math jokes 4 mathy folks
 live. laugh. learn.
Three Points in a Square
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?

This problem appeared on the MJ4MF blog in the post Fair and Square.

To examine the problem with some data, the following spreadsheet shows the results for 10,000 randomly selected cases. (If you don't like the 10,000 cases that are currently displayed, hit F9 to generate a new set of 10,000 cases.) The second page of the spreadsheet gives some details about how the simulation works.

Several days after the original post, a "proof" appeared on the MJ4MF blog in the post Fair and Square Solution. However, that proof has been variously described as "half-assed," "a lot of hand-waving," "elegant," and "very nice indeed." It's difficult to say which description is most accurate, but…

  • the simulation from the spreadsheet seems to corroborate this result;
  • Xander Henderson at Rhapsody in Numbers provided a proof by discretizing that gives the same result; and,
  • math professor Andrew Bernoff from Harvey Mudd College confirmed the answer via email (though he did not explain his method).
The "Aha" Moment

I spent the better part of a week thinking about this problem. When I finally arrived at a solution, it felt magical. Here's how I described it on the MJ4MF blog:

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 Iíll make no progress. Then later, when I least expect it, when Iím freed from the shackles 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!