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.
This cropped up on Reddit: $I = \int_0^1 \br { \Pi_{r=1}^{10}(x+r)}\br{\sum_{r=1}^{10}\frac{1}{x+r}} \dx$ Show that $I=(a)(b!)$, where $a$ and $b$ are positive integers to be found. Hm. Tricky. I have some ideas, but I’m going to tackle it a bit more brutally at first. What would happen if the …
Dear Uncle Colin, I keep getting tripped up in my mechanics questions because I can never figure out which way the frictional force has to go. Do you have any advice? Mechanics Understanding Hi, MU, and thanks for your message! This is a situation where I have One Big Trick and not much else to say. …
Someone on reddit asked how to show that $2^{15}-1$ was not a prime number, and I suddenly understood something I’d previously just ‘become used to’. In binary, $2^{15}-1$ is 111,111,111,111,111. You can break that down in groups of three (or five) – but it’s fairly …
Dear Uncle Colin, I’ve figured out the graph of $y = e^x\br{2x^2 -5x + 2}$ using algebra and calculus; the second part of the question asks about $y = e^{x^2}\br{2x^4 - 5x^2 + 2}$. I feel like there must be a quicker way than redoing all of that work! Some Heuristic Or Rule To Circumvent …
A question we looked at in class: If $f(x) = \arctan(x) + \arctan\br{\frac{1-x}{1+x}}$, find $f’(x)$. Hence, or otherwise, find a simple expression for $f(x)$. I’m not sure if I like this question, but there’s a good deal of depth to it. Have a go if you want to – I’ll …
Dear Uncle Colin, I’m playing a solo D&D game and want some strategy advice for the final battle, which consists of a series of rounds, alternating between monster attacks and my attacks. In a monster attack, it rolls nine dice and I roll eight; if it rolls more sixes than I do, I lose the …
In class, we tackled a Senior Maths Challenge problem that goes something like this: “Given that $y^3 f(x) = x^3 f(y)$ for all real $x$ and $y$, and that $f(3) \ne 0$, find the value of $\frac{f(20)-f(2)}{f(3)}$. A: 6; B: 20; C: 216; D: 296; E:2023” Have a play with it. Spoilers below …
Dear Uncle Colin, I bought a 750-piece jigsaw from a charity shop and found it had only 740 pieces! If I bought the same jigsaw again, and found it was missing ten random pieces, how likely is it I’d be able to complete the puzzle? Just Imagine Going Shopping As Well Hi, JIGSAW, and thanks for …
What is the Jordan Curve Theorem? The Jordan Curve Theorem is the sort of thing I adore about mathematics. It states (I paraphrase): A 2D loop that doesn’t cross itself has an inside and an outside (and the two are separate). That might strike you as the kind of thing that’s obvious. …
Dear Uncle Colin, Why does $1 - \frac{1}{3} + \frac{1}{5} - \dots = \frac{\pi}{4}$? My Arithmetic Doesn’t Have A Viable Answer Hi, MADHAVA, and thanks for your message! There are several ways to show this, but I think my favourite stems from the Maclaurin series for $\ln(1+x)$. Or rather, in …
An interview question that came my way and spoke to my interests: Your task is to determine which three horses out of a group of 25 horses are the fastest. You may race up to five horses at a time. How many races are required? Have a crack at it yourself, if you’re so inclined. There are …
Dear Uncle Colin, How would you factorise $2x^2 - 8x - 16$? Yonder Expression Evades Hard Algebraic Work Howdy, YEEHAW, and thanks for your message! It turns out that your quadratic there doesn’t factorise: $b^2 - 4ac$ is 192, which isn’t a square. However, there is a way1 to find where it’s equal …
This has been hanging around in my ’to write’ folder for several years. The tl;dr of the shoelace formula is: suppose you have a plane polygon with $n$ vertices the vertices of which (in order) are $(x_i, y_i)$ for $i = 0,1,2\dots,(n-1)$ then the area is $\frac{1}{2}\sum_{i=0}^n x_i …
Dear Uncle Colin, How would I go about finding $a$, $b$ and $c$ in $x^3 + y^3 = (x+y)\br{ax^2 + bxy + cy^2}$? Constants Using Brackets & Its Coefficients Hi, CUBIC, and thanks for your message! There are, of course, multiple approaches; the most obvious way is to tackle it by dividing everything …
Take a recurrence relation, like the way the Fibonacci sequence is defined: $a_n = a_{n-1} + a_{n-2}$, with $a_0 = 0$ and $a_1 = 1$. I’m going to make up an infinite degree polynomial that looks like this: $A(x) = a_0 + a_1 x + a_2 x^2 + \dots + a_k x^k + \dots$ … and figure out another …
