Today I gave my multivariable calculus class a problem — a problem I give every year, that I found… somewhere. Maybe MIT, maybe an Exeter problem set, maybe a textbook. And if you ever want to see kids work together, and do some good problem solving, this is a prime problem for that.
Up to now, we’ve been working on vectors — and they learned vector basics (read: dot product and cross product). Here’s the question.
You have any tetrahedron. Sticking outwards from each face, orthogonal to each face, is a vector with magnitude equal to the area of face it is sticking out of. Prove that if you add these four vectors together, they sum to the zero vector.
It’s such a beautiful problem. I don’t have a totally geometric way to explain why this is true (we do lots of good vector algebra), but I do enjoy watching everything all come together. To me, it almost works like magic.
I then had a student ask an amazing extension question. (If they’re asking extension questions, you know it’s a rich problem.) He said: “Will this always work for any polyhedra? What about figures involving faces which aren’t triangles?” I, of course, decided I loved the problem. And I desperately want him to work on this for his final project.
I love the idea of this student taking this problem and seeing how far he can run with it. I mean: hello, gluing tetrahedrons together! (I expect him to make some stick models, if he does it!)