*You’re visiting Carnival of Mathematics #172 hosted by Cassandra Lee. Find all Carnivals here and the previous Carnival (#171) here. Many thanks to the Aperiodical for the wonderful opportunity to host this Carnival.*

The Roman representation of the number 172 comprises the initials of my English name, the chromosomes determining my gender and the eating utensils that I use daily. Yes, that’s CLXXII. *gasps at the sudden weight on her shoulders*

Without further ado, let’s begin with something light-hearted. It’s easy to show that 172 = 2 × 2 × 43, an even composite number; it doesn’t appear special until you also learn that 172 is also a deficient number (the sum of the factors is less than the number itself: 2 + 2 + 43 = 47 < 172) and an evil number (!), which means it has an even number of 1’s in its binary representation (172_{10} = 10101100_{2}). Note that, if the units digit is relabelled the “zeroth/0-th” digit, then you can verify that the binary representation of 172 (= 10101100_{2}) has the following property:

The

n-th digit is 1 if and only ifnis prime.

Furthermore, 172 is in the lazy caterer’s sequence at *n* = 18; the maximum number of pieces you can get with 18 cuts of a circular pie is 172.

It’s a crying shame that none of our submissions talked about the maths behind the Apollo moon landing 50 years ago in July 1969, in which the Schimdt-Kalman filter is key to airplane flights landing on time. On top of that, extensions and modifications of the Kalman filter are used to track moving objects in computer vision. The maths that put the man on the moon is now used to drive your vehicles and track you down…

Now for the standard loading — the roundup of selected contributions from maths enthusiasts around the world in July 2019:

**Euler Line [in a Triangle],**by**DoubleRoot**

Three interesting points in a triangle must lie on a straight line; don’t miss the explanatory animations.**Quadrature of the Parabola Proposition 2****,**by**Ben Leis**

It’s the sequel to a previous post and I got confused reading and re-reading it, until I realised the correct proposition in the beginning of the post should be:Given a parabola and a secant AB. Construct a tangent line parallel to AB at C and a tangent line at B. Drop a parallel line to the axis of symmetry DM through C. Prove DC = CM (i.e. C is the midpoint of DM).

**Edit, 5 August 2019:**The aforementioned post has since been corrected.**Coding Stories: Exploring Factors,**by**Greg Benedis-Grab**

This article on the juxtaposition of maths and coding blew my mind. It illustrates the beautiful, never-ending process of problem-solving: a question leads to an answer which is another question in disguise which leads to another answer which leads to another question…

My favourite quote:The students were excited that a computer could solve a math problem. One of them said, “now we don’t need math anymore. We can use computers.” I love the intersection of humor and problem solving so I laughed at his joke. However, I don’t think this student realized at the time that using computers means you in fact need math even more.

**Number of Feet in a Mile,**by**John D. Cook, PhD.**John described this article to me as “[a] couple frivolous observations that lead to deeper topics”. When I read his post, it felt like climbing a typical mathematical anthill only to realise it’s the steep of a cliff, albeit one I could scale. A ten-year-old could grasp the first half, but I needed to look up several references to make sure I understood the second half…

**Danesh Forouhari**tweeted a**British Mathematical Olympiad Problem on the factorial of 34**, perhaps to challenge us.

Here’s the official solution sketch.- I’ve been learning maths at
**Murray Bourne**‘s fantastic website “Interactive Mathematics” since I was in Form 2 or 3 (Grade 8 or 9), and he’s updated his**applet on eigenvectors and eigenvalues**. “When I first studied eigenvectors as a student, I could do the algebra, but had no idea what the concept actually meant. Hopefully this applet goes some way to addressing this issue for some readers,” Murray told me in his submission, adding that there’s also a new eigenvectors calculator. (As for me, I love eigenvalues and eigenvectors because they are the first linear algebra concepts I “got” immediately when I took up university maths.) **Fraction [Best Rational Approximations of Real Numbers],**by**Shreevatsa R**, requires a browser that supports “BigInt”. It works on Chrome 75.0 but not on Safari 12, and Firefox 68+ supports it.

On Chrome, I plugged in Euler’s number,*e,*and out came this:

Thanks all for submitting!

I’ve mentioned Apollo moon landing maths above. Now for the rest of my personal findings in July 2019.

I’m glad that I’m not the only person who advocates for a playful approach to maths (from Matthew Oldridge), and maths gamification with open-ended answers (from Matthew Peterson, PhD) seems promising as an effective educational tool. Wait… did I just see two Matthews…?

Differential equations play a part in this *Mathematical Objects* podcast discussion on thermometers by the Aperiodical. On a side note, even though everyone says “podcasting is the future”, we should provide accessibility options to people with hearing loss too.

This caught my attention: “Why Mirror Symmetry Is Like Fancy Ramen” by Evelyn Lamb. In my undergraduate years, I’ve had the honour of meeting and learning from mathematicians who loved mirror symmetry and algebraic geometry/topology, and this podcast brings back fond memories. [**Edit, 16 August 2019**: Also look into Quanta Magazine‘s articles making maths research accessible to the general public. Thanks Dr Mandy Cheung for the tip!]

We’ll round off the Carnival with a video on the Dehn Invariant by the amazing Numberphile. Takeaway: **Don’t overestimate the difficulties in your field of expertise, and don’t underestimate your students — they may be smarter than you think.** Educators beware.

The next Carnival of Mathematics (#173) will be held by Peter Krautzberger in September 2019. Please submit to him your maths articles, videos, podcasts, online threads, memes you’re permitted to use, etc. here.

Thanks for dropping by!

Cassandra Lee

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