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, This is the fifth time in the last six World Cups that Nigeria have been drawn against Argentina in the group stages. What are the odds?! - Coincidences At FIFA? Unlikely! Hi, CAFU, and thanks for your message! I’m going to make some simplifying assumptions for this answer, …
In previous articles, I’ve looked at how to find $\cos(72º)$ using some nasty algebra and some comparatively nice geometry. In this one, inspired by @ImMisterAl, I try some nicer - although quite literally complex - geometry. De Moivre’s Theorem I’m going to assume you’re ok …
Dear Uncle Colin I have a percentages problem: I’m told that in an election, 95.74% of the electorate voted for the winning side. What is the minimum possible size of the electorate? - Percentages Often Lack Logic Hi, POLL, and thank you for your message! There are two possible answers to …
Owing to a spotty connection, the sound quality is a bit patchy on this one. Apologies. In this month’s installment, we are joined by Belinda Keir. We discuss: Belinda convenes the Sydney MathsJam and is on the committee for the Celebration of Mind What makes a mathematician? Why Belinda is in …
Some months ago, I wrote about a method for finding $\cos(72º)$, or $\cos\br{\frac{2\pi}{5}}$ in proper units. Almost immediately, the good people of Twitter and Facebook - notably @ImMisterAl (Al) and @BuryMathsTutor (Mark)- suggested other ways of doing it. Let’s start with Mark’s …
Dear Uncle Colin, I’m trying to solve $3^{2x+1} - 28\times 3^x + 9 =0$. I’ve split up the first term into $3 \times 3^{2x}$ but then I’m stuck! Any suggestions? Likely Overthinking Gettable Sum Hi, LOGS, and thanks for your message! You’ve made a really good start there. The …
The ever-challenging Adam Atkinson, having noticed my attention to the “impossible” New Zealand exams, pointed me at a tricky question from an Italian exam which asked students to verify that, to give a smooth ride on a bike with square wheels (of side length 2), the height of the floor …
Dear Uncle Colin Do you have any tips for sketching three-dimensional vectors? Every time we have an A-level question, my teacher says “draw a diagram!” but I don’t know how to draw in 3D. - Got A Useless Sketching Situation Hi, GAUSS, and thank you for your message! …
That @solvemymaths is an excellent source of puzzles and whathaveyou: How would you find $\sqrt[3]{\frac{1-x^2}{x}}$ when $x=0.962$, using log tables or otherwise? I would start by trying to make the numbers nicer: I note that $x=(1-0.038)$, which means we can rewrite the expression (using …
Dear Uncle Colin, I’m sitting my GCSE Maths starting tomorrow. What will the grade boundaries be? - First Exams Are Redoubtable Hi, FEAR, I’m writing this, believe it or not, in early February1. I’m not even sure this summer’s GCSE papers have been written yet, but I am going …
One of the puzzles in the MathsJam Shout looked impossible, so obviously I sat down with Mr Miller and had a go at it. I don’t have it in front of me, but it went something like: A couple hosts a party to which five other couples are invited. At some point in the evening, hands are shaken. …
Dear Uncle Colin, I had a question in an exam that gave a cubic, $f(x) = x^3 - 8x^2 + cx + d$, with roots $\alpha$, $\beta$ and $\gamma$. When plotted on an Argand diagram, the triangle formed by the three roots has area 8. Given that $\alpha=2$, find $c$ and $d$. I couldn’t get near it - any …
In Episode 56 of Wrong, But Useful, we’re joined by @zoelgriffiths (Zoe Griffiths), maths communicator from Think Maths. Zoe had her poem e, to thee, x in @chalkdustmagazine recently, and did a set about misleading statistics at @aeoud (An Evening Of Unnecessary Detail) Bad polls and fake …
It turns out, I made an error in Cracking Mathematics. Not (in this case) a mathematical or historical error, although there are plenty of those1 but an error of etiquette: my potted biography of John Horton Conway emphasised the Game of Life above the rest of his work; I imagine the book made a …
Dear Uncle Colin, I’m practicing for the Oxford PAT and have been asked how many terms of the binomial expansion would be needed to determine $(3.12)^5$ to one decimal place? I don’t really know where to start. - Knows Expansions (Binomial); Lacks Explanations Hi, KEBLE, and thanks for …
