A math teacher gets everything he wants. What could possibly go wrong?

Problem Archetypes

I bet somebody has written a book about this, but I’m unaware of it, so here goes. Stop me if you know, and put me out of my misery.

Jason Buell just posted about how interesting it is when we (or students) don’t go to the question we expect in a given situation, and how important it is for us to break set. For example, when you have nine supreme-court justices and they start shaking hands, every math teacher in the room knows to ask, “how many handshakes altogether?”

It’s vital that we learn to ask other questions. But this post is not about that.

Rather, let us observe that the “handshake problem” is an example of what I’m gonna call a problem archetype. It’s part of our mathematical maturity (I claim) that we have a fistful of these that we can bring out and use; and we do, because they’re useful. It may be that other problems have the same mathematical structure, or that it illustrates an important principle, or some other reason I haven’t thought of.

In any case, it’s part of the shared culture. We refer to it in shorthand in order to communicate with one another or to remind ourselves. It often has a name, as in, “the handshake problem,” or, to name another, “the Monty Hall problem.” (I happen to dislike the Monty Hall problem for the classroom, but I still think it’s archetypal.)

So:

What are these? Can we start a list?

What role do they actually play in problem-solving?

Are they, ultimately, a positive influence? Or do they shackle us?

Just to get things started, here are some other archetypes:

Boat in a river. Is this actually an archetypal problem, or just a common situation in problems in Algebra texts? Does that matter? We all recognize “boat-in-a-river” problems as a particular genre.

The seven bridges of Königsberg. When I first saw a map of the city, I was astonished at the shapes of the rivers. Of course, topology is topology, but still!

That problem where you cut two squares out of a chessboard, from opposite corners, and then try to cover the board with dominoes.

Speaking of chessboards, the one where you get one grain of wheat for the first square, and then double every time.