# April, 2016

Dear Uncle Colin, I’ve come across a seemingly simple question I can’t tackle: solve $x^2 + 2x \ge 2$. I tried factorising to get $x(x+2) \ge 2$, which has the roots 0 and -2, but the book says the answer is $x < -1-\sqrt{3}$ or $x > -1 + \sqrt{3}$.

## A STEP expansion

A STEP question (1999 STEP II, Q4) asks: By considering the expansions in powers of $x$ of both sides of the identity $(1+x)^n (1+x)^n \equiv (1+x)^{2n}$ show that: $\sum_{s=0}^{n} \left( \nCr{n}{s} \right)^2 = \left( \nCr{2n}{n} \right)$, where $\nCr{n}{s} = \frac{n!}{s!(n-s)!}$. By considering similar identities, or otherwise, show also that: (i)

Dear Uncle Colin, I have a triangle with sides 4.35cm, 8cm and 12cm; the angle opposite the 4.35cm side is 10º1 and need to find the largest angle. I know how to work this out in two ways: I can use the cosine rule with the three sides, which gives

## Attack of the Mathematical Zombies: $1=2$

“One equals two” growled the mass of zombies in the distance. “One equals two.” The first put down the shotgun. “I’ve got this one,” he said, picking up the megaphone. “If you’re sure,” said the second. “I’M SURE.” The second covered his ears. “SORRY. I mean, sorry.” The first redirected

## Ask Uncle Colin: A Complex Battle

Dear Uncle Colin, I’m supposed to solve $(1+i)^N = 16$ for $N$, and I don’t know where to start! — Don’t Even Mention Other Imaginary Variations — Reality’s Enough Hello, DEMOIVRE, there are a couple of ways to attack this. The simplest way (I think) is to convert the problem

## A MathsJam Masterclass: A Circle That Won’t Behave

Somewhere deep in the recesses of my email folder lurks a puzzle that looks simple enough, but that several of my so-inclined friends haven’t found easy: A circle of radius $r$, has centre $C\ (0,r)$. A tangent to the circle touches the axes at $A\ (9,0)$ and $B\ (0, 2r+3)$.

## Ask Uncle Colin: A trigonometric integral

Dear Uncle Colin, I’m trying to find a definite integral: $\int_0^\pi \sin(kx) \sin(mx) \dx$, where $m$ and $k$ are positive integers and the answer needs to be simplified as far as possible. I’ve wound up with $\left[\frac{ (k+m) \sin((k-m) \pi) – (k-m)\sin((k+m)\pi) }{2(k-m)(k+m)}\right]$, but it’s been marked wrong. — Flat

## The Mathematical Ninja and The Slinky Coincidence

“No, no, wait!” said the student. “Look!” “8.000 000 072 9,” said the Mathematical Ninja. “Isn’t that $\frac{987,654,321}{123,456,789}$? What do you think this is, some sort of a game?” “It has all the hallmarks of…” “I’ll hallmark you in a minute!” said the Mathematical Ninja. Seconds later, the students arms

## Wrong, But Useful: Episode 34

In this month’s podcast, @reflectivemaths and I discuss: Colin’s book being available to buy Number of the podcast: Catalan’s constant, which is about 0.915 965 (defined as $\frac{1}{1} – \frac{1}{9} + \frac{1}{25} – \frac{1}{49} + … + \frac{1}{(2n+1)^2} – \frac{1}{(2n+3)^2} + …$). Not known whether it’s rational. Used in combinatorics