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.

Maths in Museums: Castle Keeps and Their Shapes

I recently took time out for a quick weekend away in South Wales. The itinerary involved, as is only right, castles^[1]. I was struck by the geometric shapes of the keeps - particularly one with a circular cross-section at Tretower castle, and a larger one with a hexagonal base that forms part of Raglan castle.

The Circular Keep at Tretower © 2025 by T. Briggs is licensed under CC BY-NC-SA 4.0

Maths in Museums: Roaring Meg

Goodrich Castle, a ten-minute drive from the Welsh border in Herefordshire, England, has to be one of my favourites^[1]. It's a Norman medieval ruin that played host to an English Civil War siege by Parliamentary forces upon the Royalist forces stationed there. Despite not having its defences updated to 17th-century standards, the medieval castle stood up well to direct attacks and artillery was introduced to the conflict, with Parliamentary Army Colonel John Birch commissioning the casting of a mortar called Roaring Meg.

Roaring Meg at Goodrich Castle © 2025 by Thomas K. Briggs is licensed under CC BY-NC-SA 4.0

Maths in Museums: Bolsover Castle

For years I have been periodically prompted to think "ah yes, must go there one day" by signs for Bolsover Castle as I pass through the vicinity of Junction 29A on the M1. That day, as all things must surely do, finally came to pass on Monday 11th August, 2025^[1].

Bolsover Castle © 2025 by Tom Briggs is licensed under CC BY-NC-SA 4.0

As with all posts in the Maths in Museums series, it is not for me to tell you how or when to visit, how much it costs, or even much about the history of the site (though some will surely creep in). All of that can be left to English Heritage, who are its current custodians. Our purpose here is to uncover some of the opportunities for mathematical exploration that I spotted during my visit^[2]. As with most museums and galleries, there's no interpretation at the site that will help you to structure any exploration of these ideas.

Book Review: Love Triangle by Matt Parker

Full disclosure: I received a free copy of the book on the promise that I would write this review, but you have my word that what follows is my honest appraisal! I need to take this opportunity to apologise profusely to Matt^[1].

Love Triangle^[2] is Matt Parker's n^th offering to the pop-sci shelves in your bookshop of choice, where n is a number that's a little bigger every time I look it up. He's one of the more well-known current writers of books destined for that section that have an explicit focus on maths, and his trademark informal style of unashamed explicit nerdity carried along by a current of comic sarcasm might not be for everyone, but it does seem to do the job for more people than it doesn't.

My copy of the book starts with a note...

To Thomas,

Hope you love

this book about

loving triangles.

... and I have to say at the outset that I do.

Its subtitle, The Life-Changing Magic of Trigonometry, reassures us that this book isn't a foray into romantic fiction. Instead, it's very much romantic fact: Matt Parker loves triangles, and for very good reason. He begins by introducing us to probably the most famous mathematician to non-mathematicians, Pythagoras, who almost every living Brit has heard of unless they had a highly specific pattern of absences spanning their schooldays. I'm all for exploring the history-of-maths^[3], and Parker doesn't disappoint, peppering his introduction with some quickfire examples of triangles being very useful to humans throughout their development.

Chapter 1 focuses on some key properties of triangles that allowed the ancients to get started using them to measure, build and share, prompting the birth of geometry about four thousand years ago. He provides examples of calculations involving triangles that are used to solve problems, end arguments, and while away time on holiday, and then takes us on a chain of events that begins with measuring the length of a stick and ends with calculating the size of one of the biggest things in the universe.

While Parker's prose is, I feel, accessible and entertaining^[4] to readers of all self-perceptions regarding ability in maths, he's not one to shy away from including opportunities for his more mathematically excitable audience members to get their nerd on. Anyone who's already experienced some of his other books will have noticed a penchant for using a page numbering system that makes you work to find out what number page you're on. He scratches that particular itch here too, and it took me a few pages to figure out. The theme of the book is a clue, naturally, and I won't spoil things here.

