Mathematics is the search for patterns and for why they hold

0:00Before I say anything at all, look at these. Five figures. No numbers, no labels, no question. This is exactly how your textbook opens. These sit above the title of chapter one, and the book doesn't explain them. So have a go. What would the sixth one look like? Take a moment. Count the strokes if it helps. Whatever you just did in your head, that was mathematics. We'll come back to why.

0:31Now here's the thing your textbook says on its very first page, and it's a bigger claim than it looks. Mathematics is the search for patterns, and for the reasons those patterns hold. Notice that's two things, not one. The pattern, and the reason. And notice the word search. Not a pile of rules somebody finished long ago and wrote down for you to memorise. Searching. Discovering. Explaining. All happening right now, all still unfinished.

1:04You just did the first half a minute ago, when you looked at those five figures. So where do you find these patterns? The book's answer is: more or less everywhere. In nature. In your home and your school. In how the sun, the moon and the stars travel across the sky. In shopping and in cooking. In throwing a ball. In the games you play. In the weather, and in the technology you're watching this on.

1:34Which is worth pausing on, because most of that list is nowhere near a maths book. Patterns aren't things printed in textbooks. They're things that happen, and the textbook is where we write them down. But finding a pattern is only half the job, and I want to show you why with a puzzle. Fair warning: this next one is mine, not your textbook's. But it makes the point better than anything I know.

2:04Draw a circle, and mark some points on the edge. Then join every point to every other point with a straight line. Now count the regions the circle gets cut into. One point, and there are no lines to draw. One region. Two points, one line across the middle. Two regions. Three points make a triangle. Four regions. Four points. Now we get six lines, and eight regions. Five points. Ten lines, and count them carefully. Sixteen regions.

2:40So look at what we've got. One, two, four, eight, sixteen. It doubles. Every single time, it doubles. So tell me, before I draw it. Six points. How many regions? Say your answer out loud. I'll wait. Almost everybody says thirty-two, and honestly, that is an excellent guess. It's the only sensible guess. Four doublings in a row is a lot of evidence. So let's draw it. Six points. Fifteen lines. And now we count.

3:13Thirty-one. Not thirty-two. Thirty-one. One short, and the whole beautiful pattern falls apart. Now, that guess wasn't stupid. It was the best guess available. The problem is that checking cases can never finish the job. It can only ever suggest. Five confirmations bought us exactly no guarantee about the sixth. And that's the missing half. Not what happens, but why it has to. Because a reason works differently from a check.

3:48A check settles one case. A reason settles every case at once, including the ones nobody has looked at yet. If we'd had a reason for the doubling, we'd have known whether it survived to six without drawing anything. That's why your textbook puts the pattern and the reason side by side, on the same footing. The pattern is the interesting part. The reason is the part you can trust.

4:18And here's what makes reasons so valuable. A reason can travel. It can be picked up and carried into a situation it was never discovered in. Your book gives two examples of this, and they're both enormous. Here's the first. People watched the sky. For thousands of years, they watched. And they noticed patterns in how the stars, the planets and their moons move. Then somebody asked why, and out of that came the theory of gravitation.

4:50Now look what that reason was good for. It wasn't just about planets any more. We used it to launch satellites of our own. And to send rockets to Mars, and to the Moon. Nobody watching the sky was trying to build a satellite. The reason went somewhere its pattern had never been. The second example is closer to home. Inside every living thing there's a genome. It's a very long set of instructions, written in a four-letter alphabet.

5:24For a long time it just looked like an enormous jumble. But people found patterns in it. Sequences that repeat, pieces that appear in some people and not others. And then, again, they asked why. Why does that piece sit there, and what does it do? Out of those reasons came the ability to diagnose diseases, and in some cases to cure them. Same shape of story, both times. First the pattern. Then the reason. Then something nobody expected.

5:59And always in that order. Your book also calls mathematics an art, as well as a science. Which sounds like a strange thing to say. But think about what you actually did with those five figures at the start. Nobody gave you a method. There isn't a procedure for noticing. You looked, you guessed, you tried something, and you probably threw away a wrong idea or two. Finding a pattern is a creative act. So is finding the reason behind it.

6:33There's no formula for having an idea. That's the art part. The science part is what happens next, when you check the idea honestly, and let it die if it deserves to. Like our doubling did. Now the book turns the question back on you, and it's worth taking seriously. Where does mathematics show up in an ordinary day of yours? Not in a maths lesson. In an ordinary day.

7:02Doubling a recipe. Splitting a bill. Working out if you can get to school in ten minutes. Deciding where to stand to catch a ball. Keeping score. Guessing whether it'll rain. And the book asks a second, harder question. How has mathematics moved humanity forward? These aren't questions to write an answer to. They're questions to argue about, out loud, with somebody who disagrees. So here's the stance for the rest of this chapter, and really for the rest of the subject.

7:39From here on, almost every pattern you meet comes with a question attached. Why does that happen? And that question is not extra credit. It isn't the hard bonus at the end of the exercise. It's the actual work. The pattern is where you start, not where you finish. So when you spot something, get into the habit of asking two questions instead of one. What's the pattern? And why does it have to be that way?

8:09The first one makes you a good guesser. The second one makes you a mathematician. Which brings us back to those five figures from the beginning. One stroke. Then three. Then six. Then ten. Then fifteen. You probably worked out that the sixth one has twenty-one. And you might have spotted the rule. Each time, you add one more than you added last time. Which is a lovely pattern. But now you know that's only half the job.

8:40So here's my question for you, and it's the real one. Why does it have to grow that way? What is it about the shape that forces the jumps to get bigger by exactly one each time? Don't just check it. Explain it. Tell me in the comments, and I'll see you in the next one, where we go hunting for rules.