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 read that when cars are driving at 70mph on the motorway, they take up more space than when they travel more slowly (because you need to leave a longer safe gap between them). What’s the most efficient speed for motorway travel if you want to get as many cars past a given point …
“A ninety-seventh.” The student scratched her head. “I’d call that 0.01.” A moment more’s thought. “0.0103? Probably good enough.” For the Mathematical Ninja, this was about as good as could be expected. They sighed all the same and wrote down: $0. \dot 01\, 03\, 09\, 27\, 83\, 50\, 51 \\ 54\, 63\, …
In this month’s Wrong, But Useful, we’re joined by @televisionduck, who is TD Dang in real life. We discuss: Chalkdust Issue 091 Fun spring cover with Harris spiral, Horoscope is back!, New academic webpage checklist (c.f. Colin’s old webpage, @standupmaths interview, top ten regulars, etc. Write …
Dear Uncle Colin, I have the equation of a curve, $\frac{2x+3y}{x^2 + y^2} = 9$. If I differentiate implicitly using the quotient rule, I get $\diff{y}{x} = \frac{2(x^2 + 3xy - y^2)}{3x^2 - 4xy - 3y^2}$. If I rearrange first to make it $2x + 3y = 9\left(x^2 + y^2\right)$, I find $\diff{y}{x} = …
“Here’s a quick one,” suggested a fellow tutor. “Prove that $2^{50} < 3^{33}$.” Easy, I thought: but I knew better than to say it aloud. First approach “I know that $9 > 8$,” I said, checking on my fingers. “So if $2^3 < 3^2$, then $2^{150} < 3^{100}$ and $2^{50} < …
Dear Uncle Colin, How can I tell whether $\frac{221}{391}$ and $\frac{403}{713}$ are equivalent? - Calculator Answer Not Considered Enough, LOL Hi, CANCEL, and thanks for your message! There’s a naive way to do it and a clever way. Let’s do it naively The naive way is to see whether $\frac{221}{391} …
In a recent MathsJam Shout, courtesy of Bristol MathsJam, we were given a situation, which I paraphrase: Cards bearing the letters A to E are shuffled and placed face-down on the table. You predict which of the cards bears which letter (You make all of your guesses before anything is revealed, and …
Dear Uncle Colin, I’m trying to solve $2\cos(3x) = -\sqrt{2}$, for $0 \leq x \lt 2\pi$, but the answers I find are outside the specified interval, and obviously I miss the ones that are in the interval. How would you tackle this? - Like A Puzzle, Like A Cosine Equation Hi, LAPLACE, and thanks for …
What it is Every so often, one comes across a teacher who is Properly Evil. I’ll spare names here, but I have a clear, strong memory of being introduced to the Collatz conjecture on a school trip. “Take a number, let’s say 3. If it’s odd, you treble it and add 1.” “Ten.” “And if it’s even, you halve …
Dear Uncle Colin, I’m trying to organise a tournament involving seven teams and two pitches. The following conditions must hold: Each team plays four games No pair of teams meets more than once Each team must play at most one pair of back-to-back matches How would you solve this? Bit Of Hard …
This is based on a puzzle I heard from @colinthemathmo, who wrote it up here; he heard it from @DavidB52s, and there the trail goes cold. The Mathematical Ninja lay awake, toes itching. This generally meant that a mission was in the offing. Awake or dreaming? Unclear. But the thought had implanted …
Dear Uncle Colin, How would I work out $\sqrt{\ln(100!)}$ in my head? - Some Tricks I’d Really Like In Number Games Hi, STIRLING, and thanks for your message! I don’t know how you’d do it, but I know how the Mathematical Ninja would! Stirling’s Approximation1 says that $\ln(n!) \approx n \ln(n) - n …
In class, a student asked to work through a question: Let $f(x) = \frac{5(x-1)}{(x+1)(x-4)} - \frac{3}{x-4}$. (a) Show that $f(x)$ can be written as $\frac{2}{x+1}$. (b)Hence find $f^{-1}(x)$, stating its domain. The answer they gave was outrageous1. Part (a) Part (a) was fine: combine it all into a …
Dear Uncle Colin, I’m struggling to make any headway with this: find all integers $n$ such that $5 \times 2^n + 1$ is square. Any ideas? Lousy Expression Being Equalto Square Gives Undue Exasperation Hi, LEBESGUE, and thanks for your message! Every mathematician should have a Bag Of Tricks – things …
In this month’s episode of Wrong, But Useful, we’re joined by @DrSmokyFurby and his handler, Belgin Seymenoglu. Apologies for the poor audio quality on this call. Dave’s fault, obviously1 . We discuss: The Talkdust podcast (via Adam Atkinson): Life insurance Superpermutations: new record for n …
