proudly presents  Carnival of Mathematics Number 245

Welcome to the 245th edition of Carnival of Mathematics!

Carnival of Mathematics	All Carnivals at Theorem of the Day

Welcome to the 245th edition of Carnival of Mathematics, meta-hosted by the remarkable aperiodical.com.

So 245, impress me! Well for a start, check out en.wikipedia.org/wiki/245_(number) which tells us that 245 is very far from being untouchable, being the aligot sum of 723 and a whole bunch of bigger numbers. Currently unknown to English Wiki but known to it.wikipedia.org/wiki/245_(numero) is the fact that 245 is un numero malvagio: an EVil number has, when written in binary, an EVen number of ones (as opposed to an ODd number of ones which makes the number, you guessed it, ODious). Ironically, Italian Wiki has no numero malvagio page — you go straight to Successione di Thue-Morse, which is what numeros malvagios want to do when they grow up. Meanwhile a novelty from ro.wikipedia.org/wiki/245_(num%C4%83r) is the fact that 245 is un număr rotund - all its prime divisors, 5 and 7 (twice), are smaller than its square root.

All of which goes to show that Wiki pages in different languages are not just translations of each other and some multilingual prowling around can be rewarding. And sometimes problematical: the English for "un număr rotund" is "A number k that is sqrt(k)-smooth". This is not a catchy name for a Wiki page and if you choose English from the "8 limbi" offered by ro.wikipedia.org/wiki/Număr_rotund you get palmed off with en.wikipedia.org/wiki/Round_number (until I or somebody else remembers to fix it).

Anyway, on the subject of rotundness, Carnival's postbag for the month of October was just that, thanks to everyone who shared.

October news first. You want the good news or the bad news first? OK, good news! MathsWorld London opened its doors and although I wasn't there Katie Steckles on Mathstodon made me feel that I was. The bad news is the loss of Chen-Ning (Frank) Yang (1922–2025) as reported by Peter Woit. Although a physicist, Yang's work made fundamental connections with mathematics, as witness the fact that his name is on one of the Clay Math Institute millenium prize problems. Make sure you read the comments because Woit as usual gets a lot of informed feedback from well-informed people. His blog has been seriously sidetracked by his chronicling the impact on Columbia University of the Trump administration mud slide but he also reported Some Math/Physics Items this month, including a link to advance postings on arxiv of many invited contributions for the 2026 International Congress of Mathematicians. We lost as well in October Gérard Laumon whose tremendous contributions to the Langlands programme include two doctoral students with Fields medals, plus a third, Sophie Morel, who won the inaugural AWM-Microsoft Research Prize in Algebra and Number Theory.

But this is a carnival so we have "a mission to share the joy and love that comes from doing maths" as Colin Beveridge eloquently puts it (each Wednesday morning, of which there were five in October!) in Double Maths First Thing. Peter Lynch does joy and love wonderfully in a monthly column called That's Maths for the Irish Times with October offering Music and maths are inextricably intertwined (seemed open-access when I read it but otherwise see the version on his blog thatsmaths.com). Monthly magic also from James Propp, whose Mathematical Enchantments blog has, I find, a particularly high TIL/TLDR ratio. October's enchantment was Picturing Mathematics as exemplified by (see, TIL!) "an 87 year old fractal". More magic, literally, from veteren Carnival host Ganit Charcha who presented to us the amazing card trick – “The Final 3”. Oh alright, it's not magic literally, because the post explains very clearly how the trick works. So I can say it's magic figuratively. Literally. I'll put Kit Yates's post on The Art Gallery Theorem in this paragraph as well because it's a lovely theorem with a joyful constructive proof. Kudos to Kit Yates for pitching it as a missed security opportunity for the Louvre, even if his title "could a simple maths problem have foiled the Louvre robbery?" is classic Betteridge. Kit Yate's blog "Math(s) and the real world" is monthly when it isn't weekly, so he's up there with Beveridge, Lynch and Propp. Poor Theorem of the Day really makes daily mean dally. But did manage a new post for October and it is The Hockey Stick Identity, inspired by discovering that this pretty Pascal's triangle trick is actually simple random sampling in disguise. Finally, nothing actually submitted under the mathober heading but it earns a big joy and love shout-out. Anyway, #Mathober2025 was all over mathstodon plus lots to marvel at back at the fractalkitty.com mothership.

Mathematical collections that are not scraped together by LLMs are at a premium just now. So all power to Tim Gowers' initiative Creating a database of motivated proofs. Not to say that Gowers isn't heavily invested in AI-enabled mathematical progress and the idea of this database is precisely that 'scraping' should not mean 'scraping the bottom of the barrel'. But the idea is to pool lots of real (human) wisdom about writing "structured motivated proofs" and that will be valuable long after the LLM bubble bursts and its billions drain away into artery navigating nanobots or whatever. As will OEIS which served me well in O-for-Offline form in the 1980s and will long outlast busting bubbles (within reason, pace Kathryn Bigelow). Peter Kagey adds value by talking about RSS for the OEIS which means how to get what's new in integer sequences without waiting for it to be scraped up and spat out by a data farm.

It looks as if I've been procrastinating about doing any actual maths! But a clutch of Carnival submissions invite us to get our hands dirty. Ganit Charcha again with Componendo and Dividendo a little arithmetical trick that can make immediate some quite gnarly-looking identities. Tarquin Takes, on Youtube, has a cute take on gnarly Age Problems. Finally, John D. Cook, that indefatiguable prospector for mathematical gemstones, posted about turning trig identities into Fibonacci identities, a too-good-to-be-true correspondence that appears, surprisingly, to be barely twenty years old.

And we're almost back at the beginning, because John D. Cook's gem is a little metaphore for Gérard Laumon's Langlands treasure chest: it's geometry but, watch carefully, abracadabra, it's arithmetic!

Thanks for reading, clicking through ... and submitting to future Carnivals! Indeed, submit right here to Carnival 246, to be hosted by Thomas Mathematicians'-Library Briggs at tkbriggs.co.uk.