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.
An interesting one: A triangle is right-angled at O with its legs parallel to the axes. Its hypotenuse passes through the point with coordinates $(a,b)$; what is the shortest possible hypotenuse? It’s interesting (to me) because there are several different approaches; all of them sort of …
Dear Uncle Colin, How would you prove that $7^{n+1} + 8 ^{2n-1}$ is a multiple of 57 for integer $n ≥ 1$? I’m Not Doing Unnecessary Calculations To Investigate Our Numbers Hi, INDUCTION, and thanks for your message! Well not with that attitude, you aren’t. For convenience’s sake, …
Out for a run, I listened to A Podcast Of Unnecessary Detail talking about amicable numbers, and I wondered if I could use generating functions to sum the factors of an integer. I know, now I have $1 + x$ problems. While GF underlie this, I’m not going to use them explicitly here. If I wanted …
Given that $p = \frac{3(x^2+1)}{2x-1}$, show that $p^2 -3(p+3)\ge 0$ provided $x$ is real. Well, this doesn’t look very nice. Let’s try and attack it with the sledgehammer first and then see if there are any shortcuts. $p^2$ is $\frac{9(x^4 + 2x^2 + 1)}{(2x-1)^2}$, and $3(p+3)$… well, …
Hello, Sensei! Greetings. I gather you’re here about that quartic. Indeed. $x^4 -2x^3 - x^2 +2x+1$. You seemed to think that I wouldn’t spot it was a perfect square? Oh! Did I? Gosh, that doesn’t sound like the sort of thing I would say. Are you sure you’re not confusing me… …
One of @sparksmaths’s favourite puzzles is: Two cars are driving along the motorway, one at 70mph and the other at 100mph. As they’re beside each other, they both spot an obstruction ahead and slam on the brakes. The slower car just barely stops before hitting the obstruction. Assuming …
I state this precisely as it was stated to me: $\sin(a+7) = \cos(b-10)$ In the equation above, $A$ and $B$ are constants. What is the value of $A+B$? I mean, let’s leave aside for a moment the capitalisation – although it’s a signal that this might not be the best-thought-out …
This is a proof that I’ve seen, not one I’ve come up with; I don’t have a reference, but I imagine it’s relatively well-known. It’s not the usual proof, but I like this one because it doesn’t rely on an infinite regress. The following works for 2, but also (with …
Dear Uncle Colin, Given that $\sin(x)\cos(x) = \frac{2}{5}$, how can I work out $\tan(x)$? - Tangent Ratio, I Guess Hi, TRIG, and thanks for your message! I think this calls for some problem solving out loud! I see two immediate approaches. Approach 1: double angles Whenever I see the product of …
So, there I was, happily working out that the square root of 5,100 is very close to $70 + \sqrt{2}$… OK, OK, OK, big aside here. Why is that? I’ll tell you why that is. It’s due to the binomial expansion, in a slightly surprising way: $\left( 5000 + x\right)^{1/2} \approx …
Dear Uncle Colin, Why does the difference of two squares work? - Doubtful Of Truth Status Hi, DOTS, and thanks for your message! I have several ways of convincing yourself that $(a+b)(a-b)$ is the same thing as $a^2 - b^2$. Just multiply it out This is the “obvious” way: if you multiply …
Dear Uncle Colin, I want to show that $(a^{60} - b^{30})ab$ is a multiple of 77 for all integers $a$ and $b$. Where do I even begin? Factoring Expression, Reasonable Methods Aren’t Trivial Hi, FERMAT, and thanks for your message! I would start by thinking about the possible values of $a^{60}$ …
A nice puzzle from reddit: Evaluate $\sqrt{\sqrt{2025}-\sqrt{2024}}$ The suggestion is that this is the sort of thing that’ll be in Olympiad papers in a couple of years. Fair enough. This isn’t quite a real-time solution, but it’s roughly how I thought about it and tackled it. …
Today’s problem: $\frac{a}{b} = \frac{2}{3}$ $a^b = b^a$ Find $b-a$. I’m just going to straight-up answer this below the line. My first step would be to separate the $a$ and $b$ in the second equation, to get $a^{1/a} = b^{1/b}$. I was tempted by logs for a moment, but we don’t …
You have 20 spaces, which you want to fill with numbers, in order. You will be given 20 random numbers between 0 and 999 (inclusive), each of which you must place into a space as soon as you see it. If you can place all 20 while keeping them in order, you win. What’s your best strategy? How …
