So what did she use to actually fold it? You can't just use standard printer paper for that. You can't. She actually found a massive 4,000-foot-long roll of really thin tissue paper. Basically like a giant roll of toilet paper.

3:06 Haha, a 4,000-foot roll? Where do you even unroll something that massive? She took it to a shopping mall in California, rolled it all the way down the corridors, and managed to fold it in half 12 times. Nobody in the world had ever done that before, and 12 folds is still the record for a single sheet.

3:27 That is amazing! She proved the famous seven-fold rule was completely made up. But wait, 12 folds is still a really long way from 42 folds. It is. Brittany proved the maths, but actually doing 42 folds in real life is physically impossible for anyone. We just don't have paper long enough, and it would become far too rigid to bend.

3:52 So what would happen if we could magically keep folding it? I still don't understand how it reaches the moon. Let's picture it in our heads. This is how exponential growth works. It starts so, so slow. You fold that paper once, it's two sheets thick. You fold it five times, it's about as thick as a small notebook.

4:14 Okay, a notebook is nothing. I can hold a notebook in one hand. We're trying to get all the way up to space, Mira. I know. Keep doubling it. Tenfolds, and your paper is about as thick as your handspan. Still small, right? Yeah, I'm definitely not reaching the stars with my hand. This is taking forever.

4:35 But here is where the doubling starts to run away from you. At 17 folds, your stack of paper is suddenly as tall as a house. Wait, hold on. From the width of a hand to a whole house in just seven more folds? That feels completely impossible. That is the magic of exponential growth. Every single step adds everything you had before, all over again.

4:59 By 27 folds, your stack of paper isn't a house anymore. It is a mountain. It is thicker than the height of Mount Everest. No way, Mount Everest? Just from folding a piece of paper 27 times. I'm picturing this giant snowy paper mountain. Yep, and you only have 15 folds left to go. At 30 folds, you have shot past the clouds, and you are out in space, though still below where satellites orbit the Earth.

5:31 I'm getting a little dizzy just thinking about it. The growth just exploded out of nowhere. And when you hit that magic number, exactly 42 folds, the stack of paper is hundreds of thousands of kilometres thick. It has crashed right into the moon. That is the wildest thing I have ever heard. It feels like my brain can't even hold a thought that big.

5:54 It is literally like a magic trick. It does feel like a magic trick, but it's just the plain truth of doubling. And because doubling tricks our human brains so completely, it's actually the heart of a very old legend. A legend? Like an ancient story about magic. Better than magic. It's a legend from ancient India about a wise man named Cesar, who was said to have invented the game of chess.

6:21 Oh, I love playing chess. What did Cesar do? The story goes that the king was so amazed by the brilliant new game of chess that he called Cesar to the palace and asked him to name any reward he wanted. Nice! I would absolutely ask for my own castle. Or maybe a room filled entirely with infinite ice cream.

6:42 Well, Cessa was much more clever than that. He brought out his wooden chessboard, and he asked the king for something that sounded completely tiny. He asked for one single grain of rice on the first square of the board. One single grain of rice? That's not even a crumb. You can't even make a snack out of that.

7:03 Then Cessa asked for two grains on the second square, four grains on the third square. He asked the king to keep doubling the rice for all 64 squares on the board. Oh no! I know what doubling does now. The king fell right into the trap, didn't he? He really did. The king actually laughed out loud. He thought it was the cheapest reward anyone had ever asked for.

7:29 He ordered his servants to bring a small bag of rice. But by the time they reached the middle of the chessboard, the king was bringing out heavy sacks of rice on carts. And they still had the whole second half of the chessboard to go. Exactly. To finish the 64th square, the king would have needed more rice than the entire world has ever grown in the history of the planet.

7:52 A pile of rice bigger than a massive mountain range.

7:56unknown Hahaha.

7:57 The king must have been absolutely terrified when he realised. It is just a legend, of course, so we don't know if the king ever paid up. But it shows perfectly how exponential growth sneaks up on you. It really does. It looks so completely harmless at the start. Just like a single grain of rice or one millimeter of paper.

8:17 That sneaky, quiet start is exactly why exponential growth surprises us everywhere. Have you ever heard the famous riddle about the pond of the lily pads? I don't think I have. Tell me. Imagine a quiet pond. On the very first day, there is just one tiny lily pad floating on the water. Every single day, the number of lily pads doubles.

8:44 Okay, one becomes two, two becomes four. I'm following. If it takes exactly 30 days for the pond to be completely covered in lily pads, on what day is the pond only half covered? Hmm, well, there are 30 days total, so half covered would have to be right in the middle. Day 15? Nope, it's day 29. Wait, really?

9:10 Oh, oh I see it. Because if they are doubling every single day, the day right before it's completely full is when it's exactly half full. Exactly! On day 29, the pond is only half covered. So if you looked at it, it would still seem like there's plenty of open water left. Then, boom, the very next day it is completely packed.

9:34 That is actually quite creepy, but so cool. So if you aren't paying close attention, it just takes overnight. Yes, and we see exponential growth in our real everyday lives all the time, too. Really? Like where? Think about how rumours spread on the playground. You tell a secret to two friends, then they each run and tell two of their friends.

9:59 Suddenly, by lunchtime, the whole school knows. Oh wow, yeah. It's like my friends are the squares on the chessboard. It is also exactly how germs spread, which is why washing your hands is so incredibly important. One single germ becomes two, two become four, and so on. So this isn't just about reaching the moon or ancient kings making mistakes.

10:25 This math is literally happening in my school and on my hands right now. Exactly! And now that you know its real name, exponential growth, you will start spotting it hiding everywhere you look. Alright Finn, time for the quiz. Are you ready? Okay, I'm ready. First question. How many times would you have to fold a piece of paper to reach the moon?

10:51 Next one. Who is the student who set the record by folding a piece of paper 12 times? Question 3. What kind of paper did she use to break the record? Here's number four. What game was the king playing in the ancient Indian legend about Rite? Last question. In the riddle, if a pond is full of lily pads on day 30, what day is it half full?

11:28 Great job! Let's see how you did. The first answer is you would have to fold it exactly 42 times. For question two, the record-breaking student was Britney Galavan. Number three, the answer is, she used a massive roll of tissue paper rolled out in a mall. Question 4. The legend is about the invention of chess.

11:51 And the final answer is, the pond is half full on day 29. I still just cannot get over that paper mountain fin. It just doesn't feel like it should be true, even though I know the math works. Haha, I know! I'm definitely going to try and fold my homework 42 times tonight and tell my teacher it's stuck on the moon.

12:13 Good luck with that. Hey, if you enjoyed going all the way to the moon with us today, please leave this show a five-star review. Just scroll down on your app and tap the stars. Yeah, and tap follow so you never miss the next strange math idea. Oh, and we have sister shows too. History, science, and philosophy's not boring.

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