Dear Uncle Colin, How would I show that $4y - 3x + 26 = 0$ is a tangent to $(x+4)^2 +(y-3)^2 = 100$? Can I Reconcile Circle/Line Equations? Hi, CIRCLE, and thanks for your message! The simplest way (I think) is to note that if the line is tangent to the circle, solving the equations simultaneously …
What is Itô Calculus? Calculus is all well and good. Some people think it’s a pretty neat idea, and certainly plenty of decent maths and real-world applications rest on it. But it’s a bit… [dons sunglasses]… limited1. In particular, when you’re dealing with stochastic …
Dear Uncle Colin, I was asked to work out the number of ways 11 distinct objects could be distributed between three people such that each had an odd number of objects. I worked that out fine (it’s 44,286) and generalised it to $2n+1$ objects having $\frac{3}{4}\br{9^n-1}$ arrangements – …
In an idle moment, I wondered how many people in the UK1 have an A-level in maths. This strikes me as a great Fermi estimation question – you can come up with any number of answers and bounds on it and probably nobody knows the answer even roughly. (You might be able to get the number of …
Dear Uncle Colin, I have solved $16x^5 - 20x^3 + 5x + 1 = 0$ by letting $x = \cos(\theta)$ and considering $\cos(5\theta)$ – but how do I explain that there are only three roots? My Understanding Limited To Interesting Problems Like Integration; Can’t Interpret Trigonometry Yet Hi, …
A puzzle from @dmarain: As usual, have a crack if you want to. Spoilers below the line. I bet there are dozens of methods here. Approach Number 1 My first approach is to just multiply out the brackets: one multiplied by the second bracket is $(1 -x^2 + x^4 - x^6 + \dots)$, and $x^2$ multiplied by …
Dear Uncle Colin, I have the curve $y = \frac{1}{x}$ and need to rotate it by 45 degrees clockwise. I can’t figure out what curve I get! Any clues? Where Asymptotes Lie, Like I Said Hi, WALLIS, and thanks for your message! I can think of several ways to do this. Let’s look at some! Polar …
I suppose this ought to be a Dictionary of Mathematical Eponymy post, but it isn’t. So there1. The legend is that Richard Feynman, as a child2, was discussing trigonometry with his friend Morrie Jacobs in the Jacobs family leatherworks, when Morrie divulged that $\cos(20^o) \cos(40^o) \cos(80^o) = …
Dear Uncle Colin, I’m told that $x^2 \tan(y) = 9$, for $0<y< \piby 2$. I have to show that $\dydx = \frac{-18x}{x^4+81}$, and that there’s a point of inflection at $x = 27^{1/4}$. Where do I even start!? - I Might Plot Loci… I Can’t, It’s Tricky Hi, IMPLICIT, and …
What is the Haynsworth Inertia Additivity Formula? Start with a Hermitian matrix, $\bb{H}$ – which is one that’s equal to its conjugate transpose $\bb{H^*}$1. Haynsworth defined the inertia of a Hermitian matrix to be a triple, $\text{In}(\bb{H}) = \br{ \pi(\bb{H}), \nu(\bb{H}), …
Dear Uncle Colin, How do I compare two numbers with irrational exponents? I want to know whether $2^{\sqrt{3}}$ is larger or smaller than $3^{\sqrt{2}}$. - How’d You Power A Thing Irrationally? Aaargh! Hi, HYPATIA, and thanks for your message! There’s a sensible way to do this and an …
“Find all of the five-digit numbers that are reversed when multiplied by (a) 9 and (b) 4.” I can’t say I’m a fan of this type of puzzle, but a student asked if I’d help them prep some number theory, so here we are. How I went about it Well, with a great deal of bad …
A couple of months ago, I asked about this plot, and promptly forgot about it. This wasn’t a great What’s My Plot, in honesty. The broad class of thing was far too easy to recognise, but the specific thing far too hard. I think the best response was by @realityminus3: Astrophotography of …
Dear Uncle Colin, Can you simplify $\cosh(\arsinh(x))$? - Clumsy Oaf Shunning Hyperbolics Hi, COSH, and thanks for your message! I can see two ways of doing this that I think are ok. In both cases, let’s let $y = \cosh(\arsinh(x))$. Let’s do the clumsy way first: Let $\sinh(A) = x$, so …
This could almost be a DOME entry, but I’m probably the only person who calls it this. “To quote one of the modern, white-ball good coaches, Simon Katich: in a chase, you subtract the amount of balls [remaining] from the amount of runs [required] and divide it by six. That’s the …