Dear Uncle Colin How would you prove that the area under the curve $y=\sin(x)$ from $x = 0$ to $x=\pi$ was exactly 2? ‘Cause I Realised Calculus Lacked Explanation Hi, CIRCLE, and thanks for your message! The standard way is just to know that the integral1 of $\sin(x)$ is $-\cos(x)$, plug in …
Via reddit, a challenge to solve: $y = y’ + y’’ + y’’’ + \dots$ Once you’ve stopped running away and hiding, I’ll show you the solution they suggested. The trick is to notice that $y’ = y’’ + y’’’ + y^{(4)} + \dots$, …
In the course of solving a puzzle, I had cause to assert that $\pi <\sqrt{10}$. I mean, that’s just true: I know that $\sqrt{10} \approx 3.16$ and $\pi \approx 3.14$; I also know that $\pi < \frac{22}{7}$ and that $\left(\frac{22}{7}\right)^2 < 10$. But these beg the question. How …
A post on Hacker News descended into an argument in the comments1 about whether random(random()) and random() * random() gave different distributions. Now, I know better than to wade into a Hacker News discussion unless it’s about mental arithmetic and can name-drop Colin Wright. However, this …
Dear Uncle Colin, As a challenge, I need to work out $1024 - 512 + 256 - \dots + 1$, but I’m getting mixed up on the fractions and signs. Can you help? Colin, Help! Answer Largely Lacking for Evaluating Negative Geometric Expression Hi, CHALLENGE, and thanks for your message! I’ve got …
A trick I learned from Colin Wright: Take the first nine powers of 2 and place them in lexical order; then put a decimal point after the first digit. OK, Colin, here you go: 1.28 1.6 2.56 3.2 5.12 6.4 These numbers are approximately $10^{0.1}$, $10^{0.2}$, etc., up to $10^{0.9}$. Huh! Isn’t …
Dear Uncle Colin, I have to work out 17.5% of 354 in my head. How would you go about it? Various Arithmetical Tricks Hi, VAT, and thanks for your message! The traditional method is to take 10% of your base number (35.4), then add on half (17.7) and half again (8.85) to get 61.95. However, …
On reddit, somebody asked: why does $\frac{7}{3} + \frac{7}{4} = \frac{7}{3}\times\frac{7}{4}$? My first thought: that’s a neat variation of $2+2=2\times 2$. My second: when does that hold true? It’s not too tricky to solve algebraically: $x + y = xy$ $x = xy - y$ $x = y(x-1)$ $y …
Dear Uncle Colin, I’m writing a handbook for a game in which a character’s block percentage is a function of the number of defence points they have. I know the following values: | Defence points | Block percentage | |80| 11.7| |280| 31.8| |480| 44.4| |680| 53.1| |880| 59.4| |1080| 64.2| …
Dear Uncle Colin, I’m told that $x \sin^3(\theta) + y \cos^3(\theta) =\sin(\theta)\cos(\theta)$ and that $x\sin(\theta) = y\cos(\theta)$. I need to work out $x^2+y^2$. How would you approach it? - Simply Impossible, Need Explanation Hi, SINE, and thanks for your message! I certainly hope it …
Say, following on from that thing about Laplace transforms, it reminded me that I typically wave my hands at the ODE characteristic equation trick. You know the one. You have something like $2y’’ + 5y’ + 3y = 0$ and you magically turn it into $2\lambda^2 + 5\lambda + 3=0$, …
The quadratic expression $a^2 - b^2$ factorises as $(a-b)(a+b)$. Similarly, the expression $a^2 + b^2$ factorises over the complex numbers as $(a -bi)(a+bi)$. And while I’d always sort-of-known that $a^3 - b^3 = (a-b)(a^2 + ab + b^2)$, I hadn’t quite in my head made it $(a-b)(a-\omega …
Dear Uncle Colin, I need to factorise $x^3 - 4x^2 - 60x + 288$. What’s your best approach? - Cubics Are Really Dangerous And Nasty Hi, CARDAN, and thanks for your message! That’s an awkward one: 288 has a lot of factors! My best approach is to differentiate first and find turning …
“That’s going to be a quick one,” I thought. It did not turn out that way. An email from friend and hero Rob Eastaway, asking if I can run a Monte Carlo simulation for him. The problem: Imagine you’re playing a game like Snakes and Ladders but without snakes or ladders. Each …
What’s this? A format innovation? Haven’t had one of those in years! I think it’s adequately explained by the title. Let’s go! 1 If we integrate $\int \frac{1}{1-x}\dx$, we get $-\ln |1-x|+C$. Where did the minus come from? It comes from the chain rule, or integration by …