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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.

Apéryodical: Roger Apéry’s Mathematical Story

This is a guest post by mathematician and maths communicator Ben Sparks.

Roger Apéry: 14th November 1916 – 18th December 1994

100 years ago (on 14th November) was born a Frenchman called Roger Apéry. He died in 1994, is buried in Paris, and upon his tombstone is the cryptic inscription:

\[ 1 + \frac{1}{8} + \frac{1}{27} +\frac{1}{64} + \cdots \neq \frac{p}{q} \]

Apéry's gravestone - Image from St. Andrews MacTutor Archive

Apéry’s gravestone – Image from St. Andrews MacTutor Archive

Roger Apéry - Image from St. Andrews MacTutor Archive

Roger Apéry – Image from St. Andrews MacTutor Archive

The centenary of Roger Apéry’s birth is an appropriate time to unpack something of this mathematical story.

Solomon Golomb (1932-2016)

“I’m proud that I’ve lived to see… so many of the things that I’ve worked on being so widely adopted that no one even thinks about where they came from.” Solomon Golomb (1932-2016)

Solomon Golomb, who died on Sunday May 1st, was a man who revelled in the key objects in a recreational mathematician’s toolbox: number sequences, shapes and words (in many languages). He also carved out a distinguished career by, broadly speaking, transferring his detailed knowledge of the mathematics behind integer sequences to engineering problems in the nascent field of digital communications, and his discoveries are very much still in use today.

Learning to play Go


We bumped into Robert at the last MathsJam conference. He spoke so enthusiastically about Go, and how easy it is to get started, that we asked him if he could write a guide for someone who wants to get into the game. Here it is!

This article is designed for those who want to learn Go or read about a way of teaching it.

Go is an ancient game from Asia (being deliberately vague here). The reasons I love this game are the simplicity of the rules and the fact it has an effective handicap system. The rule system means it is great to introduce to people who may not have a background in games. The handicap allows beginners to play world champions and for both to have a challenging game.

Guest post: Sequence Numbers

This is a guest post, sent in by David, who’s discovered an interesting property of numbers, and is looking for collaborators to take it further.

Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost. Rigour should be a signal to the historian that the maps have been made, and the real explorers have gone elsewhere.

– W. S. Anglin

A few years ago I saw a post on a website that showed that the inverse of 998,001 produces a decimal expansion that counts, using three digit strings, from 000 to 997 without error.

\[ \frac{1}{998,\!001} = 0.0000010020030040050060070080090100110120130\ldots \]

I immediately thought that this had to be a hoax. I decided to work it out to prove it was a hoax – after all some people put anything they want on the web whether it is true or not.

How many ways to shuffle a pack of cards?

This is an excerpt from friend of The Aperiodical, Matt Parker’s book, “Things to Make and Do in the Fourth Dimension”, which is out now in paperback.


There’s a lovely function in mathematics called the factorial function, which involves multiplying the input number by every number smaller than it. For example: $\operatorname{factorial}(5) = 5 \times 4 \times 3 \times 2 \times 1 = 120$. The values of factorials get alarmingly big so, conveniently, the function is written in shorthand as an exclamation mark. So when a mathematician writes things like $5! = 120$ and $13! = 6,\!227,\!020,\!800$ the exclamation mark represents both factorial and pure excitement. Factorials are mathematically interesting for several reasons, possibly the most common being that they represent the ways objects can be shuffled. If you have thirteen cards to shuffle, then there are thirteen possible cards you could put down first. You then have the remaining twelve cards as options for the second one, eleven for the next, and so on – giving just over 6 billion possibilities for arranging a mere thirteen cards.