One of the running themes of the book is that everything is triangles and Parker demonstrates this across ten chapters, each beginning with a single idea sparking a stream-of-consciousness power-walk through the triangular landscape, zig-zagging between real-world applications and theoretical and conceptual understanding. Chapter 2 starts with a game of pool, explores how rainbows work, and ends up at the end of the dinosaurs. Chapter 3 explores six laws of triangles, including the famous one that Pythagoras almost certainly did not originate, and one attributed to a mathematician known as Heron. Chapter 4 looks at the various applications that result from approximating complicated surfaces by breaking them down into meshes of triangles, which leads nicely into Chapter 5's rummage through ideas and uses of mathematical tilings and... whatever the 3D version of tilings are called.

Much of this appears to merely be a vehicle for Parker to get from one groan-worthy pun to the next, but far from being a criticism I feel that this smooths the way through a journey which, even if it doesn't involve a swim through the waters of deep and potentially scary maths, at least stops off for a paddle by the shoreline at a number of points.

These early chapters lay down basic ideas of geometry and trigonometry that allow Parker to take us a little deeper into their applications, and he does so partly with reference to theory (I particularly liked his assertion that 'the power of trigonometry is not memorising equations, but rather knowing you can look them up and having faith they will solve all your problems'), but he always supports this with examples, many of which are surprising and lifted from contexts that may initially seem distant from maths including, in Chapter 7, basketball and DOOM (the computer game).

Chapter 8 helps us to explore our place in the world - very literally - and Chapter 9 puts things into perspective, both themes beginning with ancient (chapter 8) and surprisingly recent (chapter 9) historical contexts leading to more modern concerns like GPS, Guns 'n' Roses, impossible footballs, and some groundbreaking CGI scenery that provides a backdrop to the exploits of Matthew McConaughey and Anne Hathaway. Chapter 10 makes waves - sine waves - by mentioning just some of the ways that they contribute to such diverse endeavours as tracking monkeys and detecting collisions between black holes.

Love Triangle: The Life-Changing Magic of Trigonometry^[2] is a book that many of those already singing in the maths choir will undoubtedly enjoy, but it could also form a decent entry point for someone less confident in their own mathematical ability but open-minded and wishing to dip a toe into the rivers of mathematics. There are mathsy diagrams from time to time; even the occasional equation, but each is gently explained and clearly related to real-world contexts. It does assume that a reader will have a certain amount of resilience if they run into a topic that might be tricky to absorb first time around, but whenever Matt takes us into a tougher climb, a rest-stop is never far away, and his indefatigable sense of humour is a constant background reminder that it's not a problem if we don't get the hang of everything on the first pass.

My review is done, but I'd like to mention two things that make an appearance in the conclusion to Love Triangle: first was the surprising reference to a topic that has seen me falling down many a late-night rabbit-hole of nerdity: Brian May's guitar. And lastly, that giant of mathematics communication, Adam Atkinson; a reference made poignant by his departure to the Great MathsJam in the Sky just a few short months after publication. There is a page of heartfelt tributes to Adam here: gathering4gardner.org/remembering-adam-atkinson/

Finally, as a treat for anyone who read this far, a single line early in Love Triangle prompted me to create a crossword about triangles. Let me know how you get on!

Footnotes

1. This makes it sound like it's a bad review, which is naughty of me, because it very much is not. I'm apologising to Matt because I received my copy of the book before it was published in August 2024 and it's now April 2025. It has been, to say the least a bit of a year. [back]
2. If you intend to buy the book I'd recommend finding a local independent bookshop and buying it from them because they need and deserve some support. If, however, you'd rather buy it from Amazon then I'd appreciate it if you did so after clicking this link to get there as I'll get a cut (without increasing the price you'd pay if you didn't click that link to get there). [back]
3. And the mathematics-of-history. For many of the historical stories explored in Love Triangle it's difficult to decide which label fits best. [back]
4. Not to mention inclusive: it has prompted a small amount of ire (in the form of negative reviews) from the kind of people who like to complain about things being woke, so it's almost worth buying it immediately just to spite them. [back]

Maths Trails at Cultural and Heritage Sites

Many museums, galleries, and other heritage sector sites offer trails for visitors (including booked school groups and members of the public). Trails are a great way to help visitors of all stripes to structure a visit in an engaging way, and can be used to provide additional - or different - context, and to draw attention to particular themes.

