## 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.”

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• 1. Squares and Triangles | Rhapsody in Numbers  |  January 16, 2011 at 1:58 am

[…] Vennebush over at Math Jokes 4 Mathy Folks just posted an interesting problem: Three points are randomly chosen along perimeter of a square. What is the probability that the […]

• 2. xander  |  January 16, 2011 at 2:01 am

After a few hours of banging my head against the wall, I think I have a viable solution. Because it took a lot of words to get there, and because I don’t want to post spoilers right here, I will simply provide a link to my own thought process. I eagerly await your solution.

• 3. Fair and Square Solution « Math Jokes 4 Mathy Folks  |  January 19, 2011 at 6:20 am

[…] that post, I mentioned that I had an elegant solution. If you want to think about the problem a little […]

• 4. Wild About Math bloggers 1/28/11 » Fun Math Blog  |  February 4, 2011 at 10:20 am

[…] formula and identity. I’ll leave you today with a very elegant problem posted by Patrick at Math Jokes 4 Mathy Folks. Three points are randomly chosen along the perimeter of a square. What is the probability that […]

• 5. Wil Maddeaux  |  July 23, 2011 at 2:01 pm

Does “within” include being on the lines of the triangle? It puzzles me that this hasn’t been mentioned.

• 6. venneblock  |  July 27, 2011 at 9:16 am

Since it’s a geoemtric probability problem involving area, I don’t think it matters if you count points on the perimeter or not. The answer should be the same.

• 7. What If? | Math Jokes 4 Mathy Folks  |  March 30, 2015 at 7:33 am

[…] of my favorite problems, which was discussed in the post Fair and Square in 2011, is the […]

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