### “Pariah Moonshine” Part I: The Happy Family and the Pariah Groups

Being a mathematician, I often get asked if I’m good at calculating tips. I’m not. In fact, mathematicians study lots of other things besides numbers. As most people know, if they stop to think about it, one of the other things mathematicians study is shapes. Some of us are especially interested in the symmetries of those shapes, and a few of us are interested in both numbers and symmetries.

### Footballs on road signs: an international overview

I’m an old fashioned manager, I write the team down on the back of a fag packet and I play a simple 4-4-2.

• Mike Bassett, England Manager

I’m very much like Mike Bassett: I like standing on the terraces, I like full-backs whose main skill is kicking wingers into the ad hoardings, and – most of all – I like geometrically correct footballs.

### Stirling’s numbers in a nutshell

This is a guest post by researcher Audace Dossou-Olory of Stellenbosch University, South Africa.

In assignment problems, one wants to find an optimal and efficient way to assign objects of a given set to objects of another given set. An assignment can be regarded as a bijective map $\pi$ between two finite sets $E$ and $F$ of $n\geq 1$ elements. By identifying the sets $E$ and $F$ with $\{1,2,\ldots, n\}$, we can represent an assignment by a permutation.

### A new aspect of mathematics

This is a guest post written by David Nkansah, a mathematics student at the University of Glasgow.

Around the fourth century BC, the term ‘Mathematics’ was defined by Aristotle as the “science of quantity”. It’s my own experience as a young mathematician to say this definition, although correct in its own right, poses a problem for those who do not truly know what mathematics is. It fails to highlight the true creativity of the subject.

Human inspiration and imagination are essential ingredients in mathematics. Regarding creativity, one could say, with merit, that in a sense mathematics is an art. Before proceeding to outline similarities between sketching mathematical proofs and painting on a canvas, it is important to know what fundamental premises mathematical proofs are built on.

### Circular reasoning on Catalan numbers

This is a guest post by researcher Audace Dossou-Olory of Stellenbosch University, South Africa.

Consider the following question: How many ways are there to connect $2n$ points on a circle so that each point is connected to exactly one other point?

### Square wheels in an Italian maths exam

There have been various stories in the Italian press and discussion on a Physics teaching mailing list I’m accidentally on about a question in the maths exam for science high schools in Italy last week.

The paper appears to be online.

(Ed. – Here’s a copy of the first part of this four-part question, reproduced for the purposes of criticism and comment)

The question asks students to confirm that a given formula is the shape of the surface needed for a comfortable ride on a bike with square wheels. (Asking what the formula was with no hints would clearly have been harder.) It then asks what shape of polygon would work on another given surface.

What do people think? Would this be a surprising question at A-level in the UK or in the final year of high school in the US or elsewhere? Of course, I don’t know how similar this question might be to anything in the syllabus in licei scientifici.

The following links give a flavour of the reaction to the question:

6 hours, 1 question out of 2 in section 1, 5 out of 10 in section 2. My own initial reaction is that if I had to do this exam right now I’d do question 2 in section 1 but I’ve not actually attempted question 1 yet.

### Dani’s OEIS adventures: triangular square numbers

Hi! I’m Dani Poveda. This is my first post here on The Aperiodical. I’m from Spain, and I’m not a mathematician (I’d love to be one, though). I’m currently studying a Spanish equivalent to HNC in Computer Networking. I’d like to share with you some of my inquiries about some numbers. In this case, about triangular square numbers.

I’ll start at the beginning.

I’ve always loved maths, but I wasn’t aware of the number of YouTube maths channels there were. During the months of February and March 2016, I started following some of them (Brady Haran’s Numberphile, James Grime and Matt Parker among others). On July 13th, Matt published the shortest maths video he has ever made:

Maybe it’s a short video, but it got me truly mired in those numbers, as I’ve loved them since I read The Number Devil when I was 8. I only needed some pens, some paper, my calculator (Casio fx-570ES) and if I needed extra help, my laptop to write some code. And I had that quite near me, as I had just got home from tutoring high school students in maths.

I’ll start explaining now how I focused on this puzzle trying to figure out a solution.