I had a fascinating conversation on Twitter the other day about, I suppose, different modes of solving a problem. Here’s where it started: I intended it as a throwaway comment, but it got some interesting responses. @colinthemathmo (Colin Wright) pointed out: … which, as Colin’s messages are wont to …
Dear Uncle Colin, If I didn’t have a calculator and wanted to know the decimal expansion of $\sqrt{2}$, how would I be best to go about it? Roots As Decimals - Irrational Constant At Length Hi, RADICAL, and thanks for your message! There are several options for finding $\sqrt{2}$ as a decimal …
Stefan Banach was one of the early 20th century’s most important mathematicians - if you’re at all interested in popular maths, you’ll have heard of the Banach-Tarski paradox; if you’ve done any serious linera algebra, you’ll know about Banach spaces; if you’ve read Cracking Mathematics (available …
Dear Uncle Colin, I have to find the points $A$ and $B$ on the curve $x^2 + y^2 - xy =84$ where the gradient of the tangent is $\frac{1}{3}$. I find four possible points, but the mark scheme only lists two. Where have I gone wrong? I’ve Miscounted Points Like I Can’t Infer Tangents Hi, …
Every so often, I see a tweet so marvellous I can’t believe it’s true. Then I bookmark it and forget about it for months, until I don’t know what to write next. An example is @robjlow’s message from June: a) Isn’t that lovely? and b) Hang on, is it really true? …
Dear Uncle Colin, I’m told that $2^a \equiv 9 \pmod{11}$. How do I find $a$? - Powers And Stuff, Calculating And Learning Hi, PASCAL, and thanks for your message! As so often, there are (at least) two reasonable ways to tackle this: a brute force way and an elegant way. Brutally The numbers 2 …
Sometimes, someone dies and you think “it’s a pity their time came.” And sometimes, someone dies and you think “oh no! We needed them.” Hans Rosling (for me) was in the second camp: someone using maths for social good, someone combining graphic design, storytelling and numbers to make the world a …
A Hanging Rope Dear Uncle Colin, I’m designing a small cathedral and have an 80-metre long rope I want to hang between two vertical poles. The poles are both 50 metres high, and I want the lowest point on the rope to be 20 metres above the ground. How far apart should I place the poles? …
In this month’s installment of Wrong, But Useful, Dave and I are joined by @honeypisquared, who is Lucy Rycroft-Smith in real life. We discuss: Mathematical board games, including The Mind Camel Cup Qwinto Number of the podcast: Lucy doesn’t like numbers so we don’t have one. Does your collection of …
I’ve been doing some work on STEP recently - maths exams used mainly for entrance at Cambridge and Warwick, who want some way to differentiate between very good A-level candidates. When I was in Year 13, I had an interview - in fact, two interviews - at Cambridge; at one of them, I overheard …
Dear Uncle Colin, I keep forgetting how to integrate $\sec(x)$ and $\cosec(x)$. Do you have any tips? - Literally Nothing Memorable Or Distinctive Hi, LNMOD, and thanks for your message! Integrating $\sec(x)$ and $\cosec(x)$ relies on a trick, and one the average mathematician probably …
For 2019, I’m trying an experiment: every couple of weeks, writing a post about a mathematical object that a) I don’t know much about and b) is named after somebody. These posts are a trial run - let me know how you find them! The chief use of the Ackermann function, these days, is to …
Dear Uncle Colin, I have the simultaneous equations $3x^2 - 3y = 0$ and $3y^2 - 3x = 0$. I’ve worked out that $x^2 = y$ and $y^2 = x$, but then I’m stuck! - My Expertise Relatedto1 Simultaneous Equations? Not Nearly Enough! Hi, MERSENNE, and thanks for your message! There are a couple of …
Because I’m insufferably vain, I have a search running in my Twitter client for the words “The Maths Behind”, in case someone mentions my book (which is, of course, available wherever good books are sold). On the minus side, it rarely is; on the plus side, the search occasionally …
Dear Uncle Colin, I have been given the series $\frac{1}{2} + \frac{1}{3} + \frac{1}{8} + \frac{1}{30} + \frac{1}{144} + …$, which appears to have a general term of $\frac{1}{k! + (k+1)!}$ - but I can’t see how to sum that! Any ideas? - Series Underpin Maths! Hi, SUM, and thanks for …