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.
In this episode of Wrong, But Useful1: We’re joined by @ajk_44, who is Alison Kiddle from NRICH in real life. We ask Alison: how long has NRICH been going? How do you tell which problems you’ve covered before? Colin’s number of the podcast is 13,532,396,179 (he mistakenly calls it …
I imagine, if one put one’s mind to it, one could acquire copies of this year’s paper online - however, many schools plan to use it as a mock for next year’s candidates. In view of that, and at the request of my top-secret source, I’m not sharing the actual questions used. …
Dear Uncle Colin, I’ve been asked to solve Chebyshev’s equation using a series expansion: $(1-x)\diffn{2}{y}{x} - x\dydx + p^2 y = 0$ assuming $y=C_0 + C_1 x + C_2 x^2 + …$. I end up with the relation $C_{N+2} = \frac{C_N \left(N^2 -p^2\right)}{(N+2)(N+1)}$, but the given answer …
“Counting is hard. This is what I keep saying.” - @realityminus3 It all stemmed from an arithmetic series problem with a known sum, but an unknown number of terms. As these things are prone to do, it led to a quadratic equation; as those things are prone to do, that led to two possible …
This is a rolling update post for responses to this morning’s post. The Admirable Adam Atkinson has emailed to suggest an answer I hadn’t considered: 100. “I could imagine many programming languages would say 100. You start with the first term, 100. discover that the …
Dear Uncle Colin, I’ve been given $u = (2\sqrt{3} - 2\i)^6$ and been told to express it in polar form. I’ve got as far as $u=54 -2\i^6$, but don’t know where to take it from there! - Not A Problem I’m Expecting to Resolve Hello, NAPIER, and thanks for your message! I fear …
Someone recently asked me where I get enough ideas for blog posts that I can keep up such a ‘prolific’ schedule. (Two posts a week? Prolific? If you say so.) The answer is straightforward: Twitter Reddit One reliable source of interesting stuff is @WWMGT - What Would Martin Gardner …
Dear Uncle Colin, I’m told that $z=i$ is a solution to the complex quadratic $z^2 + wz + (1+i)=0$, and need to find $w$. I’ve tried the quadratic formula and completing the square, but neither of those seem to work! How do I solve it? - Don’t Even Start Contemplating A Robust Trial …
It turns out I was wrong: there is something worse than spurious pseudocontext. It’s pseudocontext so creepy it made me throw up a little bit: Yeah, I know: things were different, 250-odd years ago, but still, ew. I managed to put my disgust to one side for long enough to solve the thing, …
Dear Uncle Colin, I recently had to decompose $\frac{3+4p}{9p^2 - 16}$ into partial fractions, and ended up with $\frac{\frac{25}{8}}{p-\frac{4}{3}} + \frac{\frac{7}{8}}{p-\frac{4}{3}}$. Apparently, that’s wrong, but I don’t see why! -- Drat! Everything Came Out Messy. Perhaps Other …
In this month’s episode of Wrong, But Useful, @reflectivemaths1 and I are joined by consultant and lapsed mathematician @freezingsheep2. We discuss: Mel’s career trajectory into ‘maths-enabled type things that are not actually maths’, although she gets to wave her hands a …
There is a danger, when your book comes plastered in praise from people like Art Benjamin and Ron Graham, that reviewers will hold it to a higher standard than a book that doesn’t. That would be unfair, and I’ll try to avoid that. What it does well This is a book with plenty to …
Dear Uncle Colin, In an answer sheet, they’ve made a leap from $\arctan\left(\frac{\cos(x)+\sin(x)}{\cos(x)-\sin(x)}\right)$ to $x + \frac{\pi}{4}$ and I don’t understand where it’s come from. Can you help? -- Awful Ratio Converted To A Number Hello, ARCTAN, and thank you for your …
Last week, I wrote about the volume and outer surface area of a spherical cap using different methods, both of which gave the volume as $V = \frac{\pi}{3}R^3 (1-\cos(\alpha))^2(2-\cos(\alpha))$ and the surface area as $A_o = 2\pi R^2 (1-\cos(\alpha))$. All very nice; however, one of my most beloved …
Dear Uncle Colin, One of my students recently attempted the following question: “At time $t=0$ particle is projected upwards with a speed of 10.5m/s from a point 10m above the ground. It hits the ground with a speed of 17.5m/s at time $T$. Find $T$.” They used the equation $s = vt - …