One possible use of a trail activity is to encourage the use of mathematical skills to analyse whatever the trail guides them towards, or to highlight the role of mathematics in the objects and stories encountered where the exhibitions themselves may not have been designed with this in mind.

Prompted by a post in a TMiP chat group and a similar conversation on a GEM discussion list I thought I'd collate some of the examples and supporting resources in one place: I know that users of museums and galleries often seek out maths-related activities and events; and an increasing number of sector organisations are looking to develop their own, so I hope that this compilation might be useful for people in either camp.

Maths trails in action

These are maths trails out in the real world that I'm currently aware of, listed in alphabetical order of the organisation that offers them:

Edinburgh: The University of Edinburgh's Maths Outreach team created this digital interactive Discover Edinburgh's Mathematical History trail: browse from the comfort of your own home, or fire up your phone (assuming you've got a good data plan) and march across the streets of Edinburgh. (Google Maps link)

Museum on the Mound (Edinburgh): I'm told that The Money Maths at the Museum trail is popular with school groups, and that they occasionally bring it out for public use, e.g. during Maths Week Scotland. If you'd like them to offer a maths trail for general visitors, it wouldn't hurt to let them know. (Google Maps link)

National Museum of Scotland (Edinburgh): The A World of Maths Trail is downloadable as a .pdf. It's targeted at teachers to use to print for their students and bring to enrich a school trip to the site, but there's absolutely nothing to stop anyone else using it, and the teachers notes available from that page are useful outside that demographic too. (Google Maps link)

The Science Museum (London): Their Maths Activity Trail is a downloadable .pdf that aims to support visitors as they work their way through the Winton Mathematics Gallery and Science City 1550 – 1800: The Linbury Gallery. It's aimed at teachers, but as its offered as a free public download there's nothing to stop anyone making the most of it. (Google Maps Link)

Scotland (various locations plus location-independent resources): Each year an increasing number of museums and galleries offer mathematical activities including trails for Maths Week Scotland, which usually takes place towards the end of September. The Maths Week Scotland website lists activity packs and learning resources for schools (freely and openly accessible regardless of whether you represent a school or not); a community learning resource; family activities to do at home; and events offered by a host of museums, galleries, and other organisations across Scotland. Many of the events listed are exclusively offered during Maths Week Scotland, so do make sure you get in touch with your local sites and let them know that you'd like the option to experience maths in museums during the other 51 weeks of the year too!

York Minster: Their Maths Quest trail is available either as a .pdf downloaded from their website, or their Learning Resources page (which lists this along with trails on other themes) says that you can 'collect them at the Welcome Desk when you arrive.' (Google Maps link)

Your local museum or gallery: If your favourite local site isn't listed here and you'd like it to be, then get in touch with them and let them know that you want their help to explore the maths in their stories, objects and exhibitions. When I ask people who work for a museum or gallery why they don't offer resources to help visitors engage with the mathematics that's just out of site, they usually tell me it's because nobody asks them to...

So ask them to!

• Does your organisation offer a maths trail that you think should be featured here? 
• Is anything listed here out of date (e.g. a link is broken or a site no longer offers a listed trail)?

If so, post a comment or get in touch.

Resources for people interested in creating their own maths trails

University of Bath Cryptography Challenge case study (from Maths Engagement Case Studies): https://katiesteckles.co.uk/casestudies/CS_Bath-Crypto-Competition.html

Math in the City: Designing a Math Trail for High School Students (chapter 4 of Handbook of Mathematical Science Communication): https://ionicasmeets.com/Publicaties/1887_3567511-2023-Bossema-Zwetsloot-Smeets_Book_chapter_Math_trail.pdf

• Do you know about an excellent resource aimed at helping people to create maths trails that isn't listed here?
• Are any of these resources no longer available (e.g. a link is broken)?

If so, post a comment or get in touch.