Fair and Square

I recently discovered a great problem:

Three points are randomly chosen along the perimeter of a square. What is the probability that the center of the square will be contained within the triangle formed by these three points?

My colleagues and I spent more time talking about this problem than I care to mention. But when all was said and done, I arrived at a wonderfully elegant solution. As usual, I won’t post the solution now to allow you some time to think about it, but I’ll post it in a few days.

The best part about this problem was the “Aha!” moment it afforded me. The solution eluded me when I forced myself to work on it. But yesterday morning, I was thinking about the problem while walking my dog. No pencil, no paper, no agenda… just time to think. And I kid you not — the solution came to me as I was picking up feces. (I have no idea what that says about me.)

This is my favorite part of mathematics. I can literally spend hours reworking equations, drawing figures, and thinking about a problem, and I’ll make no progress. Then later, when I least expect, when I’m freed from the confines of pencil and paper, the solution gently alights in my mind like a butterfly coming to rest on a marigold.

Oh, how I love that feeling!

Here are some math jokes that involve squares:

What keeps a tree in place?
Square roots.

Where is the best location for a multiplication table?
Times Square.

And this one is more of a physics joke than a math joke, but I just love it…

Newton, Leibniz, and Pascal were playing hide-and-seek, and Leibniz was it. Pascal ran into the bushes, but Newton simply drew a box on the ground and stood in the middle of it. When Leibniz finished counting, he turned around and saw Newton just standing there.

“Newton, I’ve found you,” Leibniz said.

“No you haven’t,” argued Newton, “you’ve found Pascal.” Gesturing at the ground, he continued, “One Newton per square meter.”