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Problem solving 101

From the outside looking in, maths problem solving can seem like a kind of magic. Here is a typical image: a lone genius, peering at a vexing problem, rubs their chin, paces up and down; then a bolt of inspiration hits and the solution falls neatly into place.

And while it’s true that inspiration can strike the lucky few, for the rest of us this is no more than an illusion (and often a carefully cultivated illusion at that). In reality, problem solving is usually much more prosaic, nothing more than a careful application of well-known, and often quite elementary, techniques.

So what are these elementary techniques? In this article, I’ll look at some of the simplest and easiest to understand. Happily, they are also some of the most powerful and widely applicable. These techniques will be explained by way of example problems; I strongly encourage you to attempt the problems yourself before reading the solutions. Continue reading

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Hedy Lamarr: Hollywood star and secret inventor

The golden age of Hollywood was a time of classic movies and classic movie stars. A time of ‘frankly my dear’, ‘play it again’ and ‘whoops Mr Parson’ (I made the last one up). Yet only one star was billed as ‘The most beautiful woman in the world’. This was Hedy Lamarr: an Austrian-born actress, former wife of an arms dealer, international movie star, and occasional inventor. Her most celebrated invention was something without which today’s mobile phone and Wi-Fi technology would not be possible: frequency-hopping.  Continue reading

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The doodle theorem, and beyond…

One of the things I like about recreational maths is how we can start with a simple game, play around a bit, poke in the corners, and suddenly fall down a deep hole into some serious mathematics. In this article we start with some well-trodden ground, which some readers will find familiar. However, we quickly find that all is not as it seems, and we soon stumble over a veritable pot of gold. To see how, read on…

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Prime jewellery

I was recently given a copy of Crafting Conundrums: Puzzles and Patterns for the Bead Crochet Artist by Ellie Baker and Susan Goldstine. This was pretty exciting for me, as although I knew nothing about bead crochet (I’d never heard of it), I’m a mathematician who enjoys exploring mathematical ideas through craft. So naturally I rushed out and bought lots of beads and thread, and a very tiny crochet hook (1.5mm, if you’re really interested). Continue reading

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Roots: Pythagoras of Samos

In The Wizard of Oz, the Scarecrow shows us how intelligent he has become by (mis)quoting Pythagoras’ theorem:

“The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side.”

Homer Simpson does a similar thing when he puts on a pair of glasses and tries to convince himself that he is smart.

It would seem that the lasting legacy of Pythagoras of Samos is the formula linking the sides of a right-angled triangle. It could, however, be argued that the actual legacy of Pythagoras is much greater—it’s more than the formula used in contrived situations of ladders being rested against walls or finding the answer to that most fundamental of questions: would the pencil stick out of the top of the pot? His legacy is around us every day…

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Mathematics: queen of the arts?

In the brief tradition of Chalkdust cover articles there is a developing discussion of how mathematics and art are related.

Art is simply the making of representations. Art happens when a person has an idea or a vision that exists in their imagination (the mind’s eye) and is impelled to communicate said idea by making a visible manifestation (representation) of it in the material world. The idea or vision on its own is not art. Art occurs amid the struggle to make a representation of the idea that the artist can show to other people. Art may be relatively `fine’ or popular, conceptual or objective, highbrow or applied, yet still fall within this definition. Judgements about the quality of art are made largely by consensus among the cognoscenti in a given art milieu. These judgements are subject to change over time as the perception of works of art are always modified by the current `cultural environment’ and fashion.

The image Central Quadratic explains itself, I hope, as a celebration of analytic geometry.

The image Central Quadratic explains itself, I hope, as a celebration of analytic geometry.


So artists and mathematicians share the `having of ideas’. But what then? Mathematicians communicate their ideas—yes. But ideas in maths take the form of theorems or conjectures about numbers, space or other abstract entities. The quality of these ideas is first assessed by proof. Can the idea be shown to be true? And second, if the idea is true, is it interesting? That is, does it usefully contribute to the mass of existing mathematics? Communication of mathematical ideas may require the invention of new symbols or diagrammatic forms, etc, but these are in the nature of being a new language, not art.

