Welcome to the 195th Carnival and my second time hosting the event.

To see past entries in the Carnival Of Mathematics and future scheduled hosts, please visit The Aperiodical.

I am honored to again host the Carnival of Mathematics! I learn so much from hosting, things I usually wouldn’t be exposed to are jam packed into every Carnival Of Mathematics post. Be sure to dig in to the archive!.

Here are the entries. Enjoy!

**Bad Math Memes**

This is a video by me discussing how crazy I get when I see crazy math memes on Facebook and Twitter. Most are not educational and further separate mathematics from would-be students. We as mathematicians must do everything we can do to get people to EMBRACE mathematics, not shy away from it.

**Here’s a proof that Tolkien’s Middle-Earth is not flat**

**A 2021 problem: 20∼21 and 43×47**

By: Ed Pegg

Submitted By: Lewis Baxter

Ed Pegg noticed that 2021 = 43 x 47 which are successive primes with 20 and 21 being successive integers. He asked for similar solutions and Robert Israel quickly found the next biggest solution, a number with 36 digits. I (Lewis Baxter) found more than 1500 bigger solutions, the largest having 3011 digits. This month I certified the two primes (which are 20690 apart). Unlike other “titanic” primes they are not the value of some small arithmetic expression.

LINK

**Who Needs Trig Sub?**

By: Patrick Honner

Mr. Honner sent this link in, bragging about what his students came up with. “This was the coolest math my students produce this year,” Mr Honner gushed!

“I’ve taught this topic for many years and never thought of this approach. I’m grateful to have learned something new from my students, who never fail to impress me with their creativity. And I’m glad I gave them time and space to solve what I thought was an impossible problem! When I teach this next time, I’ll be sure to do it again. And I’ll be sure to share this ingenious integration.”

**When cubic polynomials have three real roots!**

By: Freya Holmér (via Twitter)

Holmer Breaks down how they can be solved using trigonometry. Geometrically, you can visualize it as an equilateral triangle centered directly above the inflection point, where its vertices coincide with the three roots.

**Why do perpendicular lines have slopes that are opposite reciprocals?**

**Half a year of the Liquid Tensor Experiment: Amazing developments**

By: Peter Scholze

Submitted By: Robin Whitty

“Exactly half a year ago I wrote the Liquid Tensor Experiment blog post, challenging the formalization of a difficult foundational theorem from my Analytic Geometry lecture notes on joint work with Dustin Clausen. While this challenge has not been completed yet, I am excited to announce that the Experiment has verified the entire part of the argument that I was unsure about. I find it absolutely insane that interactive proof assistants are now at the level that within a very reasonable time span they can formally verify difficult original research. Congratulations to everyone involved in the formalization!!

In this Q&A-style blog post, I want to reflect on my experience watching this experiment.”

**Singmaster’s conjecture in the interior of Pascal’s triangle**

By: Terence Tao

Submitted By: Robin Whitty

Kaisa Matomäki, Maksym Radziwill, Xuancheng Shao, Joni Teräväinen, and myself have just uploaded to the arXiv our preprint “Singmaster’s conjecture in the interior of Pascal’s triangle“. This paper leverages the theory of exponential sums over primes to make progress on a well known conjecture of Singmaster which asserts that any natural number larger than 1 appears at most a bounded number of times in Pascal’s triangle.

# #**GeometrySketchbook**

Submitted By: Sam Hartburn

#GeometrySketchbook is a hashtag that has been used for a daily maths art

challenge throughout June. A huge variety of media and art styles have been

used; if you’re interested in mathematical art you’re sure to find

something inspiring here.

**Is This Some Kind of Code? You Can Solve the …**

By: New York Times

(Paywalled)

“Erik and Martin Demaine, a father-and-son team of “algorithmic typographers,” have confected an entire suite of mathematically inspired typefaces.”

The verb “puzzle” — to perplex or confuse, bewilder or bemuse — is of unknown origin. “That kind of fits,” said Martin Demaine, an artist in residence at the Massachusetts Institute of Technology. “It’s a puzzle where the word ‘puzzle’ comes from.”