A professor - according to Reddit - asked their class how many people you’d need to have in a room to be absolutely certain two of them would have Social Security numbers1 ending in the same four digits (in the same order). 10001, obviously. How about a probability of 99.9%? The students …
In this post, I swap liberally between d-notation and ‘-notation for derivatives. Deal with it. Dear Uncle Colin, Why do we have to treat second-order ODEs differently when the auxiliary equation has a repeated root? Something Or Other Defies Expectations Hi, SOODE, and thanks for your …
I remember, with a faint feeling of dread, having to calculate the eigenvalues of a matrix. It became routine in the end, but I was recently reminded of the pain when a student asked if there was a shortcut. For a 2-by-2 matrix? Yes. It is up to you, though, whether you think it’s worth …
Dear Uncle Colin, I was trying to work out $\frac{\frac{3}{7+h}-\frac{3}{7}}{h}$, and I got it down to $\frac{\frac{3}{h}}{h}$ - but that’s not the answer in the book! What have I done wrong? - Likely I’ve Mistreated It Terribly Hi, LIMIT, and thank you for your message! I’m afraid …
This month, it’s Gathering4Gardner special, largely recorded in Atlanta with Adam Atkinson and @dragon_dodo. We discuss: Our favourite talks of the event, including: The Juggler Problem The Taxman Problem Ramanujan Sums Optical illusions Doris Schattschneider’s talk on Marjorie Rice, an …
Last time out, I looked at a problem unearthed by @mathsjem - to find the cube root of a degree-six polynomial. This led (unsurprisingly) to a quadratic: $3 + 4x - 2x^2$. When checking whether this was indeed the answer, I hit a problem: is there a simple way to cube a polynomial? Use the binomial, …
Dear Uncle Colin, I’m told that two lines through $(0,12)$ are tangent to the circle with equation $(x-6)^2 + (y-5)^2 = 17$ and I need to find their equations - but I’m getting in a muddle. Can you help? - Terribly Awkward Numbers, Getting Equations Not Trivial Hi, TANGENT, and thank you …
It’s always fascinating to see what’s going on in textbooks of the olden days, and National Treasure @mathsjem recently found a beauty of its type. Look at those whences! Check out the subjunctives! It thrills the heart, doesn’t it?1 What caught my attention, though, was evolution …
Dear Uncle Colin, I couldn’t make head nor tail of this geometry problem: “If $a:b=12:7$, $c=3$, and $B\hat{A}C = 2 B\hat{C}A$, find the length of the sides $a$ and $b$.” - Totally Rubbish In Geometry Hi, TRIG, and thank you for your message! (And don’t put yourself down like …
Glancing over sample papers for the new GCSE, I stumbled on this: Zahra mixes 150g of metal A and 150g of metal B to make 300g of an alloy. Metal A has a density of $19.3 \unit{g/cm^3}$. Metal B has a density of $8.9 \unit{g/cm^3}$. Work out the density of the alloy. I don’t think I’m …
Dear Uncle Colin, How would you work out $9^{41} \pmod{61}$? - Funky Exponential Result, Missed A Tutorial Hi, FERMAT, and thanks for your question! I think the answer is “ponderously”! There are only 61 possible answers (in fact, 60, because you know 61 is not a factor of $9^{41}$). …
A Christmas Pudding Puzzle I swear, this one came up in real life! My partner made a Christmas pudding for the most recent festive season. Delicious, it was. When it was about half-eaten, I went to microwave a portion. “Hang on,” she said: “there might be a coin in there.” As …
Dear Uncle Colin, Why is $\arcsin\br{\sin\br{\frac {6}{7}\pi}}$ not $\frac{6}{7}\pi$? - A Reasonable Conclusion Seems Incorrect Numerically Hi, ARCSIN, and thanks for your message! On the face of it, it does seem like a reasonable conclusion: surely feeding the output of $\sin(x)$ into its inverse …
One of the most famous examples of stuckness - both for maths as a whole and for a mathematician in particular - is Fermat’s Last Theorem, which states that there is no solution to $a^n + b^n = c^n$ for whole numbers $a$, $b$, $c$ and $n$ unless $n$ is 1 or 2. Pierre de Fermat wrote this down …
Dear Uncle Colin, I need to find the limit as $x$ approaches 1 of $\frac{x^{29}-1}{x-1}$. I tried factoring out $x^{28}$ but didn’t get anywhere. - Learning How Others Proceed In This Awful Limit Hi, LHOPITAL, and thanks for your message! Factoring out an $x^{28}$ is very unlikely to get you …