Welcome to the 162nd Carnival of Mathematics, the monthly round up of maths blogs organised by The Aperiodical. Next month’s Carnival will be hosted by Elias at The Math Section. You can submit items for next month here.

But before we begin, it is customary to share some interesting facts about the number 162.

- 162 is a Harshad number: it is a mutiple of the sum of its digits.
- 162 cannot be written in the form $ab+a+b$, where $a$ and $b$ are strictly positive integers.
- 162 is an abundant number: the sum of its factors (including 1 but not including 162 itself) is greater than 162.
- Issue 162 of Chalkdust will be released in Autumn 2095. If this is too long for you to wait, you can get your hands on issue 08 much sooner.

While waiting for the release of issue 08, you can read the following highlights from the last month of the internet.

When measuring the size of sets in the conventional way, the sets $\{1,2,3,4,…\}$ and $\{2,3,4,5,…\}$ have the same size. If this makes you unhappy—as *of course* these is one less thing in the second set—then you should read James Propp’s post about an alternative way to measure the size of infinite sets.

John Baez wrote about the 5/8 theorem: in a group, if the probability that two randomly chosen elements ($x$ and $y$) commute (ie $xy=yx$) is greater than 5/8, then all pairs of elements in the group must commute. If you want to know why this is true, then read John’s post.

Tai-Danae Bradley wrote en explanation of what applied category theory is. To understand how category theory can be applied, you’ll need to know something about what it is: but don’t panic, Tai-Danae has written another post telling you everything you need to know.

Jordan Ellenberg interviewed mathematician and retired American football player John Urchel. It’s a really interesting interview and a highly recommended read.

Mark Dominus wrote about how to find lines and curves that approximate data. Speaking of approximation, some guy I’ve never heard of called Matthew Scroggs posted last month about @RungeBot, a Twitter bot that he made.

Over on Twitter, Justin Lanier posted a thread about a recently discovered result involving points moving around inside circles or spheres.

John D Cook wrote about primes in the digits of $\pi$. John noticed that the number 314159 is prime while reading a post by Evelyn Lamb. This got him wondering how many different primes could be formed by the first digits of $\pi$, and whether there are more or less than you would expect.

As well as being the month in which Chalkdust issue 08 is released, October is also Black History Month in the UK and Black Mathematician Month. Over the course of the next month, we will publish articles written by black mathematicians about their work, as well as pieces that explore current initiatives tackling the lack of diversity.

With this in mind, we offer you a challenge to complete during October: pick a black mathematician, write about them or their work, then submit what you’ve written to next month’s Carnival of Mathematics. (If you don’t have your own blog to post on, why not submit a guest post to us or The Aperiodical?) If you need some inspiration before getting started, Faith Uwadiae is tweeting about one black scientist every day this month.

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