# April, 2020

## Ask Uncle Colin: Rotating vectors

Dear Uncle Colin, I need to find a unit vector in the xy-plane that makes an angle of 45 degrees with the vector $3\bi + 4\bj$. How would you do that? - Don’t Enjoy Maths Of Integer Vectors Rotating Enough Hi, DEMOIVRE, and thanks for your message! I can think

## Some puzzles from Cav

A couple of puzzles that came my way via @srcav today: Cav’s solutions to this one are here; mine are below the line further down. Interesting angle puzzle https://t.co/UN13XwwY3o pic.twitter.com/NyaQL0H7wE — Cav (@srcav) July 8, 2019 And to this one, here Have a go yourself before you read on! I’ve

## Ask Uncle Colin: Two trig identities

Dear Uncle Colin These two trig questions are getting me frustrated! What do you recommend? Prove $\frac{\tan(2x) + \cot(x)}{\tan(2x) - \tan(x)} \equiv \cot^2(x)$ Prove $\frac{1 + \sin(2x)}{1+\cos(2x)} = \frac{1}{2}\left(1+\tan(x)\right)^2$ - I Don’t Like Equations Hi, IDLE, and thanks for your message! The great temptation here is to send you a

## Powers and remainders

Over on Reddit, a couple of “last digit” puzzles crossed my path, and I thought I’d share the tricks I used, as much for my reference as anything else. 1) Show that the last digit of $6^k$ is 6, for any positive integer $k$. There’s a standard way to prove

## Wrong, But Useful: Episode 78

In episode 78 of Wrong, But Useful, we're joined by @c0mplexnumber, who is Clarissa Grandi in real life. This month, we discuss: Clarissa's Artful maths books, available via Tarquin - the activity book and the teacher's guide Number of the podcast: $\phi$ (and 3D maths) @anniek_p's #mathartchallenge Aperiodical’s big math-off

Dear Uncle Colin, I need to find $35^{-1} \pmod {234}$, but I’m not getting the right answer. Can you help me?1 - It’s Not Very Easy Resolving Such Expressions Hi, INVERSE, and thanks for your message! We’re looking for a number $x$ such that $35x = 234n + 1$, for

## Vectors, lines and laziness

What makes a mathematician a mathematician? Scientific studies say one thing above anything else: laziness1 We will go to extraordinary lengths to avoid doing any proper work. For example, I had a situation: I had two points - call them $P$ and $Q$ - and a line with the equation

Dear Uncle Colin, I noticed that the incircle of a 3-4-5 triangle has a radius of 1, and for a 5-12-13 triangle, it’s 2. Is it always an integer in a Pythagorean triangle? Having Elegant Radius Or Not? Hi, HERON, and thanks for your message! It turns out that yes,

## Dictionary of Mathematical Eponymy: Mrs Perkins’ Quilt

“Lightly grease a 20x20cm baking tin with butter and spoon in the mixture. Press into the corners with the back of a spoon so the mixture is flat and score into 12 squares.” - BBC Good Food flapjack recipe by user nicolajlittle Hang on a minute - I thought, mid-baking.

Dear Uncle Colin, I need to find four consecutive numbers such that the first is a multiple of 5, the second a multiple of 7, the third a multiple of 9 and the fourth a multiple of 11. Can you find such a number? - Summing Up Needs Zero Intelligence