The Fix

Bert and Dave were cruising the streets of DC, Dave at the wheel. Bored, Bert reaches into the glove compartment and pulls out a map.

"Hey Dave," he says, "this is a map of DC inside the beltway."

He looks at it for a couple of seconds, crumples it up, and tosses it out the window.

Not being a litterbug, Dave screeches to a halt, and gets out of the car, muttering. Bert is undeterred. He too jumps out, runs over to the map, whips out a Ben Franklin and says: "I will bet this fifty hundred bucks that there is some point on this map that is exactly above the point in the City that it represents."

Should Dave take the bet?

UPDATE: As Wolgang noted, this is the fixed point theorem of Luitzen Egbertus Jan Brouwer. (In this case implying that any continuous map from a disk (say the inside of the beltway) to itself has a fixed point).

In session 10 of Conceptual Mathematics the authors deploy the map theory they have taught us so far to prove Brouwer's fixed point theorem - our first bit of "real math." The theorem is notable both for its simplicity and elegance but also for its deep connection to problems in the foundation of mathematics and logic. Brouwer's proof relies on a contrapositive and offers no insight into where the fixed point (or points) of the map in question might be.

A more intuitive theorem (Banach's fp theorem) applies to cases where, like Dave and Bert's map, the map is contractile. The proof for that theorem is based on the idea that if the map in question were truly up to date and accurate it would contain a tiny image of itself, lying crumpled on the ground. That image would clearly lie inside the region of DC wherein lies the whole crumpled map. If we look at the crumpled image on the image map, and so on ad infinitum, it's clear that we converge on the unique fixed point of the map from DC to its paper image.