## Carnival of Mathematics 132

Welcome to the 132nd Carnival of Mathematics, a monthly roundup of mathematics-related blog posts!

• Of course, $132 = 12 \cdot 11 = 2^2 \cdot 3 \cdot 11$. 132 is a pronic number since it is the product of consecutive integers, which means it is twice a triangular number: $132 = 2 \cdot 66 = 2 \cdot (1 + \dots + 11)$.

• 132 is the smallest number which is the sum of all of the 2-digit numbers that can be formed from its digits: $132 = 13 + 12 + 31 + 32 + 21 + 23$.
• 132 is a refactorable number, since it is divisible by the count of its divisors: 132 has 12 divisors (1, 2, 3, 4, 6, 11, 12, 22, 33, 44, 66, and 132).
• 132 is a Harshad number, since it is divisible by the sum of its digits in base 10 ($1 + 3 + 2 = 6$).
• Last but not least, as a computer scientist and combinatorics nerd, I am especially excited about the fact that 132 is the sixth Catalan number. The Catalan numbers show up all over the place in combinatorics and computer science. Binary parenthesizations, binary trees, Dyck words, Dyck paths, polygon triangulations, and permutations avoiding 123 are a very tiny sampling of the combinatorial objects which are counted by Catalan numbers. For example, here are the 132 different ways to triangulate an octagon (in general there are $C_n$ ways to triangulate a convex $(n+2)$-gon):

The Catalan numbers satisfy the recurrence

$C_0 = 1; C_{n+1} = \sum_{i=0}^n C_i C_{n-i}$

and can be expressed in closed form as

$C_n = \frac{1}{n+1} \binom{2n}{n}$.

The first few are $1, 1, 2, 5, 14, 42, 132, 429, \dots$. If you’re interested in learning more about these fascinating numbers and their history, I highly recommend the recent and quite accessible book by Richard P. Stanley.

And now, on to the carnival! I didn’t have a ton of time to put this carnival together, so in many cases, rather than trying to write a summary myself, I let the submitters speak for their own submissions in italics.

# Mathematics and culture

Let’s kick things off with the most depressing submission, just to get it over with: In If you are a mathematician… Parthasarathy collects a list of blog postings on mathematicians who died an unnatural or unusual death. (I admit it is morbidly fascinating.) The implication seems to be that this is relatively more common among mathematicians than the general populace; perhaps some statisticians should look into that.

A short poem from Gaurish of Gaurish4math about the harmonic series: Harmonic Noise.

Evelyn Lamb submitted Black Mathematical Excellence: A Q&A with Erica Walker. For Black History Month, I interviewed Erica Walker, author of a recent book about Black mathematicians in the US. Many mathematicians want math to be a more diverse and inclusive community, and listening to people like Dr. Walker can help us get there.

Dr. Nira Chamberlain writes about Black Actors and Mathematics on the Big Screen. The buzz on the social media is there is going to be a film about the real life story of African-American mathematician Katherine Johnson. So, is this the first time a Black actor/actress has done mathematics on the Big or small screen? The answer is no. Read my blog for a few examples.

# Teaching

This month there were a lot of submissions relating to teaching, spanning the entire range from elementary to college level.

Manan Shah of Math Misery? writes Single Line Multiplication — A Gimmick To Wow Your Students. As a part two on place value arithmetic, here I look at [multiplying] two numbers in one line. I tie it in as a hook to learning about polynomial expansion. Writing down the result of multiplication in one step, without writing down any intermediate products—not very practical, but I can definitely see how it would get students to think.

Simon Gregg of Following Learning writes about Five regular routines. There are some really cool math teaching activities here. The overarching theme seems to be ways to get all students involved and sharing their own individual approaches. These are ten-minute routines that I use at the beginning of maths classes that are great for encouraging students’ thinking and talking about maths. I’m using them with 8 and 9 year olds, but they’re very portable, and could be adapted to pretty well any age!

Manan Shah also writes about Subtraction — A Variation On A Theme. As the title suggests, this post is a variation on a theme for subtraction. Specifically, I look at stacked, place value subtraction. As a simple example, in non-stacked form, consider: $436 - 289$. I show that we can simply subtract place by place without having to borrow until later like so. $436 - 289 \rightarrow [2,-5,-3] \rightarrow [2, -6, 7] \rightarrow [1, 4, 7] \rightarrow 147$ as desired. This is a nice way to think about subtraction with borrowing!

