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 given a triangle ABC such that angle $B$ is $xº$, angle $C$ is $3xº$, and the sides are labelled as you’d expect. I have to prove that $a = 4b \cos(2x)\cos(x)$, but I’m going round in circles! What do you recommend? Triangle Rules: It’s Common Knowledge? …
A quick how-to, by special request of @nathanday314. Pell’s equation is anything of the form $x^2 - ny^2 = 1$, as long as $n$ is an integer but not a square number. There is also a general Pell’s equation, which looks like $x^2 - ny^2 = N$, where $n$ is still not square and $N$ is a …
Dear Uncle Colin, Can you give me an example of an antisymmetric relation? - Equivalence Sodding Classes, Heaping Endless Rage Hi, ESCHER, and thanks for your message! I struggled with the notation for relations for a long while: I didn’t realise that the R in $a\text{ R }b$ stood for …
A puzzle that came to me via @nathanday314 and @spottytable: My first thought There are $r$ red socks and $b$ blue socks, making $n$ altogether. The probability of drawing two red socks out is $\frac{r(r-1)}{n(n-1)} = \frac{1}{2}$ The probability of drawing two blue socks out is …
Dear Uncle Colin, How would I find the area between the curve $y = \frac{1}{x^2}$, the coordinate axes, and the lines $y=4$ and $x=2$? Don’t Really Appreciate What It Needs Graphically Hi, DRAWING, and thanks for your message! The sketch is the important thing here. We’ve got a shape …
What is the Gray Graph? Imagine a Rubik’s cube – just for visualisation purposes, we’re not solving it or anything. It’s got 27 cubelets (including the middle). It also has 27 lines of three cubelets parallel to the axes (nine in each direction). Here comes the graph: …
Dear Uncle Colin, I noticed that $\frac{987654321}{123456789} \approx 8.0000000729$ in base 10. On further investigation, $\frac{321}{123}\approx 2$ in base 4, $\frac{654321}{123456}\approx 5$ in base seven and a similar pattern recurs in any base: $\frac{[b-1][b-2]\dots …
Despite “appropriate use of technology” being a key part of the A-level syllabus, it is apparently Still A Thing to have to perform the trapezium rule by hand (rather than, say, asking Wolfram|Alpha like a normal mathematician would). But still. If it’s in the syllabus, I suppose …
Dear Uncle Colin, I’m told that the normal to the curve $y=ax^2 + bx + 1$ through the point $(3,1)$ is parallel to the line $3y - x + 4 = 0$ and I need to find $a$ and $b$. There’s a lot going on there and I’m a bit lost! Bit Unhappy Solving, Yuck! Hi, BUSY, and thank for your …
If there’s one thing I like better than a STEP mechanics question, it’s a STEP mechanics question that just drops out when you get the diagram right. It’s question 11 from the 2017 STEP I: a ladder in a box on a slope. I’m going to answer it twice, once using the …
Dear Uncle Colin, If you know that $a$ is an algebraic number, how do you prove that $ai$ is also algebraic? Am Lost! Get Equations But Real Analysis Is Confusing Hi, ALGEBRAIC, and thanks for your message! I’m in the same boat – my real analysis lectures were at the other end of town …
One of the places I would always fall down in A-level mechanics was on inclined plane force diagrams: I was chronically incapable of putting the correct angle in the correct place. Until, that is, I figured out… The Angle Windmill It looks like this: The process goes as follows: Draw a big …
Dear Uncle Colin, I’m told that two teams play a five-game series. The first team wins the series, but the second team wins the second game. What’s the probability that the second team also won the first game? (Assume that the probability of winning each match is 50% and the matches are …
What is the Fischer 1960 Ellipsoid? The world is round, right? If you’re not on board with that, I suggest you take a long swim. It’s round all right, but it’s not a sphere: in the 1600s, models of the planet as an ellipsoid began to be taken seriously – the Earth is slightly …
Dear Uncle Colin, I’m told that the resultant of the vectors $\bb{a} = 3\bi - 2\bj$ and $\bb{b} = p\bi - 2p\bj$ is parallel to $\bb{c} = 2\bi - 3\bj$ - and I need to find $p$ and the resultant. I don’t really know where to begin! - A Rotten Resultant, Or Worse Hi, ARROW, and thanks for …
