Saccharine Math Thoughts
So I was working on this math problem today. I'd seen it in a, like, math meme on facebook. Basically the meme was someone posted a comment asking for help with a question, and somebody else responded by just rewording the question. Like the question was "formulate this" and the answer was just "this," which was the joke. I didn't really have any strong evidence about the difficulty of the problem. It could have been part of the joke that it's actually really hard, or more likely it could just be a sort of mid-level problem required for some class, and that's why the person in the meme was asking for help. Who knows.
(The problem was generating a formula for the sum of the floor-functioned square roots of the first arbitrary n integers. It’s a fun problem. The answer is a big messy polynomial over 6. I won’t spoil it in case you’d like to try, but it‘s quite easy to plug and check values once you think you’ve got it. Any other math amateurs who want to message me about my approach or your own for it, or to swap solutions, please do!)
Anyway.
The problem involved adding up square roots. When I was a little kid I figured out some stuff about computing perfect squares, and squares (and little-ish exponents more generally) have always been one of my favorite topics in math. So I thought ok, cool, I can figure this out. I was doodling on it for maybe ten minutes before a tutoring session, then for three hours after it ended. As it turned out, the problem was, like, a real Odyssey. It required several conceptual breakthroughs for me, and many, many lines of work. I made mistakes at several points, had to check each line to find the mistake, and use various combinations of intuition and grit to correct them. And at various points the question was hanging over me of if I'd just get stuck and not be able to complete the problem. But I did.
The biggest, most exciting hurdle is something I could have gotten around easily if I looked up something really basic, but I didn't look anything up. I figured it out another way that feels way cooler. If I were a little smarter, it might seem obvious or like silly extra work. But as it was it felt great. Things aligned just right.
And...
At one point in the middle, making good progress, for some reason I remembered being at my grandparents' old house, when I was a kid. Maybe ten or twelve years old. I remembered the smell of that house. And how at certain points I'd get sheets and sheets of paper and work things out there. Just when I had a fit of inspiration about it, but at least one time in particular there I did. And I remember various other times I did math in a frenzy when I was growing up. And decided to try studying it in college, at least partially. I didn't feel committed to the idea - by the end of high school I guess I already had a feeling I wouldn't be sticking with math in a serious, dedicated, soberly responsible capacity.
But so there are two things. It feels good to really see a problem and sink my teeth into it and encounter weird obstacles and finally arrive at the super weird, unintuitive answer and have it turn out to be right, and to know that I hadn't done that for the last time yet, that I had at least this one in me, and maybe I could have more, too. And also a sadness that this feels like it might be it. That proving fairly trivial 3-hour results using high school methods and pure intuition is where my math road seems to stop. That the ceiling is low. That it's too advanced for me to explain to almost anyone I know in real life, but it'd be sort of laughably basic, maybe, for people who are really dedicated. Like "yes, that's a clever technique, and it's interesting it worked out that way, but you could also do it easier."
I wrote down my work really carefully. The question, and the solution, in detail. Like it was homework. But I didn't do that with homework, really. I gave up on actual homework. But I didn't give up on this problem. I got it. It's a crazy polynomial mess bound up in a fraction with half a dozen floor functions, but I got it. And it works. And it's right.
And maybe, alone, without anyone to teach or prove anything to, I can still be that clever math kid I used to be sometimes. And maybe that's enough.
