Carnival of Mathematics #246

Hello and welcome to December 2025's Carnival of Mathematics!

We've reached Carnival number $246 = 2 \times 3 \times 41$, and that's special for a variety of reasons, not least of which is that its digits are, in order, the first three terms in one of the first sequences most of us were introduced to; the chicken soup of all integer progressions that is the two times table.

$246$ is also the current best-known upper-bound for the minimum size of gap that exists between an infinite number of pairs of consecutive primes. It's palindromic in (e.g.) bases 5, 9, and 40; it is untouchable (which means that it is not expressible as the sum of the proper factors of any other number); and if you had a bit of string and seven of each of two colours of bead, you could make one of  $246$ different necklaces (using every bead).

And in maths history, Indiana House Bill No. $246$ was an 1897 foray into proof by legislation: it proposed to square the circle using a method that, among other things, implied that $ \pi =3.2 $.

You've taken your seats and loaded up on popcorn, so let's get started on the Carnival's...

Headline Acts

Or: Things Submitted Via The Form On The Aperiodical's Carnival Of Mathematics Page

Act I

Here's Dr Christoph Bartneck of Canterbury University talking about charts and data visualisation. The video is the very first one hosted on Mathateca's new YouTube Channel, and the talk was the public keynote at the Oceania MathsJam Gathering 2025.

Mathateca is an Ōtautahi Christchurch, NZ based charity working towards creating a public space to celebrate mathematics, and who among us isn't well up for some more of that?

In serious business, Skewray Research isn't clowning around with the basics of probability in their exploration of that 'mysterious gremlin of chaos: Randomness is an Inverse Stochastic Process.

Richard Elwes thought that Matthew Aldridge's argument 'don’t write the binomial coefficient as n! / k! (n-k)!' was was very persuasive (and so do I). Those exclamation marks mean it must be pretty important, too.

An Interval...

... and a mystery! Robin Whitty (of Theorem of the Day) wonders what's going on: 'A Plus magazine email said "Take part in our pilot study". The link was to a rather anonymous Cambridge University questionnaire. Frustratingly, nothing on Plus's website that I could see, nor on that of its host, the Millennium Mathematics Project.' The survey at the link he included has since been deactivated. Can you shed any light on this mystery study? Answers on a postcard direct to Robin, or written in lemon juice and left at the usual dead-drop.

Less mysteriously, Robin also submitted a post on Factoring Carmichael Numbers, in which Lance Fortnow discusses how the Miller-Rabin algorithm can (unbeknownst to the AI models he asked) find non-trivial factors as well as determining the compositeness of Carmichael numbers (which are, as I had to look up, composite numbers that can be mistaken for primes because they satisfy $ a^{n-1} \equiv 1\pmod{n} $ for all integers $a$ that are relatively prime to $n$). Robin has also enjoyed Gil Kalai's blog post on Computational Complexity and Explanations in Physics, which discusses their thoughts on some of the ideas raised by Scott Aaronson in a talk of the same name.

Act II

November also saw the release of Chalkdust's 22nd issue which includes, in addition to its usual features, an interview with the creator of legendary vigilante number-hero El Nombre. Printed copies are available for nothing more than the cost of P&P, or you can read it as a PDF for nowt. Donovan Young submitted his own article, A tale of $n$ cities, to the Carnival, so check that out while you're there, even if it's not the Dickens novel that I, personally, would think of first at this time of year. There's also a blink-and-you'll-miss-it moment on page 60: second-from-bottom, on the left.

Jencel Panic wants you to know about the online book about category theory that they're writing. They've submitted it here because in the last month they've finished an important chapter: Natural Transformations.

Arguably too early for December's Carnival, it was also too late for November's, so Robin thought it was worth including and I agree: Fractal Kitty's Mathtober 2025 Sketches are a case-in-point that mathematics and art do not exist in isolation from each other. They're also a bit lovely, and fairly soothing to browse.

John D Cook explores the 'intellectually and visually satisfying' Japanese Polygon Thereom, complete with some Python code for exploring the theorem more deeply, and generating your own related images.

That rounds up this months submissions, but feel free to explore some of the...

Sideshows

Or: Things I Stumbled Upon or Did Last Month That I Thought Might Be Worth Sharing

• 2025's MathsJam UK Gathering took place over the weekend of 22-23 November. This year it was held in my own stomping ground of Milton Keynes, and you can find out what you missed on the MathsJam Gathering 2025 archive. I particularly enjoyed dusting off my guitar and taking part in the MathsJam Jam for the first time in years!
• Among the many other things that I also particularly enjoyed about Big MathsJam was the fact that two of this year's talks were from regulars of the Bletchley & MK monthly MathsJam: Brigitte spoke about the mathematical poetry of the Dhananjayas, whilst David explored fair coins, logical abduction, and sea monsters. We'd love to see new faces joining us at our monthly MathsJam in Central Milton Keynes, but if it's not exactly on your doorstep, there are plenty of other locations lucky enough to have their own: find your closest monthly MathsJam here.
• Maths Week England took place from 15-23 November, with a launch event at the new MathsWorld discovery centre in London. Go to mathsweekengland.co.uk to find resources and read about this year's events, and sign up to make sure you don't miss out on next year's activities! If you represent a museum or primary school you might be interested in this.
• I've recently been made aware of the AgRoMa project, based at Newcastle University, which is studying the Roman practical mathematics tradition. I'm particularly excited by their intention to create educational materials to encourage and support teaching that bridges mathematics and humanities. Fun fact: AgRoMa is a contraction of Agrimensores and Roman Mathematics, and a groma was an instrument used by ancient Roman surveyors! 

Next time...

Thanks for joining me for this month's Carnival of Mathematics! The show's over, the curtains have closed. You don't have to go home but you can't stay here: if you're still buzzing from the experience why not head over to the Aperiodical's Carnival of Mathematics page and see where it's appearing next month, and maybe get involved: there's a form to submit any mathematical tidbits you feel might be worth featuring over the next month, and if you have a maths-themed blog they're always looking for future hosts.

As you go, do have a look around the gift shop: The Mathematicians' Library would make an excellent gift for anyone in your life with a slightly nerdy disposition.