Some Math Problems Seem Impossible. That Can Be a Good Thing | Quantized Academy - Quanta Magazine

Patrick Honner, award-winning mathematics teacher explains, Struggling with math problems that can’t be solved helps us better understand the ones we can.

Struggling with math problems that can’t be solved helps us better understand the ones we can.
Photo: BIG MOUTH for Quanta Magazine

Construct a convex octagon with four right angles.

It probably says a lot about me as a teacher that I assign problems like this. I watch as students try to arrange the right angles consecutively. When that doesn’t work, some try alternating the right angles. Failing again, they insert them randomly into the polygon. They scribble, erase and argue. The sound of productive struggle is music to a teacher’s ears.

Then they get suspicious and start asking questions. “You said four right angles. Did you really mean three?” “Are you sure you meant to say convex?” “Four right angles would basically make a rectangle. How can we get four more sides in our octagon?” I listen attentively, nodding along, acknowledging their insights.

Finally someone asks the question they’ve been tiptoeing around, the question I’ve been waiting for: “Wait, is this even possible?”

This question has the power to shift mindsets in math...

If we push further, the impossible can even inspire the creation of new mathematical worlds. To prove the impossibility of squaring the circle — a problem that’s at least 2,000 years old — we needed the modern theory of transcendental numbers that cannot be roots of integer polynomials. To solve the bridges of Königsberg problem, Euler turned islands and bridges into vertices and edges, bringing to life the rich fields of graph theory and network theory, with their many applications. Taking the square root of −1 led to an entirely new system of arithmetic. And the logician Kurt Gödel changed the landscape of math forever when he proved that it’s impossible to prove that everything that is true is true.

So the next time you’re stuck on a math problem, ask yourself: “Is this possible?” Struggling with impossibility could give you a better understanding of what actually is possible. You might even create some new math along the way.

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Source: Quanta Magazine