Sep 29

Math: When You Need to Feel Superior

From as long as I can remember, I’ve been good at math. It was just fun. The same way that Jerry West practiced basketball late into the night, I was the same way about math. Probably as a result, I’ve always hated math used as a substitute for intelligence. And it was in that context that I came upon the image above.

It was for something like joining Mensa or buying some “brain training” course of something. But I like a good math puzzle as much as anyone. So I figured I would solve it. It’s been a while since I’ve posted anything here, and it is as good an excuse as anything.

The thing about it is that it isn’t much of a puzzle. It’s just a math problem. And it takes zero creativity. All you need is to have taken a basic algebra course taught by a reasonably competent teacher. And you must have taken the course seriously enough to learn the material.

That shouldn’t have been hard because math actually is loads of fun. But the fact that someone would use it to construct such a simple, uninspiring problem does sum up everything that makes most people hate math. But without further ado, I guess I will get to it.

Why Someone Thought It Was Interesting

The two equations look similar. It’s just that one is with x and the other with x2. And so it looks like there ought to be some clever trick that allows the problem to simply pop out.

Unfortunately, that doesn’t seem to be the case. A mathematician better than I might see something brilliant. But I don’t. There are lots of ways to approach the problem, but they all end in a slog.

Solving the Equations

Sadly, this is a brute force problem. The first equation is just a quadratic that can be solved all by itself, admittedly, with two solutions.

We just multiply both sides of the equation by x, and end up with the following equation:

x2 + x – 5 = 0

Oh, how I do wish this were more interesting. There are a bunch of ways to solve this, one way or another, you get that the two values of x are:

x = (5 ± 21½)/2

If we choose the “plus” case, then x = 4.79. If we choose the “minus” case, then x = 0.209. And through the wonders of mathematics, if you put these two numbers into the second equation, you will get that:

x2 + 1/x2 = 23

Now an actual mathematical question is why this is so. What we just did was plug numbers into equations. This is a good example of why most people hate math.

Even worse is that some dimrod thought that solving this question indicated any kind of intelligence or, even worse, intellectual creativity.

But trust me: this is the kind of nonsense that your children are being taught in school. This is why the American Empire won’t last the century.

1. Kaleberg

Shouldn’t it be x^2 – 5x + 1 = 0? You got the right numeric answers, so I assume that this was a slip.

It’s an interesting enough algebra problem, but it isn’t particularly revealing. The two roots of the equation are inverses, that is 1 / 0.2087121525220801 = 4.791287847477919, which is a nice symmetry, but this doesn’t get used anywhere in the second problem.

This is a perfectly reasonable algebra problem. The whole point of algebra is like learning to read music or play scales or transpose chords. It’s a set of skills in manipulating symbols that wind up letting you do something in the real world like figuring out how much something would cost or how long it would take. It’s like the physicist Wheeler’s dialog with the rock. The rock argues it has free will, but Wheeler tosses it in the air and demonstrates otherwise.

On the other hand, that problem isn’t particularly useful as an intelligence test. It’s more of an achievement test. Have you learned enough algebra to solve it? Clever math problems usually require one to develop a particular insight or exploit some symmetry. That’s why they are fun. Using a problem like this as an intelligence metric is like Google’s silly write a program that runs first try at a job interview. An interviewer would probably learn more if the problem were to write a program that runs right on the third try since that would reflect the usual nature of the work.

BTW I use Maxima for this kind of stuff, to be honest. Life is too short for doing stuff that computers can do for us.

2. Barney

Yes, there is a more elegant way that doesn’t involve working out the 2 possible values of x and then plugging them in. This will look ugly without formatting, but I hope you can follow it:

Square both sides of the original equation:
(x + (1/x)) squared = 25
expand the left
x squared + 2 . x . (1/x) + (1/x) squared = 25
x squared + (1/x) squared + 2 = 25
x squared + (1/x) squared = 23

So it involves knowing the general identity (a+b)squared = a squared + 2ab + b squared, and figuring out that the (ab) in this case will always end up as ‘1’. Which is a reasonable test of algebraic knowledge, though not ‘brilliance’, I’d say.

3. mike shupp

Yeah, Barney gets the gold star.

The Big Trick here, the thing that seemed So Cool, wasn’t figuring out the values of x that made the first equation true so they could be plugged into the second equation. It was the X(1/x) = 1 thing,

Which is childishly obvious to anyone who remembers first semester high school algebra but …

4. James Fillmore

I hope this blog isn’t dying. It was my favorite thing on the Web for years. If it’s getting slowly retired, though, I can’t thank Mr. Moraes enough for what he accomplished. Kos, HuffPo, and other sites that lure writers in via promises of “exposure” don’t produce anything near the quantity and quality this site did. Impressive work, sir.

1. James McCollough

Couldn’t agree more. Far and away my favorite blog and political writer. Going all the way back to HH, no one else has had more of an impact on my thinking and writing. I offer my thanks as well.

Keep in mind, however, that these are tough times to write about politics (or anything topical). I mean, what do you say? It’s depressing. I wouldn’t be surprised if Frank returns to producing more content given time. If not I’ll certainly be bummed, but I’ll be understanding and thankful for all of his work.

1. Frank Moraes

Hey James! I haven’t heard from you in a while; it’s great to! James Fillmore is almost assistant editor around here just through his comments. Plus you can check out the actual articles he’s written. He’s a great guy. If humans were more like him, we would have a much better world.

1. James Fillmore

Too kind. Glad to know the fires haven’t gotten you. An old friend of mine has elderly parents in Santa Rosa, and they were absolutely terrified, but they’re OK. At least California is a rich state with a competent governor, so that’s going to help.