# August, 2020

## How many people did they ask?

I’m on a bit of a continued fractions jaunt at the moment, as they’re my current “oo! That’s a tool I haven’t played with enough, and that I think might be interesting!” One of their applications (surprisingly to me) is to answer the question “A survey firm says ‘82.43% of

## Ask Uncle Colin: On the Mediant

Dear Uncle Colin, Can you explain why $\frac{a+b}{c + d}$ is between $\frac{a}{c}$ and $\frac{b}{d}$? - Grinding Out Solid Proof Explaining Rationals Hi, GOSPER, and thanks for your message! As per usual, there are several methods to show this. I’m going to assume (since it hasn’t been stated) that $\frac{a}{c}$

## On Continued Fractions

Lately I’ve been playing around with continued fractions - I blame credit @evelynjlamb for pointing me at this post by @johndcook. I’ve done most of my learnin’ from here and from various Wikipedia pages, and I thought I’d revisit some of it for the benefit of others. In fact, I’ll

## Ask Uncle Colin: Powers

Dear Uncle Colin, I need to solve $5^{2x+2} +16\cdot 15^x - 9^{x+1} = 0$ but I’ve hit a dead end! Can you help? - Puzzling Out Wild Exponent Relations Hi, POWER, and thanks for your message! From the working you sent, it looks like you’ve picked a good strategy: you’ve

## 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