Volunteering at a math fair

Spent the day volunteering at a math fair here in Salt Lake City, talking to high schoolers and younger kids about higher math. It's a good exercise in a different kind of communication than a thesis. I always love an opportunity to teach others about what I am so passionate about.

My booth was a live version of Buffon's needle, done with toothpicks instead of needles. I taped down a set of parallel lines spaced a fixed distance $d$ apart and cut toothpicks to exactly half that length, $d/2$. Buffon's classical result says that if you drop a needle of length $\ell$ onto lines spaced $d$ apart (with $\ell \le d$), the probability it crosses a line is $$P = \frac{2\ell}{\pi d}.$$ Plug in $\ell = d/2$ and the $2$'s cancel: $P = 1/\pi$. So every toothpick drop is a tiny, physical coin flip that's secretly weighted by $\pi$.

We kept a running tally taped next to the board (total drops, and drops that crossed a line) and let kids take turns adding to it. Divide total drops by crossings and you get an estimate of $\pi$ that gets better the longer the fair runs. Watching a nine-year-old realize that flicking toothpicks onto a table is somehow "finding pi" was probably the best part of my day. A few of the older kids wanted to know why the probability worked out to $1/\pi$ at all, which turned into a nice quick sketch of where the $\sin\theta$ in the needle-crossing condition comes from geometrically, without ever writing down an integral.

By the end of the day our tally had climbed into the thousands of drops, and the running estimate had settled in close to $3.45$, which is a little high and I had to explain that even if we did 1,000,000 needle drops we cannot say for certain where the final total would end up (we have a good idea) but if we had infinite time we can say for certain it will approach $\pi$.