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.
Put four numbers on the corners of a square. I’ll pick 1, 3, 14 and 39, in that order: Between each pair, write their unsigned difference: Now you have a new square: (2, 11, 15, 38). I’m not going to keep drawing them; it’s a pain to typeset. But we repeat the process, getting …
Dear Uncle Colin, I have a triangle with edge lengths of 9, 7 and 6 units. I need to find the area of the circumcircle. How do I do that? Help! Euclidean Reasoning’s Off! Hi, HERO, and thanks for your message! I have a slightly cloggy method, and a possibly nicer one. Get your clogs on …
This is a resources post about my 2022 MathsJam talk, Two Puzzles. Puzzle statements (1). For what values of $n$ can $n$ one-ohm resistors be arranged to have an equivalent resistance of one ohm? (No funny business: current must pass through each of the resistors.) (2) For what values of $n$ can a …
You might remember James Grime from a couple posts about logarithms many years ago. Or from, you know, Numberphile, or dozens of other places online or in person. He’s a very nice chap, and I think the only person I’ve ever seen recognised in public and asked for a selfie. Anyway. James, …
Dear Uncle Colin, I know that the imaginary part of the number $\frac{x-2i}{1+i}$ is 1, and I need to find $z$, given that $x$ is real. What do I do? - Any Really Good Answers Now, Dear? Hi, ARGAND, and thanks for your message! I would start by realising the denominator: multiply top and bottom by …
What is a Vernier scale? The idea behind a Vernier scale is that using two different scales on a measuring device, you can improve your resolution. For example, standard calipers might have a single scale – you place the object you’re measuring between two reference plates; the scale is …
Dear Uncle Colin, Apparently $\sqrt[3]{2+\sqrt{5}} + \sqrt[3]{2-\sqrt{5}}$ is an integer. Really? Can A Root Define A Number? Oo! Hi, CARDANO, and thanks for your message! Let’s eyeball it first: $2 + \sqrt{5}$ is about 4.2, which has a cube root somewhere about 1.6, I’d have thought; …
Some time ago, I tackled a sock puzzle: I’m happy with my original solutions, but wanted to explore another way of doing it. It’s going to end up in a generalised Pell’s equation, so we’re looking at another example of great depth coming from a relatively simple puzzle. As a …
Dear Uncle Colin, Is there any benefit to writing the equation of a line in 3D space in the form $\br{\bb{r} - \bb{r_0}}\times\bb{d} = \bb{0}$? Doesn’t Everyone See Cartesian Are Remarkably Terse Equations, Sir? Hi, DESCARTES, and thanks for your message1! The answer is yes, the vector product …
This one came to me by way of @blatherwick_sam: Now, my … presumably Turkish? … is not even rudimentary. It might be that yours is better, but maths is the universal language, right? In any case, have a crack at it, and I’ll share how I solved it below the line where – …
What is a Ueda Attractor? It looks like this. Pretty, huh? It’s a phase solution of the Duffing equation $\ddot x + 0.05 \dot x + x^3 = 7.5\cos(t)$. It is a strange attractor – it is not periodic, but it is dense (and nearby trajectories converge into it). Why is it interesting? I know I said …
Dear Uncle Colin, The first term of a geometric progression is 1 and the second is $2\cos(x)$, with $0 \lt x \le \piby 2$. Find the set of values for which the progression converges. Read And Typed It Out Hi, RATIO, and thanks for your message! It looks a bit like you’re asking me to do your …
For the benefit of anyone who prefers text: Let $n$ be a positive integer not divisible by 2 or 5. Prove that there exists a multiple of $n$, all of whose digits are 1s. I’m going to try solving this “live” – although it’s a bit of a cheat, because I’ve got a good …
Dear Uncle Colin, How do we know the binomial coefficients are always integer? I know we can do it by a counting of sets argument, but is there an arithmetic way? Bloomin’ Isaac Newton Obviously Made It All Labyrinthine Hi, BINOMIAL, and thanks for your message! To restate the question: if you …
What is Torricelli’s Law? If you pierce a container of fluid a distance $h$ below its surface level, the fluid (ignoring viscosity) will flow through the hole at a speed of $v = \sqrt{2gh}$. This is the same as the speed at which a particle (ignoring air resistance) would attain after falling …
