# Author: Colin

## A Trigonometric Puzzle

A puzzle that came to me via @realityminus3, who credits it to @manuelcj89: $\sin(A) + \sin(B) + \sin(C) = 0$ $\cos(A) + \cos(B) + \cos(C) = 0$ Find $\cos(A-B)$. There’s something pretty about that puzzle. Interestingly, my approach differed substantially from all of my Trusted And Respected Friends’. Spoilers below

## Wrong, But Useful, Episode 74

It’s time for the @BigMathsJam Wrong, But Useful! @stecks (Katie Steckles): Brouwer’s Fixed Point Theorem: “they said it’s a theorem, so I’ve got to believe it.” Mentions @jamesgrime. @christianp (Christian Lawson-Perfect): ordering cards to generate a fractal sequence. @peterrowlett (Peter Rowlett): transforming numbers problems into graphs. Mentions @alexcorner and @wtgowers.

## Ask Uncle Colin: A proof

Dear Uncle Colin, I have to prove that, if $a > b \ge 2$, then $ab > a + b$. I can see it must be true, but I can’t prove it! Any ideas? Questioning Everything Done Hi, QED, and thanks for your message! There are several ways to go

## Dictionary of Mathematical Eponymy: The Lutz-Nagell Theorem

One of the reasons I’m writing the Dictionary of Mathematical Eponymy is to introduce myself to new ideas, and to mathematicians I didn’t know about. To things I wish I knew more about. Elliptic curves are pretty high on that list. What is the Lutz-Nagell theorem? It’s sometimes - reasonably,

## Ask Uncle Colin: Greatest Common Divisor (with Polynomials)

Dear Uncle Colin, I have to find the greatest common divisor of $x^4 - 5x^3 + 8x^2 - 10x + 12$ and $x^4 + x^2 - 2$. How do I go about that? - Extremely Unhappy, Clueless Learner In Distress Hi, EUCLID, and thanks for your message! Before we try

## A Triangle In A Square

In a currently-recent (but by the time you read this, long in the past) Chalkdust1, @cshearer42 gave a puzzle that caught my eye. One of the things I love about Catriona’s puzzles is that you usually get two-for-the-price-of-one: there’s “getting the right answer”, which is not usually hard, and there’s

Dear Uncle Colin, I’m told that $f(x) = \frac{5x-7}{(x-1)(x-2)}, x\ne 1, x\ne 2$, and need to express it in partial fractions. My usual method would be to write it as $\frac{A}{x-1} + \frac{B}{x-2}$, multiply by $(x-1)(x-2)$ and substitute $x=1$ and $x=2$ to find $A$ and $B$ - but the definition

## Constructing the square root of 6

On Twitter, @RuedigerSimpson pointed me at an episode of My Favourite Theorem in which @FawnPNguyen mentioned a method for constructing $\sqrt{7}$: draw a circle of radius 4 construct a perpendicular to the radius at a distance of 3 from the centre the distance between the base of the perpendicular and

## Ask Uncle Colin: An Exponential Limit

Dear Uncle Colin, I need to work out the limit of $\frac{2^{3x} - 1}{3^{2x}-1}$ as $x \to 0$, and I don’t have any ideas at all. Do you? - Fractions Rotten, Exponents Generally Excellent Hi, FREGE, and thanks for your message! There are a couple of ways to approach this:

## A Harmonic Conundrum

This one came from user_1312 on reddit with a heading “This is a bit tricky… Enjoy!”. What else can we do but solve it? Let $m$ and $n$ be positive numbers such that $\frac{m}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{101}$. Prove that $m-n$ is a multiple