Carnival of Mathematics 247

It is a new month, and with it comes a new Carnival! Before we enjoy the carnival in its entirety, we shall stake out the surroundings a bit. What is this number, 247?
It is a Harshad number, which means it is divisible by the sum of its digits. Indeed, that is so, for \(247=19 \times 13\). Yet another fun fact, 'harshad' means 'joy giver' in Sanskrit. An appropriate number for a carnival, one might suppose. Here is an interesting Numberphile video on Harshad numbers featuring Tony Padilla.
This is also the start of a new year, 2026, in the Gregorian calendar. A look at the OEIS tells us that there is a connection between 247 and 2026. Let \(a(n)\) be the number of ways one can write \(n\) as an ordered sum of 1s, 2s, 3s and 4s such that no 2 precedes any 1 and no 3 precedes any 1 or 2. Now 247 factors as: \(247=13\times 19\) and we have \(247 = a(13)\), \(2026 = a(19)\). More details on the sequence of \(a(i)\)'s are here: A123569.

It is now time to explore the Carnival's mainstays. Hopefully this carnival is also a harshad, like 247! :)

Formalisation of Erdős Problems

A blog post by Boris Alexeev on the value of collaboration and AI in formalisation, with particular focus on Erdős problems. This post provides a brief update on the recent developments in this field. Find it here: Post on the Xena Project.

The Equational Theories Project

Launched by Terence Tao in September 2024 (Initial post and an update, this was featured in Carnival 233), he now reports on its completion: Post on his blog.

Notes on Probability

Brian Sutin from Skewray Research has made available their notes on probability theory here: Skewray Research. This builds upon an earlier post on clowns which featured in Carnival 246, but can be read in isolation.

Winning Lotteries Twice

While it is unlikely for a specific person to win the lottery twice (about 1 in 24 trillion!), this article is explains why it is not as unlikely for the lottery to be won twice (by a non-specific somebody). Read it here: Kit Yates' substack post.

The Rado Graph

The Rado graph is a graph with countably infinite edges that contains an isomorphic copy of any arbitrary finite graph. A bold claim! Matt Parker has published a video that walks us through proving this: You can prove every possible graph is just a sub-graph.

Exploitation and Game Theory

Rodrigo Treviño has published a note in The Mathematical Intelligencer recounting their experience with an exploitative seller, and a way to retain the upper hand: How to Not Let Them Get Away With It: A Note on the Mathematics of Infinite Exploitation. Math Intelligencer (2025) (Open access).

Christmas Cracker Calculator

Laura has published an article on their blog, Letters and Words, about how a set of magic trick cards was a binary calculator in disguise! Here's the link to their article: Christmas Cracker Calculator.

Highlights of 2025

The team at Mathateca have shared a list of maths stuff that they loved from 2025 here.
Quanta magazine has published their Year in Mathematics article, part of the 2025 in Review series.

That's it for this carnival, I hope it was fun. The fun doesn't end here though. Carnival 248 will be hosted by Laura Walsh, and you can submit entries here.

My thanks go to Robin Whitty, Skewray, Mathateca and Laura Walsh who submitted entries for this Carnival of Mathematics. I am also grateful to Ioanna, Katie and the team at The Aperiodical for organising the Carnivals.