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.
You know how things escalate on Twitter sometimes? Somebody makes an off-hand comment wondering whether a number is prime and suddenly you’re neck deep in number theory? This is the story of how you might factorise 842,909 on paper. In fact, it’s the second part of the story; we join it …
Dear Uncle Colin, How would you integrate $e^x \sin(x)$ (with respect to $x$, obviously)? - Difficult Integral, Just Kan’t See The Right Answer Hi, DIJKSTRA, and thanks for your message! As seems to be the way recently, there are several ways to approach this. My favourite way One of the best …
In this month’s episode of Wrong, But Useful, we’re joined by Special Guest Cohost @macaronique, who is Angela Brett in real life. Angela recites her poem They Might Not Be Giants We discuss joy in teaching and learning, and crosswords Number of the podcast: 44, the number of derangements of five …
There is a theorem that states: if a number can be written as the sum of two squares in two different ways, it is composite. Because of Twitter, I became interested in factorising $n=842,909$. Can this be written as the sum of two squares1? How - without cheating and using a computer - could we …
Dear Uncle Colin, What’s the correct method to find the general solution of $y’’+4y’+4y=(3+x)e^{-2x}$? I’ve got the complementary function just fine (it’s $y=Ae^{-2x}+Bxe^{-2x}$), but I’m going in circles with the particular integral. - Differential …
The best science writers (as far as I’m concerned, at least) are the ones who make you feel like you’re sitting down for a coffee with a smart friend, excited about a thing they know about, a thing they’ve found out, or a thing they’ve just put together. I have never (yet) …
Dear Uncle Colin, How do people do decimal calculations like $80 \times 0.15$ in their heads? It seems impossible. - Doesn’t Everyone Seem Clever, Answering Readily These Evil Sums Hi, DESCARTES, and thank you for your message! There are several possible strategies for a question like this - …
“What are you doing here?” “Never mind that, sensei, how on earth did they catch you?” “… I don’t want to talk about it. In any case, I was only doing 5% more than the speed of light.” “Yeah, that poses more questions than it answers, in honesty.” The Powerpoint slide had the car coming to a …
Dear Uncle Colin, According to John D Cook, you can estimate the natural logarithm of a number $x$ by working out $\log_2(x)-\log_{10}(x)$. It seems to be pretty accurate - but why? Not A Problem - Interesting Exponential Riddle Hi, Napier, and thanks for your message! That is indeed a conundrum: …
In the olden days, I had a proper job1 writing software for customer survey forms. Among other things, our users could ask their customers to rate how likely it was that they would recommend the product to their friends. Typically, this would be on a ten-point scale, from extremely unlikely to …
Dear Uncle Colin, When I’m 8 metres away from a flagpole, the angle of elevation to its top is exactly 40º, according to this angle-measurey-majigger I have here. What will it be when I’m only five metres away? The Hypotenuse Eludes One; Does One Look Into Trigonometric Expressions? Hi, …
In Episode 59 of Wrong, But Useful, we’re joined by @el_timbre, who is Emma Bell in real life. We discuss: Emma’s Roots column in Chalkdust1 Number of the podcast: 64, because it’s pleasing, and Dave sings the 64 Zoo Lane theme, which is less so. Emma’s hexastix construction and the maths behind it. …
I recently had a flurry of correspondence with translators of The Maths Behind (available wherever etc., but also soon in Swedish and Korean): embarrassingly, they had caught several mistakes in the book. These things happen; we try to put them right and move on. However, it got me wondering: can I …
Dear Uncle Colin, I have a trig identity I can’t prove! I have to show that $\frac{\cos(x)}{1-\sin(x)} = \tan(x) + \sec(x)$. Strangely Excited Comment About Non-Euclidean Trigonometry. Hi, SECANT, and thanks for your message! This is a slightly sneaky one, but definitely a good one to …
You know how I often bang on about how ‘impossible’ exams are really nothing of the sort? Well, just for a change, I’m going to bang on about how sometimes exam boards get it wrong. I’m looking at the 2014 Edexcel FP2 paper (the normal one, not the (R) one or the IAL one), in …
