# Pi Day Puzzle Hunt

I’ll post more later, but here’s a sneak preview of this year’s Pi Day Puzzle Hunt. We’ve been working on this for months, and it finally happens next week. Here we go!

# I’m back

Hey everyone.

I took a two year hiatus from blogging. Life got busy and I let the blog slide. I’m sorry.

But I’m back, and my New Year’s Resolution for 2017 is to post at least once a month!

Here’s what I’ve been up to over the last two years:

• Twitter. When people ask why I haven’t blogged, I say “twitter ate my blog.” It’s true. Twitter keeps feeding me brilliant things to read, engaging me in wonderful conversations, and providing the amazing fellowship of the MTBoS.
• James Key. I consistently receive mathematical distractions from my colleague and friend, James, who has a revolutionary view on math education and a keen love for geometry. This won’t be the last time I mention his work. Go check out his blog and let’s start the revolution.

with my nerdy friends named James

• My Masters. I finally finished my 5-year long masters program at Johns Hopkins. I now have a MS in Applied and Computational Mathematics…whatever that means!
• Life. My wife and I had our second daughter, Heidi. We’re super involved in our church. I tutor two nights a week. Sue me for having a life! 🙂

family photo

• New curriculum. In our district, like many others, we’ve been rolling out new Common Core aligned curriculum. This has been good for our district, but also a monumental chore. I’m a huge fan of the new math standards, and I’d love to chat with you about the positive transitions that come with the CCSS.
• Curriculum development. I’ve been working with our district, helping review curriculum, write assessments, and I even helped James Key make some video resources for teachers.
• Books. Here are a few I’ve read in the last few months: The Joy of x, Mathematical Mindsets, The Mathematical Tourist, Principles to Actions
• Math Newsletters. Do you get the newsletters from Chris Smith or James Tanton (did you know he’s pushing three essays on us these days?). Email these guys and they’ll put you on their mailing list immediately.
• Growing. I’ve grown a lot as a teacher in the last two years. For example, my desks are finally in groups. See?

my classroom

• Pi day puzzle hunt! Two years ago we started a new annual tradition. To correspond with the “big” pi-day back in 2015, we launched a giant puzzle hunt that involves dozens of teams of players in a multi-day scavenger hunt. Each year we outdo ourselves. Check out some of the puzzles we’ve done in the last two years.
• Quora. This question/answer site is awesome, but careful. You’ll be on the site and an hour later you’ll look up and wonder what happened. Here are some of the answers I’ve written recently, most of which are math-related. I know, I know, I should have been pouring that energy into blog posts. I promise I won’t do it again.
• National Math Festival. Two years ago we had the first ever National Math Festival on the mall in DC. It was a huge success. I helped coordinate volunteers for MoMATH and I’ll be doing it again this year. See you downtown on April 22!

famous mathematicians you might run into at the National Math Festival

Now you’ll hopefully find me more regularly hanging out here on my blog. I have some posts in mind that I think you’ll like, and I also invited my colleague Will Rose to write some guest posts here on the blog. Please give him a warm welcome.

Thanks for all the love and comments on recent posts. Be assured that Random Walks is back in business!

# When will she pass me for the first time? [solution]

Recently, my dad posed the following question here:

My wife and I walk on a circular track, starting at the same point.  She does m laps in the time that it takes me to do n laps.  She walks faster than I do, so m > n.  After how many laps will she catch up with me again?

If you haven’t solved it yet, give it a crack. It’s a fun problem that has surprising depth.

Here’s my solution (in it, I refer to “mom” rather than “my wife” for obvious reasons!):

Since mom’s lap rate is $m$ laps per unit time, and dad’s lap rate is $n$ laps per unit time, in time $t$, mom goes $mt$ laps and dad goes $nt$ laps.

They meet whenever their distance (measured in laps) is separated by an integer number of laps $k$. That is, mom and dad meet when

$mt=nt+k, k\in\mathbb{Z}.$

This happens at time

$t=\frac{k}{m-n}.$

Mom will have gone

$mt=\frac{mk}{m-n}$

laps and dad will have gone

$nt=\frac{nk}{m-n}$

laps when they meet for the $k$th time.

And that’s it! That’s the general solution. This means that:

• At time $t=0$, dad and mom “meet” because they haven’t even started walking at all (they are $k=0$ laps apart).
• At time $t=\frac{1}{m-n}$, dad and mom meet for their first time after having started walking (they are $k=1$ lap apart). This is the answer to the problem as it was originally stated. Mom will have gone $mt=\frac{m}{m-n}$ laps and dad will have gone $nt=\frac{n}{m-n}$ laps when they meet for the first time.
• At time $t=\frac{2}{m-n}$, dad and mom meet for their second time (now $k=2$ laps apart).
• At time $t=\frac{k}{m-n}$, dad and mom meet for their $k$th time.

Here are two examples:

• If mom walks 15 laps in the time it takes dad to walk 10 laps, when they meet up for the first time, mom will have gone $\frac{m}{m-n}=3$ laps and dad will have gone $\frac{n}{m-n}=2$ laps.
• If mom walks 12 laps in the time it takes dad to walk 5 laps, when they meet up for the first time, mom will have gone $\frac{m}{m-n}=1\frac{5}{7}$ laps and dad will have gone $\frac{n}{m-n}=\frac{5}{7}$ laps.

