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Problem solving 101

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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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The doodle theorem, and beyond…

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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

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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

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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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Proof by storytelling

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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

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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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Analogue computing: fun with differential equations

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When it comes to differential equations, things start to get pretty complicated—or at least that’s what it looks like. When I studied mathematics, lectures on differential equations were considered to be amongst the hardest and most abstract of all and, to be honest, I feared them because they really were incredibly formalistic and dry. This is a pity as differential equations make nature tick and there are few things more fascinating than them.

When asked about solving differential equations, most people tend to think of a plethora of complex numerical techniques, such as Euler’s algorithm, Runge–Kutta or Heun’s method, but few people think of using physical phenomena to tackle them, representing the equation to be solved by interconnecting various mechanical or electrical components in the right way. Before the arrival of high-performance stored-program digital computers, however, this was the main means of solving highly complicated problems and spawned the development of analogue computers.

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Roots: the legacy of Fibonacci

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Most initial thoughts when the name Fibonacci is mentioned centre around sequences, rabbits, nature and spirals. However, the Fibonacci legacy is much more fundamental to modern scientific studies, and without his influence, mathematics—as we know it—would not exist.

The famous Fibonacci spiral

The famous Fibonacci spiral

Leonardo, of the family of Bonacci, was born in Pisa, Italy, in around 1170. It wouldn’t be until the French mathematician, Édouard Lucas, wrote extensively about the $1, 1, 2, 3, 5, 8, \ldots$ sequence in 1877 that the “Fibonacci sequence” would become more well-known. Leonardo’s father was a successful merchant and customs officer, travelling around the Mediterranean with his family in tow.
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You can count on Dirichlet

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In the days before email, mathematicians relied upon pen, paper and the postman to share ideas and communicate fiendish numerical taunts. An excited Dirichlet wrote to Kronecker in 1858:

… that sum, which I could only describe up to an error of order $\sqrt{x}$ at the time of my last letter, I’ve now managed to home in on significantly.

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A mathematical view of voting systems

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Soon after I began my undergraduate degree, Barack Obama won the 56th United States presidential election. The next day, my pure maths tutor asked me if I had followed it closely, saying that elections really interested him. I was surprised to hear this: surely the only maths involved was adding up the number of votes? In reality, voting systems hold considerable interest for mathematicians, and there are several mathematical results and theorems concerning electoral processes. The main thing that I like about the language of mathematics is that it allows us to make extremely precise statements without ambiguity and, as you’ll see, we can make precise mathematical statements about voting systems—with some surprising results.

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