Browsing category there’s more than one way to do it

Cowboy Completing The Square

Factorising a quadratic? It's nice when it comes off, but there's a lot of guesswork, and no guarantee it even factorises. Completing the square? Who has time for all that algebra? And as for the quadratic formula, or your clever calculator methods: honestly, what are you, an engineer? There is

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The Maths Behind… Cakes

"Cooking," said my friend Liz in a recent Facebook post, "is one of the activities where maths is most useful in my everyday life." She added this picture: I've got several reasons for wanting to share this. 1. It's pretty much a model answer Imagine you're in a GCSE exam,

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Several Strings of 1s

This puzzle was in February's MathsJam Shout, contributed by the Antwerp MathsJam. Visit mathsjam.com to find your nearest event! Consider the set ${1, 11, 111, ...}$ with 2017 elements. Show that at least one of the elements is a multiple of 2017. The Shout describes this one as tough; you

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An alternative proof of the $\sin(2x)$ identity

Uncle Colin recently explained how he would prove the identity $\sin(2x) \equiv 2 \sin(x)\cos(x)$. Naturally, that isn't the only proof. @traumath pointed me at an especially elegant one involving the unit circle. Suppose we have an isosceles triangle set up like this: The vertical 'base' of the triangle is $2\sin(\alpha)$

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How to invert a $3 \times 3$ matrix

So much wasted time. I spent much of my first two years at university cursing the names of Gauss and Jordan, railing at my lecturer (who grim-facedly assured me there were no more useful uses of a student’s thinking time than ham-fistedly rearranging these things), and thinking “there MUST be

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Matrix Determinants — TMTOWTDI

Oh, the days -- weeks, even -- of my university life I spent working out the determinants of matrices. The 3×3 version was the main culprit, of course, usually needing to be split down into three smaller determinants, and usually requiring a sign change in one or two that I'd

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Highest common factor and least common multiple – TMTOWTDI

A student asks: I don't get the Venn diagram method for highest common factor and least common multiple. Do you have any other suggestions? As it happens, I do. I'm assuming you're OK with finding the prime factorisation of a number using (for example) a factor tree. In this example,

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