The Flying Colours Maths blog has been running posts twice weekly since 2012, covering maths from the basics to… well, the most advanced stuff I have a clue about.
Here they all are, sorted by date. Some day, other ways to filter them will be possible.
Dear Uncle Colin, I want to find the oblique asymptote of $f(x) = \frac{2x^2 + 3x -9}{x+2}$, but I don’t want to learn polynomial division. What do you recommend? Genuine Reason: It’s Difficult Hi, GRID, and thanks for your message! I would recommend learning to divide polynomials but …
The original D/L model started by assuming that the number of runs that can still be scored (called $Z$), for a given number of overs remaining (called $u$) and wickets lost (called $w$), takes the following exponential decay relationship: $Z(u,w)=Z_{0}(w)\left({1-e^{-b(w)u}}\right)$ where the …
Dear Uncle Colin If you hang a square up by one corner and cut it in the plane perpendicular to the vertical diagonal, you get a regular hexagon. What’s the corresponding result for a tesseract? - Have You Proved Everything, Really? Hi, HYPER, and thanks for your message! Let’s start in …
What is the Lute of Pythagoras? Draw a regular pentagon (lightly). Draw its diagonals (darkly) The outline of the dark shape is a concave, equilateral decagon. Pick one of the concave1 corners and its two neighbours; the edges linking these can be two edges of a regular pentagon. Draw this regular …
If you follow me on Twitter, you might have noticed that I’m a fan of legal blogger David Allen Green – not because I always agree with him, but because even when I don’t, he sets out a clear and compelling case for why I should. He’s famed for writing in one-sentence …
Dear Uncle Colin, Someone gave me the puzzle: $ff(n) = 3n$ for all real $n$, $f(n)$ is a positive integer for all positive integer $n$ $f(n+1) > f(n)$ for all positive integer $n$ What is $f(13)$? Any ideas? Functions? Fun? No. Hi, FFN, This is a fun puzzle! Here’s how I tackled it: $ff(1) …
I say it’s legendary – I hadn’t heard of it before stumbling across this Numberphile video: That said, it’s a question with a good story behind it: it’s from the 1988 International Mathematics Olympiad (which immediately says “this is a hard question”); it …
I’m currently reading (on @drmackiver’s recommendation) Across the Board by John J Watkins, a study of the mathematical side of some chessboard-related puzzles. Among them is using knight’s tours to generate magic squares, which is the kind of useless but pleasing trick I am …
Dear Uncle Colin, My textbook claims that the RMS voltage of a sine wave is $\sqrt{\frac{1}{2}}$ because between $x=0$ and $x=\pi$, the signed area between the curves $y=\sin(x)$ and $y=\sqrt{2}$ is 0 – but I checked and it isn’t. What’s going on? Really A Disappointing …
On Twitter, @whitehughes posted a nice complex numbers problem: Have a go yourself, if you’d like to; below the line are spoilers. As Susan says, there are several ways to tackle this. Off the top of my head, I can see: An algebraic method, finding the equation of the line BC and a circle of …
Dear Uncle Colin, I have a quartic expression and I want to know whether it can be expressed as a perfect square. How would you do it? Quite Ugly Algebra – Rooting Turns Into Clean Squares? Hi, QUARTIC, and thanks for your message! Like quadratics, quartics have a discriminant. Unfortunately, …
A slight change of direction this month: this entry is named after a non-mathematician who is the subject of the thing rather than the discoverer. You might have heard of the subject. What is Nixon’s diamond? Nixon’s diamond is a logical scenario that seems to be a contradiction. The …
Here’s @nathanday314 with an unusual puzzle: What’s the problem? I’m currently working on making my problem-solving techniques more explicit, and in that spirit: my first step is to write down exactly what the problem is: We have two triangles. We’re told (at least …
Dear Uncle Colin, When I express $\frac{x^2}{(x-1)(x-4)^2}$ in partial fractions, why do I need to use three separate fractions? I accept that that’s how it works, I just want to know why! Pupils Absolutely Raging: “Two Is Adequate, Like!” Hi, PARTIAL, and thanks for your message! …
A lovely puzzle from @jsiehler on Mathstodon: A problem I wrote for a competition some time ago (but decided not to use): Let $a_n$ denote the nearest integer to $\log_{168}(927^n)$, and let $b_n = a_{n-1} - a_{n}$. The first few terms in the sequence are $2, 1, 1, 2, 1, 1\dots$, which appears to be …
