Author Archives: apgoucher

Continuing the theme of magic squares, we begin with the unique simplest non-trivial magic square. It contains the integers {1, 2, …, 9}, one in each square, such that the rows, columns and diagonals all sum to 15. Remarkably, the converse is … Continue reading →

If we provide a bog-standard computer with an infinite data store, it suddenly becomes a Turing machine: capable of answering any question answerable by any digital computer. Even quantum computers are no more powerful; they are merely faster. For example, a quantum … Continue reading →

Here’s the post I promised you a while ago. I wrote the first part of this in an e-mail on Burns’ Night of 2011, and the main idea (which Alex Bellos termed the ‘Goucherian calendar’) was mentioned in the Guardian. … Continue reading →

The Mandelbrot set, as you’re probably aware, is the set of points c in the complex plane such that iterating f(z) = z² + c repeatedly doesn’t escape to infinity. By invoking the triangle inequality, it can be shown that a point … Continue reading →

In Albrecht Dürer’s 1514 engraving Melancolia I, there is a famous 4×4 magic square. In addition to the rows, columns and main diagonals summing to 34, so do various other arrangements such as the quadrants and four corners. We can transform this into an … Continue reading →

There have been several recent items of news, either related to or inspired by the recent postings on Complex Projective 4-Space. Rather than mention them separately, I have collected them in this ‘digest’ format. Jigcypher solved I received an anonymous e-mail … Continue reading →

One of the qualities of a quintessential British mathematician is the ability to invent crazy and contrived functions to satisfy particular properties. For instance, John Conway’s base-13 function was designed as a counter-example to the converse of the Intermediate Value … Continue reading →