A numerical coincidence is an equality or near-equality between different mathematical quantities which has no known theoretical explanation. Sometimes such equalities remain mysterious and intriguing, and sometimes theory advances to the point where they can be explained and are no longer regarded as surprising.

## Archive for the 'Occasional' Category

### Numerical Coincidences

Published March 23, 2017 Occasional Leave a CommentTags: Number Theory, Recreational Maths

### Brun’s Constant and the Pentium Bug

Published March 9, 2017 Occasional Leave a CommentTags: Arithmetic, Euler, Number Theory

Euclid showed by a deliciously simple argument that the number of primes is infinite. In a completely different manner, Euler confirmed the same result. Euler’s conclusion followed from his demonstration that the sum of the reciprocals of the primes diverges:

Obviously, this could not happen if there were only finitely many primes.

### Topology in the Oval Office

Published February 23, 2017 Occasional Leave a CommentTags: Graph Theory, Recreational Maths, Topology

Imagine a room – the Oval Office for example – that has three electrical appliances:

• An air-conditioner ( a ) with an American plug socket ( A ),

• A boiler ( b ) with a British plug socket ( B ),

• A coffee-maker ( c ) with a Continental plug socket ( C ).

The problem is to connect each appliance to the correct socket, **avoiding any crossings of the connecting wires.**

### Metallic Means

Published February 9, 2017 Occasional Leave a CommentTags: Algebra, Arithmetic, Recreational Maths

with the value

.

There is no doubt that is significant in many biological contexts and has also been an inspiration for artists. Called the *Divine Proportion*, it was described in a book of that name by Luca Pacioli, a contemporary and friend of Leonardo da Vinci.

### The Beginning of Modern Mathematics

Published January 26, 2017 Occasional Leave a CommentTags: Algebra, History

The late fifteenth century was an exciting time in Europe. Western civilization woke with a start after the slumbers of the medieval age. Johannes Gutenberg’s printing press arrived in 1450 and changed everything. Universities in Bologna, Oxford, Salamanca, Paris and elsewhere began to flourish. Leonardo da Vinci was in his prime and Christopher Columbus was discovering a new world.

The picture below is of a sculpture piece called * Intuition*, which stands in front of the Isaac Newton Institute (INI) in Cambridge. It is in the form of the

*Borromean Rings*, a set of three interlocked rings, no two of which encircle each other.

### Unsolved: the Square Peg Problem

Published December 29, 2016 Occasional Leave a CommentTags: Geometry, Topology

The idiom “square peg in a round hole” expresses a mismatch or misfit, often referring to somebody in the wrong profession. It may also indicate a difficult or impossible task but, of course, it is quite simple to fit a square peg in a round hole, hammering it in until the corners are tight against the circular boundary of the hole. Since the peg may be oriented at any angle, there are an infinite number of ways to fit a square within a circle. In contract, for a boomerang-shaped hole, there is just one way to draw a square with its vertices on the curve.