I’m writing this during the delayed 2020 Olympics, and the natural question of trying to compare athletic performances across disciplines came up. Of course it would. It’s a natural question. And it reminded me that I had a bookmark about that. You can compare athletic performances …
Dear Uncle Colin, What’s the second derivative of $\frac{x^2}{x^2-4}$, and how would you work it out? - Calculus Obviously Hasn’t Evolved Nicely Hi, COHEN, and thanks for your message! There’s an ugly way and a neat way. I’m going to call your expression $y$, so that it has a …
My client was concerned. Not for the usual reason a client might be concerned - in fact, I’d just come up with a key insight that fixed our biggest problem. He was concerned that I’d come up with the insight while out for a walk with my family. He was worried I was thinking about the …
Dear Uncle Colin, I need to solve $\frac{1}{(x-10)^2} = 4$ and all the explanations I’ve seen online are much more difficult than anything we’ve done in class. How would you approach it? - Fractions Are Rightfully Execrated, Yes? Hi, FAREY, and thanks for your message! I think the …
Every so often, I get a misguided message from someone claiming that the value we use for $\pi$ is wrong. (I try not to dunk on such messages, tempting as it is. But really, if you think everyone who’s looked at it, from Archimedes to Newton, Euler, Ramanujan and Gosper, has got the same wrong …
A programming note: after a decade of two or more blog posts a week, I’m taking my foot off of the writing pedal a bit this year; the blog will be running one post a week (on Mondays) for the foreseeable future, likely alternating Ask Uncle Colin with regular posts, with Dictionary posts on …
Dear Uncle Colin, I’m fine at remembering the formulas for things like $\sin(A+B)$, but I struggle to come up with the formulas for things like $\sin(A) + \sin(B)$. Any hints? Have I Learnt Badly Equations Regarding Trig? Hi, HILBERT, and thanks for your message! I always used to loathe these …
A friend1, justifiably proudly, shared on Facebook that he’d worked out the number of dots he’d need to draw a hexagon with $n$ dots on each edge. I thought it was a nice puzzle! Before we go anywhere, I’ll clear up what exactly we mean by a hexagon with $n$ dots per side. …
Dear Uncle Colin, How can I get better at the problem-solving side of maths? - Probably Really Obvious But Looks Extremely Mysterious Hi, PROBLEM, and thanks for your message! The generally excellent @drmackiver has a simple process for solving problems: see what other people with similar problems …
Suppose you have a circular cake or pizza that needs to be cut into six pieces and you don’t have a cooking protractor. How could you cut it into – at least roughly – sixths? This is something that I’ve always done by eye, and always messed up. Until very recently, when I …
Dear Uncle Colin, I need to solve $3^{x+2} + 45\br{6^x} - 9\br{2^x}=0$ and I’m completely stuck. How would you tackle it? Producing A Sketch Could Accelerate Learning Hi, PASCAL, and thanks for your message! I think this is a question best approached by simplifying a little at a time. My first …
A puzzle that came to me by way of Barney Maunder-Taylor: A long sheet of plastic is turned into a gutter by folding up two sides. What fraction of the width should be bent up and through what angle to maximise the amount of water it can hold? (Unstated, but implied: both sides are the same length …
Dear Uncle Colin, I have to solve $5\sin(x) - 5\cos(x)=2$ for $0 \le x \lt 360º$ – I get four answers, but apparently only two are correct. Any suggestions? Blooming Impeded: Rudimentary Knowledge And Reasoning Hi, BIRKAR, and thanks for your message! This looks tasty! Let’s work through …
I am not, by nature, an origamist. While I’m all for it in principle, I get frustrated at my inability to fold straight lines, and when I do succeed at making something, I never quite know what to do with it afterwards. But the maths behind it is fascinating. What is Kawasaki’s Theorem? …
Dear Uncle Colin, I tried to work out $\int \frac{1}{\sqrt{3 - 4x - 4x^2}}\dx$ going via the complex numbers and end up with a different answer compared to the ‘proper’ way – I get an $\arcosh$ instead of an $\arcsin$. Can you sort out this witchcraft? Am Genuinely Not Expecting …