# Author: Colin

Dear Uncle Colin, Why does $0! = 1$ and not 0? - Nothing Is Logical Hi, NIL, and thanks for your message! My best explanation for this - by which I mean, the one I can get some people to accept, goes like this: $4! = 4 \times 3 \times Read More ## Dictionary of Mathematical Eponymy: The Quine-McCluskey Algorithm There was a Fields Medallist named Dan Quillen, after whom are named several things in topics I’ve never head of. Other than Quillen, so far as I can tell, the only mathematical eponyms beginning with Q relate to Willard Van Ormine Quine. I know him from Godel, Escher, Bach, where Read More ## 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 Read More ## 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 Read More ## 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 Read More ## 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 Read More ## 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 Read More ## Ask Uncle Colin: An Additive Inverse 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 Read More ## 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,