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’m told that 19,683 is the cube of an integer. How would I figure out which one? - Not A Problem I Expected, Really Hi, NAPIER, and thanks for your message! There are several approaches here, and I’m sure The Mathematical Ninja has more still. First, the …
The following is a famous puzzle set by Ben Ames Williams in the Saturday Evening Post in 1926 – I’ve borrowed the telling from a New Scientist puzzle book I’m editing. It’s in a footnote. I’m sure it’s fine. There are five sailors shipwrecked, and during the night, …
On Twitter, @robotmaths asked: Proof question: discussing with Year 12 the best method to prove $(x+y)^2 ≥ 4xy$ They can’t start with this statement, so what is your best way in to the question? This is the sort of question where what’s useful to a student approaching the end of Year 12 …
Dear Uncle Colin, How would you work out $e^{-0.56}$ without a calculator? - Tell All Your Logic Or Reasoning Hi, TAYLOR, and thanks for your message! I think most people would suggest a Taylor1 series here – either centred around 0 (which converges fairly fast) or around some sensible nearby number …
Dear Uncle Colin, I’m told that $v + 3w$ is perpendicular to $7v - 5w$ and that $v-4w$ is perpendicular to $7v - 2w$, and I have to find the angle between $v$ and $w$. I know it’s something to do with the dot product, but I’ve not been able to make it shake out. Can you help? - …
Dear Uncle Colin, Player 1 rolls a fair die with 200 sides. Player 2 rolls a fair die with 300 sides. What’s the probability that player 1 rolls a higher number than player 2? - Probably A Simple Calculation Applying Logic Hi, PASCAL, and thanks for your message! Let’s ignore for the …
The question looked fairly awful – STEP questions usually do. It was a request to sketch the function: $g(x) = \int_{-1}^{1} \frac{1}{\sqrt{1-2xt + x^2}} dt$, for all real $x$. Have a go, if you’d like! I’ll spoiler it below the line. The first insight The first thing that makes the …
Via @sqrt on Twitter1: $\left(1 + \frac{1}{x}\right)^{x+1} = \left(\frac{8}{7}\right)^7$. Find $x$. Two obvious wrong answers “Let’s make the power 7!23” But if $x=6$ then the bracket is $\frac{7}{6}$. “Let’s make the bracket $\frac{8}{7}$, then!” But if $x = 7$, …
I’m not sure where I got this question from. It asks for the value of the infinite sum: $\frac{3}{1\times 2 \times 3} + \frac{5}{2\times 3 \times 4} + \frac{7}{3\times4\times5} + \dots$ Have a go if you’d like; spoilers are below the line. — The first thing I would do is work out the …
A puzzle I saw via @HAClaphamSixth, but whose originator they don’t know: What integer pairs $(n,k)$ satisfy $n! + 8 = 2^k$? How do you know you have them all? This is not a problem I know the answer to! I shall attempt problem-solving out loud, with spoilers below the line! — Two thoughts …
Dear Uncle Colin, How can I prove that $\frac{\sqrt{6}-\sqrt{2}}{4} = \frac{\sqrt{2 - \sqrt{3}}}{2}$? - Help! I’d Probably Put A $\sin$ Under Something Hi, HIPPAsUS, and thanks for your message! I can see several ways to approach this. Very directly The left hand side is clearly positive; if …
Via @cmonMattTHINK on Twitter: In case that doesn’t show up: The function $f(x) = x + \sqrt{x}$ is one-to-one. Find its inverse. That’s a nice challenge. My approach – which is clearly a spoiler – is below the line. — I tend to start function-inverses by renaming my variables for …
All-round good egg @samholloway asked, ages ago: Seven of my friends on Facebook have birthdays today! How many friends do I have? This is a question to which Sam could probably find the answer, and possibly I could too; I don’t really spend a lot of time on Facebook and can’t remember …
I’m writing this in October 2022. While I was writing my post, the Chancellor of the Exchequer resigned from his. I’m not going to redo the analysis if there’s another election, ok? In several conversations with @NotAdric about elections and proportional representation, which are …
There’s a famous, and famously tedious, story about the number 1729 and how it became known as a taxicab number. You can look it up if you’re that interested. What’s interesting to me is the numbers themselves: numbers that are the sum of two cubes in two (or more) different ways: …