Boom! Problem solved! 🙂

# Random Walks Mural

I’ve been meaning to give the back wall of my classroom a makeover for a while. This summer I finally found some time to tackle the big project. I took down all the decorations and posters. I fixed up the wall and painted it a nice tan color. Then, I let loose the randomness!

I struggled with what the new mural would be–I’ve thought about it over the last few years. I considered doing some kind of fractal like the Mandelbrot Set. But it should have been obvious, given the name of my blog!! What you see in the picture above is three two-dimensional random walks in green, blue, and red. In the limiting case, one gets Brownian motion:

Brownian motion of a yellow particle in a gas. (CCL)

I honestly didn’t know what it was going to look like until I did it. I generated it as I went, rolling a die to determine the direction I would go each time. I weighted the left and right directions because of the shape of the wall (1,2=right; 3,4=left; 5=up; 6=down). For more details about the process of making it, here’s a documentary-style youtube video that explains all:

Actually, I lied–it doesn’t tell “all.” If you really want to know more of my thought process and some of the math behind what I did, watch the Extended Edition video which has way more mathematical commentary from me. I’ve also posted the time lapse footage of the individual green, blue, and red. Just for fun, here’s an animated random walk with 25,000 iterations:

A two-dimensional random walk with 25,000 iterations. Click the image for an animated version! (CCL)

I think the mural turned out pretty well! It was scary to be permanently marking my walls, not knowing where each path would take me, or how it would end up looking. At first I thought I would only do ONE random walk. However, the first random walk (in blue) went off the ceiling so I stopped. And then I decided to add two more random walks.

In retrospect, it actually makes complete sense. I teach three different courses (Algebra 2, Precalculus, and Calculus) and I’ve always associated with each of theses courses a “class color”–green, blue, and red, respectively. I use the class color to label their bins, to write their objective and homework on the board, and many other things.

The phrase “Where will mathematics take you?” was also a last-minute addition, if you can believe it. There just happened to be a big space between the blue and red random walks and it was begging for attention.

What a good question for our students. The random walks provide an interesting analogy for the classroom. I’d like to say I’m always organized in my teaching. But some of the richest conversations come from a “random walk” into unexpected territory when interesting questions are raised.

Speaking of interesting questions that are raised, here are a few:

• Can you figure out how many iterations occurred after looking at a “finished” random walk? Or perhaps a better question: What’s the probability that there were more than n iterations if we see m line segments in the random walk?
• Given probabilities $p_1, p_2, p_3, p_4$ of going in the four cardinal directions, can we predict how wide and how high the random walk will grow after n iterations? Can we provide confidence intervals? (might be nice to share this info with the mural creator!)
• After looking at a few random walks, can we detect any bias in a die? How many random walks would want to see in order to confidently claim that a die is biased in favor of “up” or “left”…etc?

Some of the questions are easy, some are hard. If you love this stuff, you might be interested in taking a few courses in Stochastic Processes. Any other questions you can think of?

Where will math take you this coming academic year? Welcome back everyone!

# MAA Distinguished Lecture Series

If you live in the DC area and you like math, you have no excuse! Come to the MAA Distinguished Lecture Series.

These are one-hour talks, complete with refreshments, all for free due to the generous sponsorship of the NSA. The talks are at the Carriage House, at the MAA headquarters near Dupont Circle.

Here are some of the great talks that are on the schedule in the next few months (I’m especially excited to hear Francis Su on May 14th).

I’ve been to many of these lectures and always enjoyed them. Robert Ghrist‘s lecture was out of this world (here’s the recap, but no video, audio, or slides yet) and was so very accessible and entertaining, despite the abstract nature of his expertise–algebraic topology.

And that’s the wonderful thing about all these talks: Even though these are very bright mathematicians, they go out of their way to give lectures that engage a broad audience.

Here’s another great one from William Dunham, who spoke about Newton (Dunham is probably the world’s leading expert on Newton’s letters). Recap here, and a short youtube clip here:

(full  talk also available)

So, if you’re a DC mathophile, stop by sometime. I’ll see you there!

# Math on Quora

I may not have been very active on my blog recently (sorry for the three-month hiatus), but it’s not because I haven’t been actively doing math. And in fact, I’ve also found other outlets to share about math.

Have you used Quora yet?

Quora, at least in principle, is a grown-up version of yahoo answers. It’s like stackoverflow, but more philosophical and less technical. You’ll (usually) find thoughtful questions and thoughtful answers. Like most question-answer sites, you can ‘up-vote’ an answer, so the best answers generally appear at the top of the feed.

The best part about Quora is that it somehow attracts really high quality respondents, including: Ashton Kutcher, Jimmy Wales, Jermey Lin, and even Barack Obama. Many other mayors, famous athletes, CEOs, and the like, seem to darken the halls of Quora. For a list of famous folks on Quora, check out this Quora question (how meta!).

Also contributing quality answers is none other than me. It’s still a new space for me, but I’ve made my foray into Quora in a few small ways. Check out the following questions for which I’ve contributed answers, and give me some up-votes, or start a comment battle with me or something :-).

And here are a few posts where my comments appear:

# USA Science and Engineering Festival

If you’re local, you should go check out the USA Science and Engineering Festival this weekend. It’s on the mall in DC and everything is free.

They will have tons of booths, free stuff, demonstrations, presentations, and performances. Go check it out!

For my report on the fest from two years ago, see this post. The USA Science and Engineering Festival is also responsible for bringing to our school, free of charge, the amazing James Tanton!