My school does not have a culture of math contests. But I’m on a mission to change that.

This year we signed up for the New York Math League contests, which actually seem to be getting a dedicated small set of students taking them. (I think the biggest problem with getting a sizable number of students in our  school to take these contests is that our kids are so busy that not many kids can take the exam after school on the day we administer it. They have sports and other commitments.)

However I’ve taken it upon myself to make a huge push to have more students take the AMC 10/ AMC 12 math contests this year. How?

• I’ve spoken individually to the other 6 math teachers about speaking to their classes about the contest, and I gave them some sample problems to give their kids if they want more information. I’ve convinced a few to offer some incentive for their kids if they take the exam (example: I’m giving extra credit, which I almost never offer, and another teacher is giving a homework pass).

• The math club student leaders are making a facebook group (their idea!).

• I’m having students “register” for the contest (basically fill in the blank on a sheet which says “My name is _________, I am in the ____ grade, and I am awesome because I am going to take the AMC!”). I’m going to use this list of students to send a reminder email to the kids. I’m also going to send a letter to their parents explaining the contest, and why we in the math department are really happy that their kid is going to take it. Basically, I want to get the word out to as many people about the contest, in a few different ways.
• I’m trying to get some money so we can order  pizza for the kids after they take the exam.

I think last year we had 10 or 15 students who took the exam. I really want to ramp it up this year. And if we can do this for a few years, we might be able to develop a culture where taking the AMC is a “normal” thing to do — where kids in the accelerated track all want to take it and interested kids in the regular track are encouraged to take it. I’m trying to slowly and consciously engineer a shift in school culture. It’s hard to figure out how one can create a culture shift.And I know, it’s a really small culture shift, but in my opinion, it’s really important for our department and for our kids.

One day a few weeks ago I had a day to kill with one of my two multivariable calculus students. So I decided to talk a bit about something which intrigued me when I first learned about it.

If you have any linear polynomial (), then it is easy to come up with the algebraic solution for any and . (Obviously it is .)

If you have any quadratic polynomial (), then it is still pretty easy to come up with the algebraic solution for any , , and . (Obviously it is .)

What about the cubic? quartic? quintic? higher?

I had my student take out his laptop and we used SAGE online to see what we got. We entered something like this:

I then asked if we could, for any coefficients, find all zeros (real or complex) to the polynomials.

Before pulling out the laptops, we had already answered the first two. Which we  confirmed with SAGE:

Then I asked about the cubic.

He said he knew how to find the solutions to some cubics. Like he could graph them and if they hit the x-axis three times, he could get the solutions. Not good enough. We wanted formulas for the solutions. For any coefficients. Without graphing. Like we had for the quadratic.

He (naturally) said he didn’t know.

So we turned to SAGE and found:

So there is a formula. And it’s messier than the quadratic, which is messier than the linear [1].

So we checked the quartic.

And indeed, SAGE found a solution. I’m only copying the first few lines here since it goes on forever.

So here we are, and there’s a pretty good chance things will continue — there’ll be a solution but it’ll be more and more complicated as the polynomial degree gets higher and higher.

But when we try the quintic and the sixth-degree polynomials, we get:

Um… it won’t solve it for us? Is SAGE just not powerful enough to help us? Do the solutions get *that* much harder?

Could be.

But it isn’t that SAGE was broken. It turns out there is no “formula” to get the zeros of any given quintic or higher polynomial. Sure, you can solve some quintic polynomials. Heck, they might even be factorable to or something. But that isn’t what we’re asking. We’re asking if you have any quintic (or higher) polynomials, can you come up with an algebraic formula for the exact roots.

No.

And the reason I wanted to show my student that is because it was learning that fact in high school, the insolvability of the quintic, that got me even more interested in math. It raised the huge question: what broke down after 4? Why is 5 the magic number? Is it truly impossible for any degree polynomial greater than 5? How can anyone show that a degree 1021 polynomial won’t have a “formula” solution for its zeros? No one could explain it to me, but my math teacher swore it was true.

It seemed so crazy to me! Heck, it still does. Interesting tidbits like that lit a fire under my feet [2] to take college level algebra (Abstract Algebra) to help me understand it. It was one of the most glorious days of college when in our Abstract Algebra class we finally got to tackle and solve this problem.

Do I remember how we did it?

Sigh. No. I have Flowers For Algernon syndrome.

But I at least know that the solution is out there, and given enough time and patience, I can understand it once again.

I doubt my student got out of it the same level of “WHAT THE HECK?!” as I did, nor do I think it lit the curiosity fire under his feet. But heck if I didn’t show him that our intuition breaks down without cause sometimes, and there are answers to be found. Maybe not in our class, but in some other class if he ever wants to solve the mystery.

[1] Well, when I was in high school, I had my dad’s worn, cover-falling-off CRC Book of Mathematical Tables and Formulae. In it was the solution of how to solve any cubic, and how there is indeed a formula (like the quadratic) for the cubic. (You can see it nicely typed here.) I suspect it also had a paragraph or two about the Cardano/Tartaglia dispute.

