# The Flying Colours Maths Blog: Latest posts

## 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

## A “creative” integral

An interesting “creative” integral pointed my way by the marvellous @DavidKButlerUoA: Find $\int {\frac{1}{(1+x^2)^2}} \dx$ There are “proper” ways to do this - in his tweet, David shows a clever way to do it by parts, and suggests a trig substitution as an alternative. I want to show a third

## Ask Uncle Colin: An Arcsine Inversion

Dear Uncle Colin, I’m trying to invert $y = \arcsin\br{x^2 + 4x - 5}$, but I get stuck at $x^2 + 4x = \sin(y) + 5$. Any pointers? - All Routine Calculations Should Invert Naturally Hi, ARCSIN, and thanks for your message! Just before I dive into this, I have

## Fibonacci parity

On a recent1 episode of Wrong, But Useful, Dave noted that 33 of the first 100 Fibonacci numbers were even, 333 of the first 1000, and so on. My reaction wasn’t quite as it should have been: I said something like “well, yes, obviously”. While there’s a not-too-difficult reason behind

## Ask Uncle Colin: A Horrible-looking Limit

Dear Uncle Colin, I need to work out $\lim_{x\to\infty} \left\{ \left[ \frac{ 2^{1/x} + 8^{1/x} }{2} \right]^x \right\}$. The textbook says it’s four, but my engineering friends all get different answers. How would you work it out? - Giving Engineers Real Maths Analysis Is Naive Hi, GERMAIN, and thanks for

## Emergency Ask Uncle Colin: What happens now?

Dear Uncle Colin, I’m meant to be sitting my A-levels this summer and there’s an awful disease going around. I don’t know if school will be open next week or even if the exams will go ahead. I don’t know what that means for university. I’m scared. What do I

## The Mathematical Ninja and an Irrational Power

“The square root of two… I don’t even know how to say this. The square root of two to the square root of threeth power?” “$\sqrt{2}^{\sqrt{3}}$?” said the Mathematical Ninja. “I wouldn’t bother saying it, I’d just write it down.” “But what does it mean? I mean, I can just