A problem (of the classical train variety) came my way, and I thought it would be good to share my thoughts about how I solved it on here as well as on Twitter: Read through the question to get a sense of it. What am I looking for? The (expected) speed of the train What would help with that? The …
Dear Uncle Colin, I need to work out values of $a$, $b$ and $c$ that satisfy: $abc=14$ $a+5b+5c=-12$ $5bc+ab+ac=-3$ How would you do it? - Some Nasty Equations? Ask… Kolin1? Hello, SNEAK, and thanks for your message! There’s a standard and dull way to do this, and a more exciting way. …
A puzzle from @cmonMattTHINK: You may wish to have a go at it. I am going to try to solve it “out loud”. First thoughts My main question is “with or without replacement”? I infer without, but I’m not certain about that. I’m going to roll with it and see what comes …
Dear Uncle Colin, My classmate claims he can work out $\int_0^\pi \cos^{2020}(x)\dx$. I think that’s far too many cosines to integrate. What do you think? Trigonometric Ratio, Exponentiated Excessively - That’s Hardly Realistic Enumeration, Eh? Hi, TREETHREE, and thanks for your message! …
What is Sang’s Logarithmic Method? If you’re building a bridge – for example, over a canal – you might think “I know! I’ll use an arch!” This is a great solution, so long as the thing that needs to go over the canal is perpendicular to it. For most of the history of …
Dear Uncle Colin, I have a question from my childhood. I know the answer, but not how to get it. Can you explain? Here it is: Oxford and Cambridge Universities decided to join forces and send a combined rugby union squad on a tour. Enough players were chosen from both universities to ensure the …
There’s a standard proof that $\sqrt{2}$ is irrational: Suppose $\sqrt{2} = \frac{p}{q}$, where $p$ and $q$ are coprime1; Then $q\sqrt{2} = p$, so $2q^2 = p^2$. Hence, $p^2$ is even – which can only be the case if $p$ is even. So, letting $p = 2r$, $2q^2 = 4r^2$, or $q^2 = 2r^2$. That …
Dear Uncle Colin, I’ve been told that $(ax+b)(bx+a) \equiv 10x^2 + cx + 10$, with $a$ and $b$ positive integers, and I need to find the possible values for $c$. How?! - This Ridiculous Identity Puzzle Looks Easy Hi, TRIPLE, and thanks for your message! The first thing I would do would be to …
I’m not much of a cyclist. I have a bike, and I occasionally take it out for a longish ride, but generally I struggle to tell the difference between a peloton and a catamaran. However, a question from @tzed250 on Twitter caught my attention: is there a way to rate a climb? The problem If …
I don’t know that anyone apart from me calls these Rice pentagonal tilings, but I think everyone ought to. What is a Rice pentagonal tiling? In 1975, Martin Gardner wrote a Scientific American column about the kinds of polygons that can tile the plane. He reported an assertion from a 1967 …
Dear Uncle Colin I’ve been asked to find the sum of the infinite series $\frac{3}{2} + \frac{5}{4} + \frac{7}{8} + \dots + \frac{2n+1}{2^n} + \dots$. How would you go about it? Some Expertise Required In Evaluating Sum Hi, SERIES, and thanks for your message! I can see a couple of ways of …
A lovely puzzle via @Sheena2907: For those who need it: Let $f$ be a function such that $f(1) = 1$ and $f(m+n) = f(m) + f(n) + mn$. Find $f(20)$. As with many of my favourite puzzles, it’s a twofer: first, get the answer; then, dig deeper. And as with pretty much all of the puzzles I discuss …
Dear Uncle Colin, I am trying to work out a probability using a binomial distribution and I’m getting a very different answer when I use a normal distribution: Binomially: $X \sim B(380, 0.3)$ $P(X \ge 152) = 1 - P(X \le 151)$ $\dots = 1 - 0.9999789688$ $\dots \approx 0.00002103$ Normally: $Y …
There are, for possibly obvious reasons, not many mathematicians whose names begin with Q. All the same, I think I’ve found one with a reasonably accessible theorem to his name. What is Qvist’s Theorem? Qvist’s theorem states that, given any oval in a finite projective plane of …
“The end,” as @alison_kiddle is fond of saying, “is not the end.” When you’ve finished solving a problem, it’s almost always going back and figuring out what extra value you can get from it. Good questions to ask include: “How do I feel now?” …