[2] Godel’s incompleteness theorem was another one.

My sister is a teacher too. And she’s smart. And sometimes she poses questions which stump me. She posed a good physics problem on Facebook a while ago.

In case you can’t tell, the three fixed, point masses have masses 1, 4, and 9. She wants to know where you can place a mass so that it won’t move! So that the net gravitational force on it is nil.

Just in case you forgot Newton’s Law of Gravitation between two bodies:

Before starting, I thought this problem would be so easy. If the three masses were equal, we’d have a simple geometry problem. Since they aren’t, it turns out we have something more tricky. I thought the solution would come easy. For me, it didn’t. But I think I got an approximate solution.

Just so we can compare solutions, let’s put our masses on the cartesian plane as below:

As you can tell, I placed the three points on the unit circle.

I don’t want to give much away, so I’m just going to leave you to it. Throw your thoughts in the comments below. If you’re dying for the answer, I’ve hidden what I got on this site somewhere in some not-hard-to-find spot.

If you get stuck, look after the jump for some encouragement.

Just a note: I don’t know if I got the right answer… I think I did, when checking it, but I’m not totally sure. I got tired of working it. That’s why I wanted to throw this up there to see if anyone could corroborate, and also to see your approaches!

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So we’re off for Thanksgiving today (phew!) and after a few really great days, this two-day week was pretty much a bummer. The one fun mathematical thing I’ve worked on is a problem involving geometry. Ew, I know. But I got really into it, and it got me thinking of about a million other extensions, questions, methods of attack, etc. I was thinking in so many different ways — about symmetry and about limiting and degenerate cases and about angles and such.

(1) You have two circles, radius 3 and radius 5, tangent to each other. You want to draw a third circle tangent to the given two circles. In fact, you realize there are an infinite number of these circles. So the first question is: what is the locus of all points which is comprised of the centers of these infinite circles?

If you want a small hint, go after the jump for a picture. (In case it wasn’t clear, we are taking about circles which are externally tangent to each other.)

(2) Generalize the problem to being given two circles with radius a and radius b (instead of radius 3 and radius 5).

(3) Can you find a third circle tangent to the given two circles such that the centers of the three circles forms a right triangle (if possible).

(4) What if we ask a similar question about spheres? If you are given two spheres of radius a and radius b, what is the locus of all points which is comprised of the centers of spheres tangent to the given two spheres?

So if you are bored over your Thanksgiving holiday, you might want to have some fun with this. I’ve solved the first two. I haven’t had time to think about the third yet, though I know the solution won’t be (too) hard. The fourth one? Eh, I anticipate it to be pretty tough. But having solved the first two will definitely help! [update 5 minutes after posting: Eh, nevermind, I think I know the answer to the fourth one… not really hard!]

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A couple o’ days ago, I posted a question about how to come up with a set of parametric equations equivalent to an implicit equation. It seemed to me like the general solution to this broad question would be like differential equations. There would be certain tools you could pull from the toolbox, once you saw what “kind” of equation you were dealing with. There isn’t a one-size-fits-all algorithm for solving differential equations (at least, not that I learned).

I got to thinking… didn’t I learn how to convert between parametric and implicit equations some time years ago? And, in fact, the answer was: yes. I took a class on algebraic geometry. The book we used (one of my favorite math textbooks when I was an undergrad) was:

The way this course was designed (it was officially a “seminar”) was that each day, two students would “teach” a section from the book to the rest of the class. We somehow made it through the whole book. It was a great experience, having to learn a section well enough to teach it the my classmates. The class was — however — a bit of a failure. The desks were in a row, people rarely asked questions, and no one engaged with each other. (Much like most of my college math classes, actually.) For something so student-based, it was strange that I didn’t make a single friend in that class. Plus, there was no “teaching” us how to teach well. Some some of us were great teachers, but most of us sucked. I can’t say what I was, really. I don’t remember. Regardless, I remember thinking: this textbook was incredible because I pretty much had to teach myself the subject. (I have to give major kudos to the instructor because he forced me to learn an entire course by reading a textbook.)

So upon reminiscing about this class and this book, I pulled it down from my bookshelf.

I am so dumb.

I will revise: I am so dumb now.

I look at the pages, and read theorems like Theorem 9 on page 241

“Two affine varieties and are isomorphic if and only if there is an isomorphism of coordinate rings which is the identity on constant functions.”

and see words like “Nullstellensatz,” and wonder how I ever got to the point where this stuff made sense, and how I got to the point where I see a bunch of gibberish now. Seriously, it’s disturbing. I mean, I don’t expect to be able to pick up a book I learned from years ago and know everything in it, but I do expect that it is in a language I can read.

I’ve figured it out. I am Charlie in Flowers for Algernon.

I don’t know how to feel about the loss of my mathematical mind, besides sad.

Maybe I’ll try teaching myself some math again, to either prove to myself I still have it somewhere in me, or to know that my brain has truly atrophied to a giant anti-intellectual morass.