Mathematical Insight in 4 Easy Steps: Huh, Wow, Duh, Wow

Sometimes I think about mathematics in terms of this sequence: Huh, Wow, Duh, Wow. You can stop at any point, but getting to that last wow is really worth the trip.

First you are oblivious. You are marinating in the Slough of Uncaring as Netflix flickers and the unseen stars wheel overhead. But then you notice an odd fact, or maybe someone brings it to your attention. Huh? you say. What’s going on here? There is something there that glows and sparkles. It draws you in.

Example: Take a number, say 7, and square it: 49. Take the numbers on either side of 7, 6 and 8, and multiply them together and you get 48. That’s one less than the square. Maybe you stop here and carry on with your life. So what. Big deal. But maybe you say: Neat! Are there more examples of this?

Does it work with other numbers? 10 squared is 100, and 9 times 11 is 99. Yes!

How about a huge number? Let’s see. 2558 squared is 6,543,364. 2557 times 2559 is 6,543,363. Sure enough! That’s actually kind of amazing!

This is the first wow, the little wow. If you get off the bus here, you’re still left with a sense of the magic of numbers, the tingling sensation of recognizing beauty. But of course there’s more. There’s always more.

The duh comes when you realize that this observation follows from polynomial multiplication.

(n+1)(n-1) = n2 – 1

Or maybe you realize that you can visualize these products as straightforward geometry. You can see the square that represents 49. Now take one square away and shift the top row to the side. The remaining rectangle is now clearly 6 times 8, or 48.

Well, duh. It’s not magic! I can see the rabbit at the bottom of the hat. It seemed interesting at first, but now I can see that it’s not the least bit interesting. In fact, it’s obvious. It’s inevitable! It would only be interesting if it weren’t true. I don’t know why I bothered to pick up a pencil.

The second wow is subtle and vast. Maybe you see how your observation ties neatly together with some other elements of number theory that you happen to know. Or you might prove something related. Sometimes it’s just the realization that there exist dozens of ways to see how obvious it is. The little wows keep piling up until they overwhelm the duhs. Ultimately, the big wow comes when you realize that you get to live in a world where things like this are possible, where in fact they happen all the time, so long as you keep your eyes open. When you see that the world is stitched together from glowing threads that are both inescapable and miraculous. Obvious and mysterious.

A clever person stuck at Duh can do real damage. They might find terrible pleasure in smiting people with the Duh stick every time they hear a little “Wow.” Armed with the Duh stick and the suffocating net of Well Actually, they can snuff out curiosity and drown delight. There is no school-level mathematical proof or observation that isn’t Duh-obvious to somebody. BUT there is no mathematical proof or observation that isn’t Wow-delightful when approached gently and mindfully. Some days I am amazed that multiplication commutes.

The n2 – 1 example I gave above actually happened to me when I was in high school. I have a vivid memory of surfing rapidly through all four stages of insight. I was lucky to have a good math teacher who could help me past the duh to see one corner of the shining sky.

Never cheat someone out of the wow they deserve. Most wows are little wows, flowers. But every now and then you can help someone navigate through a thicket of duhs to the big wow, a shower of fireworks in the night sky, or the glowing aurora borealis. That’s when your mouth hangs open in wonder. And you think: it’s been there all this time.

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