What is Pascal's triangle?

Welcome to Maths 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. You know, Finn, sometimes people think you need lots of complicated rules to create something really magical in maths.

Yeah, it can definitely seem like you'd need a whole long list of complex steps to build a really big, fascinating pattern.

But the most beautiful ideas in maths usually start with just one tiny, simple rule. The kind of rule you can explain in a single sentence. Like what? What kind of rule are we talking about? Well, imagine building a pyramid. But we are not using giant stone blocks, and we are not standing in the hot desert of ancient Egypt.

We are building a pyramid entirely out of numbers on a piece of paper.

Okay, so I just write numbers in a big triangle shape. Where do I even begin?

You start by writing a single number one at the very top. That is the peak of the pyramid. Just a one floating there.

Haha, a pyramid with just a one at the top. Got it. That's a very small pyramid mirror.

But then you build the next row underneath, and the rule to build this whole thing is incredibly simple. The outside edges are always ones, and every number on the inside is exactly the two numbers directly above it put together.

Wait, that's the only rule. Just add the two numbers right above it. I don't have to multiply or divide anything.

That is the entire rule. You never need to learn a harder one. But that one tiny simple rule creates one of the richest, most mind-bending patterns in the entire history of mathematics.

Just from putting two numbers together, that's hard to believe.

Yes! It makes a shape completely packed with hidden secrets, and today we are going to explore what's inside.

So if I just keep adding the two numbers above, row after row, it grows downwards into a giant triangle forever.

Exactly right. Today it is mostly known as Pascal's Triangle, named after a brilliant French thinker named Blaise Pascal. Oh, I think I've heard that name. When did he live? He wrote a very famous book all about this number pyramid in the year 1654, and it was eventually printed in 1665. He was completely fascinated by it.

So he was the very first person to invent it. Actually, no. And that is one of the most incredible parts of this story. This shape is so fundamental to how the universe works that people all over the planet kept independently discovering it long before Pascal was even born. It's like the triangle was just waiting out there to be found.

Like who? Who found it first? Let's travel back in time. In ancient India, around 2000 years ago, a brilliant scholar named Pingala saw this exact pattern. But he wasn't doing maths equations on a chalkboard. He was studying the rhythms and beats of Sanskrit poetry.

Wait, poetry?

It was hiding in poetry? Yes, the way short and long syllables combine to make a rhythm follows the exact same pattern as this triangle. That is so weird. Who else found it? Around the year 1000, in Persia, the mathematician Al-Kharaji described it perfectly. And so did the famous Persian astronomer and poet Omar Kayan.

Another poet. It seems like people who love patterns keep finding it.

Exactly. And the story travels even further. In China, around the year 1050, a mathematician named Jiaxian discovered it. Later, in the year 1261, another Chinese mathematician named Yang Qui wrote it all down in a beautiful book. So it should probably have all their names on it. It's like a worldwide puzzle.

I agree. And in fact, if you go to a maths class in China today, it is still called Yang Qui's Triangle.

Okay, so a bunch of absolute geniuses from history all love this triangle. But I don't get why. If it's literally just numbers added together row after row, what's so special about it?

Because it's not just a stack of numbers, Finn. It is a mathematical treasure map. As the numbers get bigger going down the rows, familiar things start hiding inside. What kind of things?

I want to see where the treasure is.

Picture the triangle in your mind. At the top is a one. Under it is a one and a one. If you slide your finger down the very outside edge on either side, every single number is just a one. It's an endless wall of wands.

Okay, an endless wall of wands is pretty boring so far. I wouldn't call that treasure.

But move your finger exactly one diagonal line inside and read those numbers going down. Suddenly, you're looking at the regular counting numbers, perfectly in order. One, two, three, four, five, and on and on forever.

Wait, really? Just from adding the two numbers above? The regular counting numbers just naturally appear in a straight line.

Yes! Every time you build a new row, the counting numbers just keep marching down that diagonal. But keep going. Move your finger one more diagonal line inwards. Now you find what mathematicians call the triangle numbers.

What are triangle numbers? I've never heard of those.

Think about when we go bowling. You know how the ten pins are set up at the end of the lane in a perfect flat triangle shape.

Oh right. There is one pin at the very front, then two behind it, then three behind that, and four in the back row.

