Probability questions from Tanton

Confession: I still haven’t figured out how to use twitter. (Feel free to follow me @mrchasemath, though!) I always feel like I’m drinking from a fire hose when I get on the site–I can’t keep up with the twitter feed, so I don’t even try.

But when I do, I love seeing what people are posting. Here’s a great math problem from James Tanton. He always has such interesting problems!

Feel free to work it out yourself. It’s a fun problem! Here are my tweets that answer the question (can you follow my work?):

It’s hard to do math with 140 characters! :-)

Here’s his follow-up question which has still gone unanswered. My approach to the first problem won’t work here, and I want to avoid brute-forcing it. (Reminds me of my last post!) Any ideas?

Let us know in the comments…or tweet @jamestanton!

Four ways to compute a probability

I have a guest blog post that appears on the White Group Mathematics blog here. (My first guest post!) Here’s a taste:

One thing I love about math, and particularly combinatorics and probability, is the fact that many methods exist for solving the same problem.

Each method may have its advantages. The advantage might be conceptual (as in “this makes most sense to me”) or the advantage might be computational (as in “this is the fastest way to do it”).

Discussing the merits of different methods is exactly what math class is for!

For example, check out this typical probability question that could appear in a Precalculus course:

The Texas Ranger pitching staff has 5 right-handers and 8 left-handers. If 2 pitchers are selected at random to warm up, what is the probability that at least one of them is a right-hander?

In fact, it’s one I use in my own Precalculus course and it generated a great class discussion. In teaching it this past year, I ended up showing students four ways to do the problem this year! Here they are…

For the epic conclusion of this post, visit White Group Mathematics. :-)

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

quora iconI 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.

USA-Science-and Engineering-Festival LogoThey 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!

I ♥ Icosahedra

Do you love icosahedra?

I do. On Sunday, I talked with a friend about an icosahedron for over an hour. Icosahedra, along with other polyhedra, are a wonderfully accessible entry point into math–and not just simple math, but deep math that gets you pretty far into geometry and topology, too! Just see my previous post about Matthew Wright’s guest lecture.)

A regular icosahedron is one of the five regular surfaces (“Platonic Solids”). It has twenty sides, all congruent, equilateral triangles. Here are three icosahedra:

icosahedron coloringsHere’s a question which is easy to ask but hard to answer:

How many ways can you color an icosahedron with one of n colors per face?

If you think the answer is n^{20}, that’s a good start–there are n choices of color for 20 faces, so you just multiply, right?–but that’s not correct. Here we’re talking about an unoriented icosahedron that is free to rotate in space. For example, do the three icosahedra above have the same coloring? It’s hard to tell, right?

Solving this problem requires taking the symmetry of the icosahedron into account. In particular, it requires a result known as Burnside’s Lemma.

For the full solution to this problem, I’ll refer you to my article, authored together with friends Matthew Wright and Brian Bargh, which appears in this month’s issue of MAA’s Math Horizons Magazine here (JSTOR access required).

I’m very excited that I’m a published author!

Matthew Wright visits RM

Dr. Matthew Wright paid our students a visit this past Friday and gave them a gentle introduction to topology and the Euler Characteristic. This is a topic given little to no treatment inside the traditional K-12 math curriculum, so our students welcomed the opportunity to learn some ‘college math.’ IMG_20140404_111835821He had our students counting vertices, edges, and faces of various surfaces in order to compute the Euler Characteristic. Students discovered that the Euler Characteristic is a topological invariant.

IMG_20140404_111817820In his talk he also walked the students through a proof that there are only five regular surfaces, using the Euler Characteristic. This is more difficult than the typical proof, but elegant because the proof doesn’t appeal to geometry. That is, the proof doesn’t ever require the assumption that the faces, angles, or edges are congruent. In this sense, it is a topological proof.* Very cool indeed!

Matthew's topology guest lecture at RMHSBio: Matthew Wright went to Messiah College and then went on to received his MS and PhD from University of Pennsylvania, where his thesis was in applied and computational topology. He was a professor at Huntington College for two years but is now at the Institute for Mathematics and its Applications at the University of Minnesota for a postdoctoral research fellowship. His hobbies include photography and juggling. On a personal note, Matthew was my roommate in college, and I had the privilege of being his best man in his wedding, as well!

For more about Dr. Wright, visit his website at http://mrwright.org/.

* This proof also appears in the book Euler’s Gem by Dave Richeson.