# June, 2018

## Ask Uncle Colin: An Unclear Inequality

Dear Uncle Colin, I solved $(x+1)(x-2)(x+3)>0$ by saying there were three possibilities, $x+1>0$, $x-2>0$ or $x+3>0$. The middle one gives $x>2$ and that's the strictest, so that was my answer - but apparently it's wrong. Why is that? - Logical Expressions Seem Silly Hi, LESS, and thank you for your

## A Varignon Vector Masterclass

I recently listened to @mrhonner's episode of @myfavethm, in which he cited Varignon's Theorem as his favourite. What's Varignon's Theorem when it's at home? It states that, if you draw any quadrilateral, then connect the midpoints of adjacent sides, you get a parallelogram. Don't believe it? Try Mark's nifty geometry

## Ask Uncle Colin: Argentina vs Nigeria

Dear Uncle Colin, This is the fifth time in the last six World Cups that Nigeria have been drawn against Argentina in the group stages. What are the odds?! - Coincidences At FIFA? Unlikely! Hi, CAFU, and thanks for your message! I'm going to make some simplifying assumptions for this

## $\cos(72º)$, revisited again: De Moivre’s Theorem

In previous articles, I've looked at how to find $\cos(72º)$ using some nasty algebra and some comparatively nice geometry. In this one, inspired by @ImMisterAl, I try some nicer - although quite literally complex - geometry. De Moivre's Theorem I'm going to assume you're ok with complex numbers. If you're

## Ask Uncle Colin: A Polling Percentage

Dear Uncle Colin I have a percentages problem: I'm told that in an election, 95.74% of the electorate voted for the winning side. What is the minimum possible size of the electorate? - Percentages Often Lack Logic Hi, POLL, and thank you for your message! There are two possible answers

## Wrong, But Useful: Episode 57

Owing to a spotty connection, the sound quality is a bit patchy on this one. Apologies. In this month's installment, we are joined by Belinda Keir. We discuss: Belinda convenes the Sydney MathsJam and is on the committee for the Celebration of Mind What makes a mathematician? Why Belinda is

## $\cos(72º)$, revisited: a geometric method

Some months ago, I wrote about a method for finding $\cos(72º)$, or $\cos\br{\frac{2\pi}{5}}$ in proper units. Almost immediately, the good people of Twitter and Facebook - notably @ImMisterAl (Al) and @BuryMathsTutor (Mark)- suggested other ways of doing it. Let's start with Mark's method, which he dissects in his book GCSE

## Ask Uncle Colin: Something powerful

Dear Uncle Colin, I'm trying to solve $3^{2x+1} - 28\times 3^x + 9 =0$. I've split up the first term into $3 \times 3^{2x}$ but then I'm stuck! Any suggestions? Likely Overthinking Gettable Sum Hi, LOGS, and thanks for your message! You've made a really good start there. The piece

## Square wheels on a round(ish) floor

The ever-challenging Adam Atkinson, having noticed my attention to the "impossible" New Zealand exams, pointed me at a tricky question from an Italian exam which asked students to verify that, to give a smooth ride on a bike with square wheels (of side length 2), the height of the floor