# The Flying Colours Maths Blog: Latest posts

## Eye to Eye

A nice observation from Futility Closet: Draw two circles of any size and bracket them with tangents, as shown. The chords in blue will always be equal. I’m hardly going to let that pass by without a proof, now, am I? Spoilers below the line. My proof Definitions Let the

## Ask Uncle Colin: A Circle In 3D

Dear Uncle Colin, I know a 3D circle passes through $A(4,-4,5)$, $B(0,4,1)$ and $C(0,0,5)$ and I need to find its centre and radius. I could do it in 2D, but I’m a bit stuck here! Can’t Interpret Radius/Centre, Looked Everywhere Hi, CIRCLE, and thanks for your message! There are, as

## A puzzle from BoingBoing

I get a lot of my problems-to-solve from Reddit, since if someone’s posted it there, there are probably thousands of people with the same difficulty. This one isn’t from Reddit, but from @frauenfelder, one of the high-heidyins at BoingBoing. Out of the 25 homework problems, there was one that she

## Ask Uncle Colin: The Area In Between

Dear Uncle Colin, I have the graphs of $y=\sin(x)$ and $y=\cos(x)$ for $0 < x < 2\pi$. They cross in two places, and I need to find the area enclosed. I’ve figured out that they cross at $\piby 4$ and $\frac{5}{4}\pi$, but after that I’m stuck! - Probably A Simple

## Dictionary of Mathematical Eponymy: Trémaux’s Algorithm

I recently had the chance to employ this one, but didn’t manage to: it turns out that four three-to-six-year-olds are not especially interested in putting down markers and following rules, they just want to run around the maize maze and say “maize maze” and make “amazing” jokes1 What is Tremaux’s

## Ask Uncle Colin: A Sum With An Unknown

Dear Uncle Colin, My textbook gives me an arithmetic sequence that starts $3 + 8 + 13 + \dots$, and asks me to find where the sum is 1575. I’ve got it down to $\frac{-1 \pm \sqrt{63,001}}{10}$, but I don’t know how to work out that square root without a

## The Mathematical Ninja and Logs Base 2

The student’s shoulder twitched slightly as he said “So I need to work out $\log_2(10)$…” and the crash of the cane against the table reminded him that the calculator was off-limits. “I think you can estimate that yourself,” said the Mathematical Ninja. “Uh… ok. There’s a change of base formula,

## Ask Uncle Colin: Factorising

Dear Uncle Colin, How would you go about factorising $6x^2 - xy - y^2 + 7x - y + 2$? - Argh! Getting Nowhere. Expression Simplification Impossible. Hi, AGNESI, and thanks for your message! I have, historically, not been a fan of these. However, I’ve recently come across a method

## Summing with Generating Functions

A nice challenge puzzle via Reddit: Find $\sum_{n=1}^{\infty} \frac{n2^n}{(n+2)!}$ There was a video attached to it that I didn’t watch, something about telescoping sums, but the moment I saw this, I thought: generating functions! Why would I think such a thing? The thing that jumped out at me was the

## Ask Uncle Colin: Why does the $ac$ method work?

Dear Uncle Colin, A while back, you shared an easy way to factorise nasty quadratics. Why does it work? Dutifully Indulging Students’ Curiousity & Reasoning In Maths. I’m Not A Nasty Teacher! Hi, DISCRIMINANT, and thanks for your message! Let’s start by recapping the method in the post, go through