One of my major achievements of 2020 – apart from solving the Rubik’s cube blindfold, which I’m still very smug about – was completing the Advent of Code challenge. It’s pitched just about right for me: enough to get my teeth into, enough to make me feel clever when I …
Dear Uncle Colin, Someone has challenged me to solve $\int \sqrt[3]{3x-1+x\sqrt{8x-3}} \dx + \int \sqrt[3]{3x-1-x\sqrt{8x-3}} \dx$. It looks… gnarly. Any ideas? Getting Nowhere, Asking Researchers Like You Hi, GNARLY, and thanks for your message! This one caused me a bit of head-scratching …
Suppose, for a moment, that the Monty Hall problem isn’t one of the most heavily-trodden pieces of popular maths of the last three decades, and that – contrary to all good sense – you have no idea of what to do when faced with the same problem. Since you’ve been living under …
Dear Uncle Colin, The question tells me that $5x^2 + 4x + 4 + 9y^2 +12xy =0$ and that I have to find $xy$. I’ve tried plotting it in Desmos, but nothing shows up! What should I do? Dramatically Overthinking Things Hi, DOT, and thanks for your message! I think there are at least a couple of …
A wrinkle that came my way via my excellent friend @realityminus3 (Elizabeth A. Williams in real life); I paraphrase slightly: Show that $\frac{12x^2 + x - 16\sqrt{x}}{4x\sqrt{x}} = 0$ (for $x>0$) can be rearranged as $x = \br{\frac{4}{3} - \frac{\sqrt{x}}{12}}^{2/3}$. Now, Elizabeth is the best …
Dear Uncle Colin, I’m supposed to find the coefficient of $x^0$ in the expansion of $\left(x^2 + \frac{1}{x}\right)^{15}$ – I couldn’t figure it out. Any advice? - Combinations Of Numbers Seem Tough And Nasty, Though Hi, CONSTANT, and thanks for your message! As per, I can see …
The list of things named after Leonhard Euler on Wikipedia runs to about 1500 words, and, I would hazard, omits several such things. So how to settle on one? I’ve come down on one of the greatest “low barrier, high ceiling” problems there is: it’s a conjecture so simple, you …
Another plot, another challenge: A bit more of a traditional plot, with several good answers all along the same lines: @OneDavidAtATime on Twitter was (I think) first off the mark: “I see the same function twice, which could be… a ratio? Something like $y=\frac{kx}{(a-x)(a+x)}$, $a$ and $k$ …
Dear Uncle Colin, I’ve differentiated using the quotient rule and come up with $\frac{\frac{14x}{7x^2+6}x^7 -7x^6\ln\br{7x^2+6}}{\br{x^7}^2}$. That’s… not as simple as I’d like it to be. Any hints? Quite Ugly Outcome That I’d Expect Neater, Thanks Hi, QUOTIENT, and …
My excellent friend @realityminus3 asked: tl;dr What would I buy, having read all of the comments and thoughts? I’d probably pick up Alcock and Cummings to build up my intuition, and then [Rudin]1 as a reference. I’d be tempted to download [Thomson et al.]2 as well, seeing as it’s …
Dear Uncle Colin I know that $\br{\frac{x^2}{4}} + \br{\frac{y^2}{2}}=1$ is an ellipse, but I have $\br{\frac{x^2}{4}} + \br{\frac{y^2}{4}}=1$. Is that also an ellipse? Explain Loci Like I’m a Precocious Student (Elementary) Hi, ELLIPSE, and thanks for your message! The short answer is yes, …
GeoGebra has a very useful function called Asymptote: if you have something like $f(x) = \frac{3x^2+4x+3}{x-1}$, typing Asymptote(f) in the input bar gives a list of the linear asymptotes: $\{ y=3x+7; x=1 \}$. Very nice, very useful. But something like $f(x) = \frac{2x^4 + 3x^3 + 2x + 4}{x^2 +3x + …
Last week, I invited my followers on Mathstodon, LinkedIn and Twitter to ascertain the logic behind this plot: Before I reveal the answer and the winners, let me go through some honourable mentions in the responses: @bbarber_ suggested a projection of the positive orthant of the 4D lattice …
Dear Uncle Colin, Which is larger, $\sqrt{11}$ or $5-\sqrt[3]{5}$? Calculate Larger Of Similar Expressions Hi, CLOSE, and thanks for your message! I can see two ways to do this without a calculator. The first way is the more brutal, but the second requires spotting The Exact Right Thing to work …
This puzzle came to me via Futility CLoset Two people are stranded on an island with only one banana to eat. To decide who gets it, they agree to play a game. Each of them will roll a fair 6-sided die. If the largest number rolled is a 1, 2, 3, or 4, then Player 1 gets the banana. If the largest …