Posted in inequalities, puzzles.

The square root of 2 is 1.41421356237… Multiply this successively by 1, by 2, by 3, and so on, writing down each result without its fractional part: 1 2 4 5 7 8 9 11 12... Beneath this, make a list of the numbers that are missing from the first

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Posted in ask uncle colin, surds.

Dear Uncle Colin How does $\sqrt{9 – \sqrt{17}} = \frac{\sqrt{34}-\sqrt{2}}{2}$? I tried applying a formula, but I couldn't make it work. – Roots Are Dangerous, It's Chaotic A-Level Simplification Hi, RADICALS, and thank you for your message! Square roots of square roots are not usually trivial, but this one can

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Posted in podcasts.

In this month’s episode of Wrong, But Useful, Colin and Dave are joined by Special Guest Co-Host @pecnut, who is Adam Townsend in real life. Adam studies the behaviour of sperm in mucus, and chocolate fountains1 – when he’s not editing @chalkdustmag – Issue 6 of which is available now

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Posted in number theory, proof, there's more than one way to do it.

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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Posted in ask uncle colin, proof, trigonometry.

Dear Uncle Colin I'm stuck on a trigonometry proof: I need to show that $\cosec(x) – \sin(x) \ge 0$ for $0 < x < \pi$. How would you go about it? – Coming Out Short of Expected Conclusion Hi, COSEC, and thank you for your message! As is so often

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Posted in geometry.

On a recent1 episode of Wrong, But Useful, Dave mentioned something interesting2: if you take three regular shapes that meet neatly at a point – for example, three hexagons, or a square and two octagons – and make a cuboid whose edges are in the same ratio as the number

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Posted in ask uncle colin, trigonometry.

Dear Uncle Colin, I'm normally pretty good at simultaneous equations, but I can't figure out how to solve this for $a$ and $b$. $\cos(a)-\cos(b) = x$ $\sin(a)-\sin(b) = y$ – Any Random Circle Hi, ARC, and thanks for your message! This is, it turns out, a bit trickier than it

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Posted in trigonometry.

What's that, @pickover? Shiver in ecstasy, you say? Just for a change. Shiver in ecstasy. The sides of a pentagon, hexagon, & decagon, inscribed in congruent circles, form [a] right triangle. pic.twitter.com/Uastgc7SJo — Cliff Pickover (@pickover) May 20, 2017 That's neat. But why? Let's suppose the circles all have radius

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Posted in geometry, ninja maths, quadratics.

Dear Uncle Colin, I've got a funny square and I can't find $x$. Can you help? – Oughta Be Simple, Can't Unravel Resulting Equations Hi, OBSCURE, and thanks for your message! You're right, it ought to be simple… but it turns out not to be. It is simple enough to

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Posted in algebra.

This puzzle presumably came to me by way of @ajk44, some time ago. Thanks, Alison! The problem, given here, is to find the equations of two lines that complete a square, given: Two of the lines are $y=ax+b$ and $y=ax+c$ One of the vertices is at $(0,b)$. The example given

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