# Maybe the twitter/blogging department needs its own journal…

Even though I had almost nothing to do with this, I found it to be so cool that I wanted to write about it here…

A few days ago, Allison, of Infingons, etc, wrote this awesome post:

Putting myself in my kids’ shoes.

The post describes how she experienced many of the emotions students must feel in problem solving as they wrestle with challenging problems. In Allison’s case, the problem she was thinking about was:

You have a square dartboard. What is the probability that a randomly-thrown dart will land closer to the center of the dartboard than to an edge?

Of course, that problem set a bunch of math teachers on the blog-o-twitterverse buzzing, and soon they were coming up with all sorts of cool drawing like this:

and this:

(images from Square Dartboard Probability by Mr. H).

The comments to Allison’s original post also were bubbling with beautiful handwritten descriptions of paths toward a solution.

And it was at this point that I made my tiny contribution to this endeavor, but awakening the Rundquist, and his awesome Mathematica powers with a tweet.

Which got an almost instant response:

And as always, Andy delivered, big time.

But wait, there’s more! Andy was kind enough to share exactly how he did this in Mathematica with the following screencast:

And still, there’s more. If you watch the screen cast, you’ll see Dan Anderson’s tweet describing how he solved the problem computationally using vpython, generating this image:

Wait! We are still not done. Mr. H comes back with a full theoretical solution for the generalized problem of a dartboard with *n* sides (warning: beautiful calculus and LaTeX behind link). Then he goes and shows how this solution agrees with all of Andy’s calculations.

Why does this win the award for coolest thing I’ve seen on the internet in a long time?

**It is a collaboration**between high school and college teachers, across disciplines (math and physics), spanning an entire continent (from California to Connecticut, Minnesota to Texas).No one who worked on this problem, as far as I know, has ever met face-to-face.- In addition to Allison’s beautiful post describing the process of working on a problem, you can actually see, in the notes of Ms. Cookie, more writing in many places than math.
**This is how you think about math.** **It’s a beautiful marriage of both the computational and analytical approach to mathematics.****It’s a clear example of going well beyond what the problem asks for, and exploring with playful curiosity.****It’s the whole game of mathematics**. Geometry, calculus, probability, Monte-carlo simulation, symmetry, computation, trigonmentery—this problem has it all, plus collaboration, writing, argument and verification.**It’s unfinished**. If you check any of the comments of the previously linked posts, you’ll find many questions, and puzzles from the major players that could easily turn into a tangent and later a major addition to this problem.- To the best of my knowledge,
**it has never been done before**. It might be someone has worked out the solution to the original problem, and perhaps the general solution to the*n-*sided polygon, but I doubt that anyone has ever explored this problem as deeply or in as many ways as was done this past week. **It’s manageable.**This isn’t a problem from the fringes of modern mathematics that requires a PhD to understand. Anyone who reads it can get an intuitive feel for the problem and begin to imagine its solution.

So when, where and how do you publish this?

For those who complain that twitter or the internet is making us stupid or shallow, I use this as Exhibit A for the defense (and I’ve got dozens more examples just like this). This type of collaboration—bringing together so many varied approaches and tools—is only possible because of the internet and the ease with which it allows the communication of ideas. So how do we get others to see this? How do we convince the teachers who thing they are too busy for twitter, or fear that sally’s web surfing can’t possibly lead to deep engagement like reading an 19th century novel can?

Most importantly, ** this is exactly what I want my students to be doing.** I don’t want my students just watching videos on mindless procedures and solving cookie-cutter problems for badges. I want them collaborating with peers across the globe to solve challenging problems that stretch their imaginations, call on their unique skills, and allow them amaze me with more than what I ask for—more than I even thought possible.

How do we do this? I don’t have the complete answer, but I’m pretty certain that the chances of it turning up in my twitter feed or google reader vastly exceed the chances of me finding it by myself in a book.

This was definitely a lot of fun. For me one of the most interesting things was how I coded it right with my first try but I didn’t believe the results. The problem was I had in my head that the limit should be 1/3 and not 1/4 because I made the mistake of thinking about the ratio of those darts that were closer to the center to those that were further instead of closer to total. This had me going ’round and ’round for a while, but having other solutions to look at helped convince me that I was right after all. It was also interesting, as I noted in my comment to Mr. H’s post, that I had to use cartesian coords for my random variables instead of polar. You can hear me say something like “but this stuff didn’t work though that’s interesting” in the screencast. Thanks to Mr. H I found out about the very cool Bertrand paradox having to do with how you choose your random variables. I also got an email from a math prof buddy of mine who said “of course, the Jacobian of that transition isn’t constant!” to explain my (apparently rookie) mistake.