In my view then, art and mathematics share the magical process of `idea getting’ but essentially differ in where they go with those ideas. If maths is to be considered an art, it would have to be a sort of `super-art’ or art `to a higher power’. Easier, I think, to class mathematics as the science of number, space, shape and structure, etc—the abstract entities that exist in our minds.

Imagine an intelligent alien’s perception of our arts and our mathematics. Our art would be more or less incomprehensible, depending on how alien the being was; but our maths would be as true for the alien as it is for us. Furthermore, good mathematics will not diminish with time or go out of fashion.

There is an affinity between some mathematicians and some artists. Certainly, it is a most pernicious error that scientific and artistic talent exclude each other—an idea unfortunately common among school counsellors. The common ground between art and science/maths that leads us to the `getting of ideas’ is the activity we call play. The thing of it—the thrilling thing, the magical thing—is the moment when one discovers a new idea, or pattern, or conceptual framework, or whatever: the eureka moment! And are these moments not usually approached through playing in the mind with new combinations and orderings of existing mental constructs?

In ray tracing, each ray is used to decide the colour of a pixel on the image plane.

In ray tracing, each ray is used to decide the colour of a pixel on the image plane.

Spheres was created using ray tracing.

Spheres was created using ray tracing.


I had one of my most memorable eureka moments sometime in 1971 while sitting on a dead tree in Epping forest. At that time, I had been collaborating on an automatic projective line drawing program with `hidden line removal’, going where Autocad later arrived. I was considering algorithmic approaches to colouring surfaces in projective drawings. I realised that if I thought of objects in the scene as being represented mathematically as arrays of vertices and planes in some coordinate space, then I could solve for the equation of the line going from an eyepoint through a particular pixel in the image plane and into the scene (as in the diagram). From the equation of the line, I could find the closest surface along the path and then compute the colour and illumination value for that pixel based on the defined colour on the surface, along with its relationship to any light source or other light-emitting surfaces. And so I had invented ray tracing—the foundation of all computer generated synthetic imaging for special effects in cinema, television and gaming. Of course, I neither invented it first nor alone—and I certainly had neither the persistence nor vision to pursue ray tracing to practical or rewarding development. But its discovery was a thrill, as were the few simple pictures I made using the technique in a primitive manner on the pen plotter available.

As spheres have become my most persistent motif, I will end with two more related works that play on the division and articulation of spherical surfaces: Sphere Architecture and Star Sphere.

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[ Pictures: Central Quadratic: Used with permission from UCL Art Museum, University College London; Spheres, Sphere Architecture and Star Sphere: Used with permission from John Crabtree ]

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Proof by storytelling

There are many equivalent ways of defining the binomial coefficients $\binom{n}{r}$ (pronounced ‘$n$ choose $r$’).  In this article, though, $\binom{n}{r}$ is defined simply as the number of ways of choosing a subset of $r$ things from a set of $n$ things.  Note that this definition does not give us a way to calculate $\binom{n}{r}$; and, if you already know how to evaluate the binomial coefficients, you should put the formula out of your mind and let yourself be surprised by the beautiful way in which identities can be proved without resorting to bashing out messy fractions of factorials.

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The Buckingham π theorem and the atomic bomb

On 16 July 1945, the first nuclear test, ‘Trinity’, was carried out and with it the nuclear age began. The explosion was huge, but the actual calculation of the amount of energy released was rather difficult due to the large number of physical and chemical processes involved in the detonating reaction; even the rough estimates were far from accurate. It was not until the publication of the photographs of the explosion that scientists became aware of its magnitude. With just these photographs and some clever mathematical arguments, British physicist GI Taylor,  Soviet physicist LI Sedov and Hungarian–American mathematician John von Nuemann estimated independently an energy of about 17 kilotons of TNT.  Taylor published this result in 1950, with the US Army not at all thrilled that this sensitive piece of information was now in the public domain. Although the estimates of Taylor, Sedov and von Neumann required the use of some complex mathematics, dimensional analysis and the Buckingham π theorem allow us to come to the same conclusion with a minimum amount of knowledge of physics.

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