Dear Uncle Colin, I was asked to work out $\tan\br{\theta + \piby 2}$, but the formula failed because $\tan\br{\piby 2}$ is undefined. Is there another way? - Lost Inna Mess, Infinite Trigonometry Hi, LIMIT, and thanks for your message! In fact, there are several ways to approach it! Basic geometry …
An excellent puzzle I heard from @panlepan (I paraphrase, as I’ve lost the tweet): When you move the final digit of 142857 to the front, you get 714285, which is five times as large. What is the smallest positive integer that is doubled when the last digit moves to the front? There are two …
Dear Uncle Colin, Apparently, the volume of a tetrahedron with three edges given by the vectors $\vec{AB}$, $\vec{AC}$ and $\vec{AD}$, is $\frac{1}{6} \left| \vec{AB} \cdot \br{\vec{AC}\times\vec{AD}} \right|$. Where does that come from? - Very Obviously Lacklustre Understanding of My Exam Hi, …
A nice prompt from @shahlock, some time ago: Stand back, everyone: I’m going to apply Bayes’s Theorem. A prior Let’s assume that, before we knew anything about the teams, we could have believed equally well in every possible value for $p_A$, the probability of team A winning. If …
Dear Uncle Colin, I’m trying to solve $2\cos(3x)-3\sin(3x)=-1$ (for $0\le \theta \lt 90º$) but I keep getting stuck and/or confused! What do you recommend? - Losing Angles, Getting Ridiculous Answers, Nasty Geometric Equation Hi, LAGRANGE, and thank you for your message! There are a couple of …
Some time ago, I had a message from someone who - somewhat oddly - wanted to find a centre of rotation (with an unknown angle) without constructing any bisectors. (Obviously, if it was a right-angle rotation, they could use the set-square trick; if it was a half-turn, the centre of rotation is …
Dear Uncle Colin, I’m stuck on a trigonometry question: find $\cos\br{\frac{1}{2}\arcsin\br{\frac{15}{17}}}$. Any bright ideas? - Any Rules Calculating Some Inverse Notation? Hi, ARCSIN, and thanks for your message! That’s a nasty one! Let’s start by thinking of a triangle with an …
In this episode of Wrong, But Useful, we are joined by freelance mathematician @becky_k_warren, formerly of NRICH Becky likes sharing maths with people who “don’t like maths” and the #beingmathematical twitter chat Number of the podcast: 157, which is the middle of a sexy prime …
So there I was, merrily teaching the factor and remainder theorems, and my student asked me one of my favourite questions: “I accept that the method works, but why does it?” (I like that kind of question because it makes me think on my feet in class, and that makes me feel alive!) …
Dear Uncle Colin, I had to find the $n$th term of a quadratic sequence (1, 6, 17, 34, 57). I remember my teacher saying something about a table, but I couldn’t figure it out. Can you help? Struggles Expressing Quadratics Using Educator’s Notation - Concrete Explanation? Hi, SEQUENCE, and …
When the redoubtable @cuttheknotmath (Alexander Bogomolny) poses the following question: … you know there must be Something Up. Surely (the naive reader thinks) the one with two heads out of three is the one with a probability of two heads in three? But equally surely (thinks the reader who …
Dear Uncle Colin, Please can you settle an argument? I say, if you toss a coin three times, the probability of getting all heads is one in four, because the only possibilities are HHH, HHT, HTT and TTT. My friend says it’s one in eight, being $\frac{1}{2}\times \frac{1}{2} \times \frac{1}{2}$. …
“Your calculator has broken, leaving you with only the buttons for $\sin$, $\cos$, $\tan$ and their inverses, the equals button and the 0 that starts on the screen. Show that you can still produce any positive rational number.” When this showed up on Reddit, I knew I was in for a) a …
Dear Uncle Colin, I solved $(x+1)(x-2)(x+3)>0$ by saying there were three possibilities, $x+1>0$, $x-2>0$ or $x+3>0$. The middle one gives $x>2$ and that’s the strictest, so that was my answer - but apparently it’s wrong. Why is that? - Logical Expressions Seem Silly Hi, …
I recently listened to @mrhonner’s episode of @myfavethm, in which he cited Varignon’s Theorem as his favourite. What’s Varignon’s Theorem when it’s at home? It states that, if you draw any quadrilateral, then connect the midpoints of adjacent sides, you get a …