Exactly! If you count those pins, you get the numbers one, three, six, and ten. Those are the exact numbers of things you need to build a perfect flat triangle. And those exact numbers are sitting right there, perfectly in a row, inside the big number pyramid.

Wait, so the giant number triangle is secretly hiding a pattern of other smaller triangles inside itself? That is brilliant!

But here is where it gets really strange. The triangle doesn't just hold pretty numbers, it actually holds the answers to real-world choices.

What do you mean by choices? Like picking a flavour of ice cream?

Sort of. Imagine you have five friends, and you want to pick exactly two of them to play a board game with you.

Okay, I can picture my five friends. There's a lot of different pairs I could make.

If you try to guess exactly how many different pairings you could make, it feels really hard to work out in your head.

Yeah, I probably lose track of who I already counted.

But if you look at the sixth row from the top, which begins with one, then five, and find the third number along, the answer is just sitting right there, staring at you.

Wait, no way. It just tells you the answer.

Yes, it says ten. There are exactly ten ways to choose two friends from a group of five. The triangle perfectly maps out all the possible combinations for choosing things.

That feels like actual magic. It's literally just a pyramid of added numbers. How does it know about my friends?

That is the absolute wonder of it. The triangle is like a perfect map of all the ways things can be grouped together. And there is an even better visual secret hiding in the numbers.

What happens?

If you take a bright coloured crayon and colour in only the odd numbers, leaving the even numbers blank, a massive, perfectly repeating pattern of tiny triangles magically appears inside the big one. It looks like a beautiful infinite mosaic called the Serpinsky triangle.

Whoa! So it's not just maths, it's basically a piece of art too.

That beautiful hidden structure is exactly why the French thinker Blaise Pascal was so utterly obsessed with it in 1654. He realised this triangle could help figure out games of chance. Games of chance?

Like rolling dice or flipping coins.

Exactly. Imagine you are flipping a coin just for fun, and you want to know the exact chances of flipping three heads in a row.

That's really hard to guess. Sometimes you get lucky and sometimes you don't. It feels totally random.

It feels random to us, but the numbers in the triangle tell you exactly how the chances break down. It maps out all the possible paths the coin could take as it flips through the air.

So it's not just sitting quietly on a piece of paper, it's actually describing how things happen in the real world when we play games. That's the real magic.

It shows us that chance isn't just total chaos. Even when things feel completely random, like a spinning coin or tumbling dice, there is a deep, hidden order to the universe. And that order looks exactly like this triangle.

Okay, but here's my biggest question. How far down does the pyramid go? Does it ever hit the bottom and run out of numbers?

Nope. You could keep building it forever. The numbers get unimaginably huge, growing into the thousands, millions, and billions.

And the pattern never breaks.

It never, ever breaks. You could build it past the moon, and the counting numbers, the triangle numbers, and the choices would still be perfectly lined up. Alright, Finn, it is time for the quiz. Let's see what you remember about this magical shape.

Okay, I'm ready. Hit me with the questions.

First question. What shape do you get when you build this number pattern? Next one, which mathematician wrote about this pattern in China in the year 1261? Question three. Blaise Pascal wrote a famous book about it in 1654, but what country was he from? Here's number four. What shape appears if you colour in only the odd numbers?

Last question. What kinds of games did Blaise Pascal use the triangle to understand? Great job! Let's go through the answers and see how you did. The first answer is it builds a massive triangle or pyramid of numbers. For question two, it was the mathematician Yang Hui, and it's still called Yang Kui's triangle in China today.

Number three, the answer is he was from France. Question four. You get a massive, repeating pattern of smaller triangles hiding inside, called the Sia Pinsky triangle. And the final answer is he used it for games of chance, like flipping coins and rolling dice.

I am definitely going to try drawing this triangle later and colouring in the odd numbers. I want to see that hidden mosaic pattern for myself.

It is absolutely beautiful when you draw it out. The fact that all of this incredible richness comes from just adding two numbers together is amazing to me.

It really makes you wonder what else is hiding in plain sight. Just waiting for us to notice the rules. I'll be looking at everything a bit closer now.

If you loved exploring this number pyramid with us today, please leave us a five-star review. Just scroll down and tap the stars.

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And if you want a shout out on the show or just want to say hi, email us at hello at kidopoly.com. We'd love to hear from you.

See you next time on Maths is not boring.

Bye!

What is Pascal's triangle?

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