Ilona Vashchyshyn of logs and reflections writes Against complacency (What have we learned, and where do we need to go from here?). I wrote this short reflection after Dan Meyer’s post about Ed Begle, where he suggested that much of what Begle wrote in the 1970s about mathematics education still holds true today. “Mathematics education,” wrote Begle, “is much more complicated than you expected even though you expected it to be more complicated than you expected.” This is undoubtedly true. But as someone on the cusp of their career, I found it disheartening to think that we’ve made little progress since the 1970s, and frankly, pretty hard to believe (which is not exactly what Meyer was suggesting, but I think there is a good chance it could be interpreted by some in this way). In this post, I ask mathematics education researchers and teachers to reflect on just how far we’ve come (and how far we still have to go).

From Matt Dunbar of Magical Maths: Could This Be The Best Pythagoras’ Theorem Lesson Ever?. Use of models as pupil exercises to explore and discover mathematics through geometry, algebraic proof and in nature.

Katie shared Joel David Hamkins’s post, Math for nine-year-olds: fold, punch and cut for symmetry!. Fun cutting and folding activities with nine-year olds, getting them to think about symmetry. I suspect these would work with older students as well. Heck, I want to do these activities!

Comic #43 — Messages from Manan Shah. This is one of my math education related comics. It is about the contradiction in messages we send to (young) students of mathematics.

From Lisa Winer at eat play math: Teaching (and Learning) Grit by Having Students Solve the Rubik’s Cube. I never knew how to solve the Rubik’s Cube, let alone teach how to. But it has been a great experience, even leading us to submit a Rubik’s Cube mosaic that my class created into a competition.

The Arctic Circle Theorem, from Mike Lawler at Mike’s Math Page. [I] learned about this theorem talking to a graduate student at MIT the other day. The theorem itself is almost mind blowing – a random tiling of a shape called an Aztec Diamond is likely to have features that are completely regular. Seemed like a really neat idea in math to share with kids. Luckily I found some software on line that made sharing the idea really easy! It was fun to hear what my kids had to say about the tilings.

Sue VanHattum’s Favorite Course (to teach): Calculus, at Math Mama Writes. Five reasons why I love teaching calculus: story, history, science, proof, and I get to keep learning!

# Grab bag

Katie submitted Evelyn Lamb’s post on How to Sew Like a Mathematician. “Bias tape” is made from strips of fabric cut diagonally, and Evelyn explores how to make it by first sewing fabric into a torus. Impractical? Perhaps. Awesome? Definitely.

Craig S. Kaplan at Isohedral writes about A new near miss. I’ve long been fascinated by the idea of “near misses”, specifically in the world of polyhedra, but more generally in mathematics. The post chronicles the recent discovery of a new near miss. The solid is fairly innocuous, but I think it can inspire further investigations by others. Evelyn Lamb also submitted a link to this post, writing: Craig Kaplan made a polyhedron that doesn’t exist. I love this post he wrote about how the messiness of the real world enabled him to find this nonexistent shape. Since reading the post, I’ve enjoyed thinking about “near misses” and the role of approximation and real-world messiness in creativity.

Yenergy submitted a link to Topological Magic: Infinitely Many Primes, by Tai-Danae Bradley at math3ma. Grad student Tai-Danae quickly goes through some topological definitions to explain an alternate topological proof for the infinitude of primes. Plus colorful pictures!

Yenergy of Baking and Math writes about Connecting hyperbolic and half-translation surfaces, part I (definitions). This is the first of a two-part series connecting hyperbolic surfaces with a different kind of surface. SO MANY PICTURES and gifs! Interesting math that may not be written down anywhere.

Last but not least, John Cook at The Endeavor writes about The empty middle: why no one is average. A contest to find the woman closest to the average woman found that nobody was close to average on all nine measures. A little probability shows why this is to be expected.

And that wraps things up for the 132nd Carnival of Mathematics! Thanks to everyone for their great submissions. The next carnival will be hosted by Matthew at Chalkdust Magazine. As usual, head over to the CoM site on the Aperiodical for a link to the submission form.

Assistant Professor of Computer Science at Hendrix College. Functional programmer, mathematician, teacher, pianist, follower of Jesus.
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### 6 Responses to Carnival of Mathematics 132

1. Evelyn says:

The “If you are a mathematician…” link is broken. (Just a heads-up. Feel free to delete/not approve this comment.)

• Brent says:

2. yenergy says:

With minimal searching I didn’t find the correct “If you are a mathematician…” link but I was led to this amazing page: http://www.kellenmyers.org/deaths.html which is maybe related!

3. Wally says:

“132 is the smallest number which is the sum of all of the 2-digit numbers that can be formed from its digits: 132 = 13 + 12 + 31 + 32 + 21 + 23.”

Actually 123 is even smaller. 132 may be the smallest even number with this property.

• Brent says:

123 is smaller, but it does not have this property: you can make all the same 2-digit numbers out of its digits, which still sum to 132, not 123.