# Essays on recreational mathematics

Young men should prove theorems, old men should write books.

G.H. Hardy

At King Edward VII school, King's Lynn, in the early 1970s, I had the benefit of an inspiring maths teacher, Harry Thornton. The half a dozen of us with similar interests met after school in Harry's back room, where he would introduce this 'Maths Club' to a variety of mathematical problems, puzzles and games. Many were picked from the monthly column in Scientific American, written at the time by Martin Gardner, who inspired an interest in recreational mathematics in so many people over many years.

I went on to study mathematics and theoretical physics at Cambridge University in the mid-1970s. My PhD in radio astronomy at the Cavendish Laboratory, Cambridge, in 1980, focused on the cosmological evolution of radio sources. Shortly afterwards, as a research fellow with the European Space Agency, it was the mathematical elegance underpinning the principles of space astrometry that led to my interest in the Hipparcos mission.

Returning to my curiosity in mathematics, this project started in 2018 as an informal input for a local school. Later, throughout 2020, I wrote these short weekly 'essays' on just a few of the more recreational topics that have interested or entertained me over the years. In this more structured form, I hope they may be of interest to others exploring the endlessly fascinating and beautiful world of mathematics, and that they might inspire a few young minds in the process.

And I dedicate this small collection of 52 essays to the memory of Harry Thornton.

## 1. The Basel problem

The 'Basel problem' was to find an exact sum for the series in which each term is the square of the corresponding term in the harmonic series. It was famously tackled by Jakob Bernoulli, professor of mathematics at Basel University in the 17th century. The answer connects pi, prime numbers, and the Euler product.

6 January 2020

For the more mathematical

## 2. Magic squares

A magic square is an n x n square grid, filled with positive whole numbers in sequence. The sums of the integers in each row, column and diagonal are identical. This sum is referred to as the 'magic constant', or magic sum. Magic squares have many curious properties, and similar concepts also exist in more than two dimensions.

13 January 2020

Why do they exist?

## 3. Möbius bands and topology

There is perhaps no better illustration of the seemingly abstract mathematical field of topology than the Möbius band. Take a strip of paper, and join the ends together. What would happen if you were to pierce the middle of the band with scissors, and cut around its full length? Things get more puzzling if you give the band a twist first!

20 January 2020

Fun with paper and scissors

## 4. Nim

Games which contain a strong element of mathematics, and in which probability or chance playing a greater or lesser role, abound. These range from the trivial, such as noughts-and-crosses, to games of almost infinite complexity such as chess. Nim is one of the oldest, and one of the simplest to play whilst still being engaging.

27 January 2020

Use mathematics to win!

## 5. Prime numbers

Carl Friedrich Gauss stated that "The problem of distinguishing prime numbers from composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic". Two hundred years later, prime numbers remain as mysterious and perplexing as ever.

3 February 2020

A most mysterious class

## 6. How many primes

How many prime numbers are there less than any given number, N? This apparently straightforward question is of great importance in mathematics. Is there some law, or some expression, which predicts the number of prime numbers up to a given value? Searches for this 'law' leads to some remarkable results.

10 February 2020

For the more mathematical

## 7. Almost integers

An 'almost integer' is, as its name suggests, a number that is not an integer, but is almost an integer. Almost integers are considered to be interesting when they arise in an unexpected context. So far, you may be thinking, this is not very interesting... But these two pages contain some beautiful results, and will convince you otherwise!

17 February 2020

For the more mathematical

## 8. Constrained writing

Letter frequency analysis has been used for centuries to break simple cryptograms and ciphers, and became important in the 15th century with the development of movable typeface. It is a small step from here to enter the field of constrained writing, a literary technique in which a text follows some sort of algorithmic rule, usually lexical in nature.

24 February 2020

Playing with letters and words

## 9. Integer sequences

A sequence is an ordered list of numbers whose terms can be described in some well-defined way. Well-known examples abound, including powers of two, the prime numbers, the Fibonacci sequence, and many others. Indeed, as of 2020, the Online Encyclopaedia of Integer Sequences includes more than 300,000 of them.

2 March 2020

For those that like order

## 10. Pierre de Fermat

Pierre de Fermat, 1607-1665, was a trained lawyer at the Parlement de Toulouse in France, for whom mathematics was more of a hobby than a profession. He is best known for his principle of light propagation (that light travels between two given points along the path of shortest time), and for the celebrated Fermat's Last Theorem in number theory.

9 March 2020

The life of a mathematician

## 11. The number pi

The most common and widely known definition of pi is as the ratio of the circumference of a circle to its diameter, independent of the circle's size. This simple definition of pi gives little clue as to its appearance as one of the most important, ubiquitous and surprising numbers in all of mathematics.

16 March 2020

A very important number

## 12. Calculating pi

There are two main challenges in calculating the digits of pi to ever-increasing accuracy. One is formulating a defining series which converges rapidly. The other is in the computational power, storage, and organisation required to perform the calculations. But these efforts also have numerous practical benefits.

23 March 2020

More useful than you might think

## 13. Memorising numbers

Individuals who can recall large quantities of information seem to possess superior memories. And while some people display this even in the apparent absence of an encoding strategy, evidence suggests that a key component of a superior memory is the development of mnemonic skills.

29 March 2020

Impress your friends