Essays on recreational mathematics

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.

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.

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!

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.

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.

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.

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!

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.

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.

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.

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.

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.

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.