Dear Uncle Colin, I’m trying to solve the system of equations $2x^2 + y^2 = 18$ and $xy = 4$. I’ve substituted for $y$ and ended up with a quartic. How do I solve that? - Regarding A Polynomial, Having Solutions Or Not? Hi, RAPHSON, and thanks for your message! Your approach seems pretty …
Via @MathsRH on twitter: In case you can’t see that, it’s a challenge to come up with a function that matches a given picture, which I’ll describe in a moment; if you can see it, working out the key points is part of the problem-solving, so the description might constitute a …
Not that sort of conundrum, obviously. Like everyone else on the planet, and presumably others, I got sucked into the craze of Wordle and its spinoffs during the pandemic. The one I turn to first? Countle. It’s the number games from Countdown, about which I’ve written before here and at …
Regular readers – and Twitter followers – will know my standard response to fake maths expressions such as $6 \div 2(1+2)$: The correct answer is “write the bloody thing properly”. It’s deliberately ambiguous; just because you can write something in mathematical notation …
Another in an occasional series of “problem-solving aloud” – this one is from Loren C Larson’s Problem-solving Through Problems. 4.2.22 If $a, b, c,$ and $d$ are positive integers, show that 30 divides $a^{4b+d} - a^{4c+d}$. I raised my eyebrows at this; it surprises me that this …
Dear Uncle Colin I’m told that $x$ satisfies $x^3 - x-1=0$ , and I need to find the value of $\sqrt[3]{3x^2 - 4x} +x \sqrt[4]{2x^2 +3x + 2}$. Obviously, I could solve the cubic and plug the answer in, but I don’t think that’s in the spirit of it. - Can Anyone Reduce Down Algebraic …
Once in a while, it does me good to tackle a problem “out loud” (or at least ad-lib the typing), just so I can talk through my thought process unfiltered. Today I’m going to look at question 102 from @ridermeister’s excellent collection of university entrance style problems: …
One of my favourite factorising tricks1 is the sum-of-two-squares-in-two-ways one. For example, $221 = 10^2 + 11^2 = 5^2 + 14^2$. (Now, if you’re sharp, you might spot that $221 = 15^2 - 2^2$, so it’s $13 \times 17$. But we’re doing this another way, ok?) Instead, we’re going …
A twitter thread here asked whether there was an explanation for why the discriminant of a quadratic is unchanged when you reverse the order of the coefficients. (I suspect Evariste Galois might have some ideas, but I still haven’t read up on that.) Instead, here’s my argument: Suppose …
Chris Sangwin pointed me at the following photo: … with the comment that that was an awful lot of significant figures. It’s hard to fathom why a car park would give its prices to the nearest semimillieurocent, honestly, but presumably there’s a reason behind it. So, the first …
Dear Uncle Colin, I want to jump over the Grand Canyon on a motorbike. How fast would I need to go? Extreme Velocity, Extreme Landing Hi, EVEL, and thanks for your message! According to Wikipedia, the Grand Canyon varies in width from four miles to 18 miles. I’d hesitate to recommend you go …
“Which is larger: $\left(\frac{6}{5}\right)^{\sqrt{3}}$ or $\left( \frac{5}{4}\right)^{\sqrt{2}}$? Bulgarian Mathematical Olympiad 2018, question 31 The immediate reaction of most mathematics seeing this question is revulsion. Irrational powers? what foolishness is this? Digging deeper - or …
It always struck me as cool, in A-level questions, when you’d see two completely different triangles with the same hypotenuse – for example, a $(4,7,\sqrt{65})$ triangle and a $(1,8,\sqrt{65})$ triangle. But it’s not just cool – it’s also useful. If you can write a …
This is a guest post by James Grime, all-round maths superstar, with whom it is impossible to have a conversation without someone demanding a selfie or an autograph. He disagrees with my telling of the story of the Truchet debacle and I’m happy to let him set the record straight. If you want …
What is the Wallis Sieve? Well, since $\sin(x) = x - \frac{1}{3}x^3 + \frac{1}{5}x^5- \dots$, it clearly has a factor of $x$ you can divide out: $S(x) = \frac{\sin(x)}{x} = 1 - \frac{1}{3}x^2 + \frac{1}{5}x^4 - \dots$. Where does that expression have zeros? Wherever $x$ is a non-zero multiple of …