In February, I was throwing together a geometry test for my 12-year-olds. I wanted a standard angle-chasing problem, but – and here’s the trick – I’m lazy. So I grabbed a Google image result, checked that I could do it in my head, and pasted it into the document.
But when I started writing up an answer key, I ran into a wall. Wait… how did I solve this last time? I trotted out all the standard techniques. They weren’t enough. A rung of the logical ladder seemed to have vanished overnight, and now I was stuck, grasping at air.
Eventually, some colleagues and I solved it by with industrial-strength tools: the Law of Sines, the Law of Cosines, and some fairly sophisticated algebra. Yet the problem looked so elementary, I felt sure that our approach was overkill—that we were bashing down the door, whereas a defter hand could simply pick the lock.
Apparently, I’d actually chosen a famous gem of recreational mathematics, born in 1922 from the mind of Robert Langley, and known since as “Langley’s Adventitious Angles.”
And, as I suspected, the niftiest solution requires no trigonometry or algebra, just a single ingenious move: construct a line here, at a 20o angle to the base.
It’s a masterstroke. It pops the lid right off of the problem. And, at least to me, it’s utterly un-guessable. If you had a thousand monkeys with a thousand typewriters and a thousand protractors, you’d get full verses of Shakespeare long before any of our furry friends stumbled upon this solution.
Sometimes the general techniques fail, and you need a sneaky trick. That’s life.
To me, a “trick” is a disposable insight, a funny little key that opens one particular closet door. And a “technique” is something more useful: a skeleton key, fitting the lock for a whole hallway of rooms.
But is the difference always so clear?
There’s a certain algebraic move that I first encountered in high school, which you might call “multiplying by the conjugate.”
You use it here, to simplify an expression with square roots:
And here, with complex numbers:
And here, again with square roots, to solve a limit (without recourse to the heavy machinery of L’Hopital’s Rule):
And here, to boil down a trigonometric expression:
At first glance, it feels like a trick: a little too slick, a little too cute. It’s hard to imagine it blossoming into a general technique. But it eventually proves useful in a shocking variety of situations, cutting across the whole secondary mathematics curriculum. This funny little key, which barely looks like it should open a single door, turns out to open hundreds.
Ultimately, I can’t find any categorical difference between a trick and a technique. They’re both problem-solving innovations, lying along a continuum that runs from “almost never useful” to “useful all the darn time.” A technique is simply a trick that went viral, and a trick is simply a technique that fizzled out after a single use.
It makes me wonder how to define the job of the mathematician (and of her little spiritual cousin, the mathematics student). Is it to master existing techniques? To cook up new tricks from scratch? Or to recognize how the tricks of today can be nurtured and expanded into the techniques of tomorrow?
Yes to all of that. But of all the aspects of the mathematician’s job, I think the ability to birth truly new ideas is overrated. Leaf through the pages of history, from Pythagoras to the present, and you’ll find a stunning supply of tricks, techniques, and everything in between. In the five millennia we’ve been working on this, humanity has accrued a remarkable wealth of mathematical ideas. Yes, we want our fresh young minds to make new deposits into the bank of our knowledge; but we also want them to know how to access the existing funds.
I would never, in a million years, come up with the trick to solve Langley’s Adventitious Angles. But I don’t need to. Someone else already has.
By the way, as promised, here’s the rest of the solution to Langley’s Adventitious Angles: