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.
Factorising a quadratic? It’s nice when it comes off, but there’s a lot of guesswork, and no guarantee it even factorises. Completing the square? Who has time for all that algebra? And as for the quadratic formula, or your clever calculator methods: honestly, what are you, an engineer? …
Dear Uncle Colin, I need to calculate $\int x^3 (x^3+1) (x^3 + 2)^{\frac 13} \dx$ and it’s giving me a headache! Can you help? I’ve Blundered Using Parts, Rolled Out Fourier Expansions… Nothing! Hi, IBUPROFEN, and thanks for your message! That’s a bit of a brute, but it can be …
In this month’s installment of Wrong, But Useful, Colin and Dave are joined by mathematical editor and proofreader @samhartburn. We apologise for the sound quality. We’ve done the best we can. Sam enjoys @robeastaway’s Maths On The Go with her primary-school children. Dave plugs …
Another geometry puzzle from @solvemymaths: I enjoyed this one – no solution immediately jumped out at me, and I spend a great deal of time looking smugly at a way over-engineered circle theorems approach I can no longer remember. Let’s label the apex of the triangle P, and the octagons …
Dear Uncle Colin, I’m told that the three terms $a_1 = \log(2)$, $a_2 = \log(2\sin(x)-1)$ and $a_3 = \log(1-y)$ are in arithmetic progression and I need to find the range of possible values for $y$. I don’t really know where to start! - Logarithmic Arithmetic Progression Lacks A Clear …
When RITANGLE advises you to use technology to answer a question, you know it’s going to get messy. So, with some trepidation, here goes: (As usual, everything below the line may contain spoilers.) It’s easy enough to do this in Geogebra - but somehow a little bit unsatisfactory to move …
Dear Uncle Colin, I was wondering: given a quadratic function with real coefficients, what complex arguments lead to real answers? - Researching Equations And Lines Hi, REAL, and thanks for your message! This turns out to be simpler than I expected: if you have a quadratic $f(z) = az^2 + bz + c$ …
By way of @ajk_44 at NRICH, a belter of a puzzle: You have 27 small cubes - three each of nine distinct colours. Can you arrange them in a cube so that each colour appears once on each face? (Alison has created a Geogebra widget for you to play with, should you want to do that before reading on to …
Dear Uncle Colin, I’m told that a rectangular box has a surface area of 64cm2, and I have to find the maximum possible volume. How would I do that? - Can Uncle Bring Obviousness Into Differentiation? Hi, CUBOID, and thanks for your message - I certainly hope I can! We have two relevant …
If you flick to the back of an old A-level formula sheets, you might spot a list of random digits like this one from an MEI book: Why on earth would you want random numbers? There are all sorts of reasons one might want a set of random numbers: perhaps you have a large set of data and want to pick …
Dear Uncle Colin, I was asked to complete the square on $f(x) = 2x^2 + 13x + 20$. I started by halving everything, which makes it cleaner, but the solution manual disagrees. What gives? - Have Always Loathed Functions Hi, HALF, and thanks for your message! Dividing a quadratic by the $x^2$ …
In this month’s episode of Wrong But Useful (#53), brought to you by the power of Lemsip and a day in bed, Colin and Dave are joined by Special Guest Co-Hosts @sean_jamshidi and #NikiWithoutTwitter, who are Sean Jamshidi and Niki Kalaydzhieva from Chalkdust Magazine1 in real life. Niki is an …
“Is Hamburg that much further north than London?” I furrowed my brow. Hamburg, to the best of my knowledge, is not that much further north than London. But here it was, written in stone (on the side of Durlston Castle in Swanage.) (I’ve transcribed the sign at the bottom of this …
Dear Uncle Colin, How would you calculate $\cos(72º)$ by hand? - Pointless Historical Inquiry Hi, PHI, and thanks for your message. There seems to be an awful lot of degree use around at the moment, and I’m not very happy about it. But still, in the spirit of answering what the question asks, …
Last time, we looked at an ‘impossible’ question from a New Zealand exam, which was (of course) nothing of the sort. The second question highlighted as a brute was this one: (Full exam is here.) But, you see, I look at that and think… part (iv) isn’t straightforward on its …