What is the volume above a plane, and inside a sphere of radius $r$, such that the radius of the circle where the two intersect is $R \sin(\alpha)$? What is this spherical sector’s curved surface area? I’ve lost the precise wording of the question that drove a small cabal of MathsJammers …
Dear Uncle Colin, I’m trying to sew a traditional football in the form of a truncated icosahedron. If I want a radius of 15cm, how big do the polygons need to be? -- Plugging In Euler Characteristic’s Excessive Hello, PIECE, and thank you for your message! Getting an exact answer to that …
“What are the ch…” “About 11.7%,” said the Mathematical Ninja. “Assuming $X$ is drawn from a Poisson distribution with a mean of 9 and we want the probability that $X=7$.” “That’s a fair assumption, sensei,” pointed out the student, …
Dear Uncle Colin, I have a pair of parametric equations giving $x$ and $y$ each as a function of $t$. I’m happy with the first derivative being $\diff{y}{t} \div \diff{x}{t}$, but I struggle to find the second derivative. How would I do that? - Can’t Handle An Infinitesimal Nuance Hi, …
Eagle-eyed friend of the blog @robjlow spotted an error in Uncle Colin’s last answer. As I’m forever telling my students, making errors is how you learn; Rob has graciously delivered a lesson for us all. Thanks for keeping me honest! Recently, Uncle Colin gave a couple of ways to see …
Dear Uncle Colin, What is $\lim_{x \to \infty} \left\{ \sqrt{x^2 + 3x} - x\right\}$? - Raging Over Obnoxious Terseness Hi, ROOT, and thanks for your very brief question. My approach would be to split up the square root and use either a binomial expansion or completing the square, as follows: …
In this month’s installment of Wrong, But Useful, our special guest co-host is @mathsjem (Jo Morgan in real life) from the indispensable resourceaholic.com. We start by talking about resourceaholic.com and how Jo manages to fit such a punishing blog schedule around being a nearly-full-time …
“Arr, that be a scurvy-lookin’ expression!” said the Mathematical Pirate. “A quartic on the top and a quadratic on the bottom. That Ninja would probably try to factorise and do it all elegant-like.” “Is that not the point?” “When you got something as …
Dear Uncle Colin, Help! My calculator is broken and I need to solve - or at least approximate - $0.1 = \frac{x}{e^x - 1}$! How would you do it? -- Every $x$ Produces Outrageous Numbers, Exploring New Techniques Hi, ExPONENT, and thanks for your message! That’s a bit of a beast, and …
“Sensei! I have a problem!” The Mathematical Ninja nodded. “Bring it on.” “There’s a challenge! Someone has picked a five-digit integer and cubed it to get 6,996,364,932,376. I know it ends with a six, and I could probably get the penultimate digit with a bit of …
After several months of high-intensity development, we’re very happy to announce the launch of Ninja|Alpha. Go ahead! Ask it anything you like.
Dear Uncle Colin, When I differentiate $y=2x^2 + 7x + 2$ and apply the $nx^{n-1}$ rule, why do I only apply it to the $2x^2$ and the $7x$ but not the 2? -- Nervous Over Rules, Mathematically A Liability Hi, NORMAL, and thanks for your message! There are several ways to answer that, but I’ll …
“Isn’t it somewhere around $\phi$?” asked the student, brightly. “That number sure crops up in a lot of places!” The Mathematical Ninja’s eyes narrowed. “Like shells! And body proportions! And arrawk!” Hands dusted. The Mathematical Ninja stood back. …
Dear Uncle Colin, I have an equation $3y, \dydx =x$. When I separate and integrate both sides, I end up with $\frac{3}{2}y^2 = \frac{1}{2}x^2$, which reduces to $y = x\sqrt{\frac{1}{3}}+c$. With the initial condition $y(3) = 11$, I get $y = x\sqrt{\frac{1}{3}}+11-3\sqrt{\frac{1}{3}}$, but apparently …
I’m a big advocate of error logs: notebooks in which students analyse their mistakes. I recommend a three-column approach: in the first, write the question, in the second, what went wrong, and in the last, how to do it correctly. Oddly, that’s the format for this post, too. The question …