Certainly it's artificial to restrict oneself to primitive concepts, unless of course you don't know about eigenvalues yet or forgot it all. :) And it is an exercise from the earliest chapter, so we're supposed to stay in simple-land.

Did you look at my solution (http://news.ycombinator.com/item?id=3129023)? There's no higher-level theory in it. Is it brute force? Not sure. At least it doesn't require solving any element-wise equations.