Dear Uncle Colin, In a recent contest, I was told that $a$, $b$ and $c$ were real numbers such that $a-7b+8c=4$ and $8a + 4b -c = 7$. I had to find $a^2 - b^2 + c^2$ and couldn’t see a way in. Can you? - Puzzle Lacks Answer - Nothing Evident! Hello, PLANE, and thanks for your message! There …
To-may-to / tomato; potato / po-tah-to; impossible exam / underprepared students. This time it’s the hapless Kiwis who are making Downfall parody videos and complaining that their practice papers hadn’t prepared them for stuff on the syllabus. Never mind; the formidable @solvemymaths has …
Dear Uncle Colin, I’m working on finding horizontal asymptotes for rational functions. I normally do that by division, but my teacher wants me to do it by rearranging - and I don’t really know what’s going on there! Can you explain? - Horizontal Asymptotes Leaving Me Outwitted …
The Mathematical Ninja sniffed. “$4\sin(15º)$? Degrees? In my classroom?” “Uh uh sorry, sensei, I mean $4\sin\br{\piby{12}}$, obviously, I was just reading from the textmmmff.” “Don’t eat it all at once. Now, $4\sin\br{\piby{12}}$ is an interesting one. You know …
Dear Uncle Colin, I have to work out $851 \div 37$ without pen, paper or calculator. How would you do it? - Simple Mental Arithmetic Looks Easy Hi, SMALE, and thanks for your message! I have three ways to tackle it. Brute force and estimation It’s fairly obvious that the answer is between 20 …
In this month’s episode of Wrong, But Useful, we are joined by maths communication superhero @stecks, who is Katie Steckles in real life. We discuss: Katie’s numerous and various activities as a maths freelancer, including maths busking and being a mathematician-in-residence at the …
A question that frequently comes up in the insalubrious sort of place a mathematician might hang around is, what is that value of $0^0$. We generally sigh and answer that the same way every time. It was nice, then, to see someone ask a more fundamental one: what is $0 \div 0$? The short answer is, …
Dear Uncle Colin, How come $0.3^{0.3} > 0.4^{0.4}$? - Puzzling Over It, Some Surprisingly Ordered Numbers Hi, POISSON, and thank you for your message! It is a bit surprising, isn’t it? You would expect $x^x$ to increase everywhere, at first glance. Why it doesn’t We can see that this …
“Cooking,” said my friend Liz in a recent Facebook post, “is one of the activities where maths is most useful in my everyday life.” She added this picture: I’ve got several reasons for wanting to share this. 1. It’s pretty much a model answer Imagine you’re …
Dear Uncle Colin, I need to find the largest solution to $e^x + \sin(x)=0$ and I don’t really know where to start. Any ideas? - Some Options Look Virtually Equal Hi, SOLVE, and thanks for your message! That is something we in the trade call ’not a very nice equation to solve’. …
So do we. We are aware of, and unhappy about, the predominance of heterosexual, cisgendered white men in mathematics, and we want to play a part in addressing that. Since starting the Special Guest Co-Host feature on Wrong, But Useful, we have had eleven SGCHs, of whom eight are women and three men. …
Dear Uncle Colin, I keep being told my work needs to be tidier, but I’m not really sure how to go about that. Do you have any suggestions? - My Exercises Skew Scruffy Hello, MESS, and thanks for your message. I totally feel your pain. In honesty, this is like asking Bob Dylan for advice on how …
“It is like a cave that seemed to be absolutely sparkling with jewels. Maths glittered like a Christmas tree at me.” - Amanda Spielman, at the Education Select Committee, October 30th 2017 What an image. What an image. Caves are dark, and cold, and slightly frightening, but buried deep …
Dear Uncle Colin, Does the Law Of Large Numbers contradict the fact that the Gambler’s Fallacy is a fallacy? - Got A Monkey Backing Liverpool In November’s Games Hi, GAMBLING, and thanks for your message! The short answer is: no. I imagine you want a bit more detail than that, though! …
The summer of 2003 was a glorious one in Fife: blazing sunshine, warm-but-not-too-warm, a lovely fresh breeze blowing in off of the North Sea. At least, I’m told it was glorious. I was stuck in room 226, preparing two versions of my PhD thesis as my supervisor had decided a copy with narrower …