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 need to work out - or at least estimate - the middle value in a row of Pascal’s triangle. Is there a quick way? Strange Times - I’m Really Lucky I’m Not Guessing Hi, STIRLING, and thanks for your message! It turns out there’s a really nice way to get a good estimate using …
On reddit, an interesting question: Given one hour, an unlimited amount of chalk, and an unlimited amount of blackboard space, how many (correct) digits of $\sqrt{10}$ could you find? (without any calculation aids, obviously). At the moment, the idea of having a free hour to do anything, let alone …
Dear Uncle Colin, How big a group of people do you need so that the probability of two of them being married is more than 50%? - Counting Out Unmarried People Leveraging Exponential Sums Hi, COUPLES, and thanks for your message! Im going to model this as a graph theory problem, rather than a …
What is the Dubins path? Geometrically speaking, the shortest way to get from point A to point B is along a straight line. But what’s the shortest route if you have directional restrictions? Suppose you have a car moving in a 2D plane and a restriction on how sharply it can turn, a maximum …
Dear Uncle Colin, Which is larger, $\sqrt{3}+\sqrt{11}$ or $\sqrt{5}+\sqrt{8}$? No calculator! Roots Are Difficult (I Calculated Anyway, LOL) Hi, RADICAL, and thanks for your message! If the Mathematical Ninja was nearby – and who can say, they might be – I would probably work out $\sqrt{11}$, the …
At MathsJam, I was pointed at a puzzle from the New Scientist, which I’ll paraphrase as: You have a long, thin cake of length 1. Two candles are places at random1 points on the top of the cake, and the cake is cut (perpendicular to its edges) at a third random point. What is the probability that the …
Dear Uncle Colin, I was working on a MAT question that asked about finding a subset, $S$, of the 2D-plane, and a point $P$ such that no point in $S$ was the closest to $P$. I had no idea where to start! Omitted Point, Euclidean Norm Hello, OPEN, and thanks for your message! My first answer wasn’t …
I heard it from @benjamin_leis, and he says he heard it from @CMonMattTHINK, and I love it: The number of integer solutions to $x^2 + xy + y^2 = a$ appears to be a multiple of six for all $a \in \mathbb{Z_+}$ . Why? How good a puzzle is this? I started my swimming class on Thursday and decided to …
Dear Uncle Colin, The puzzle asks: you have two glasses with equal volumes of water and juice. You take a tablespoon of water and mix it into the juice; you take a tablespoon of the juice-water mix and mix it into the water. Is there more water in the juice, or juice in the water? Apparently the …
A day off ill, and the chance to look muddle-headedly at a tweet from @trianglemanscd long ago: I’m a geometer! And I like problem-solving! There are loads of valid approaches here, and I’m going to talk of three. A really boring approach Step 1: go to Google maps Step 2: find the scale of the map …
Dear Uncle Colin, I’ve got two circles ($x^2 + y^2 = 3^2$ and $(x-10)^2 + y^2 =5^2$) and I want to find the equations of the common tangents. I’ve been stuck for ages! - Tackling A New Geometry Exercise, Need Tuition Hi, TANGENT, and thanks for your message! I can see two fairly nice ways to …
I’m cheating a bit with this one; the Borromean rings are named after the whole of the Borromeo family, and the Clélie curve is (unusually) named after a mathematician’s first name. But this is my dictionary, and I’m going to claim a twofer here. Who was Clelia Grillo Borromeo? Clelia (or Celia) …
Dear Uncle Colin, How do you remember the values for the trig functions in the unit circle? - Can’t Abide Stupid Trigonometry Hi, CAST, and thanks for your message! The short answer is, by knowing what sine and cosine mean on a unit circle, knowing a couple of special triangles, and knowing a bit …
This is based on a Mathematical Note written by George Osborn: link to DOME.1 Start by defining a function, $G$, such that $G(n, 1+x) = \frac{n^x}{\Pi_{i=1}^{n}\left(1+\frac{x}{i}\right)}$. For example, $G(3, 1 + 1) = \frac{3^1}{\left(1+ \frac{1}{1}\right)\left(1 + \frac{1}{2}\right)\left(1 + …
Dear Uncle Colin, Can you show that 4 is the largest determinant of a 3 by 3 matrix made of 1s and -1s? - Just A Crisis Of Brainy Ideas Hi, JACOBI, and thanks for your message! Pluggety and chuggety Let’s let the matrix be $\matthreethree{a}{&b}{&c}{d}{&e}{&f}{g}{&h}{&i}$. …
I was a little surprised to see a continued fractions conjecture on Wikipedia, stating: $e = { 3 + \frac{-1}{4 + \frac{-2}{5 + \frac{-3}{6 + \dots}}}}$ Obviously, my first thought was, “that doesn’t look too hard to prove”. My second thought was “I imagine it already has been”, and such is the case: …
Dear Uncle Colin, Which is larger, $x=1000^{1001}$ or $y=1001^{1000}$? - Calculating Oversize Massive Powers And Realising… Eurgh Hi, COMPARE, and thanks for your message! These feel like they ought to be a similar size! In fact, the first one, $x$ is the larger. Let’s show that in several ways. …
To get from our house to the usually-empty car park my kids like to practice their bike-riding in, we have to cross at least one road. Traffic flows one way along Cromwell Road (northbound), and at the junction, either carries on or turns right into Highland Road (which is also a one-way street, …
Dear Uncle Colin I gather UEFA is toying with the idea of using the Swiss System for future Champions Leagues. Without wanting to sound cheesy, that seems a bit cuckoo. What’s it alp… I mean all about? Strange Wayto Implementa Soccer System Hi, SWISS, and thanks for your message - this …
One of the traditional grumbles about divisibility rules is that in some cases - I’m looking at you, seven - there’s very little benefit to knowing the rule over “just” performing the division. I have one exception to this rule, and while it’s still some work, it has two Very Nice Aspects to it: The …
Dear Uncle Colin, Why is the limit of $\frac{\sin(2x)}{x}$ and $\frac{\sin(x)}{x}$ different, as $x\to 0$? Changing the argument of $\sin$ only changes the period of the function, so how come the value changes as well? - Limits Hitting Origin: Pain In The Arsinh, LOL. Hi, LHOPITAL, and thanks for …
So, we got to Z in the Dictionary of Mathematical Eponymy. What now? Well, in the course of my research, I found that there were more than 26 people with things named after them. Shocking! In particular, despite my efforts, the original list was rather heavy on dead white men. I could make excuses …
Dear Uncle Colin, I have figured out a construction of a tangent line to a circle, but haven’t been able to prove that it works. Can you help? Here’s the protocol: Pick two points on the circle, A and B Draw a circle centred on B, passing through A This circle also intersects the …
Somewhat embarrassingly, I got this wrong at first: (Dammit, Jim, I’m a mathematician, not a physicist.) Have a go at it yourself, if you’re so inclined; after the line come spoilers. Kirchhoff and Ohm In the bad old days, I used to help out at a Physics homework centre – the work they were doing …
Dear Uncle Colin, What’s the 13th root of 21,982,145,917,308,330,487,013,369? I know it’s an integer. - Extremely Large Exponent, Perhaps Having A Nice Thirteenth… Hi, ELEPHANT, and thanks for your message! Thirteenth roots are a staple of mental arithmetic contests (I’m told ), but despite this …
In preparing for a problem-solving session, I came across this lovely problem from the MAT: The reflection of point $(1,0)$ in the line $y=mx$ has the coordinates: A: $\left( \frac{m^2+1}{m^2-1}, \frac{m}{m^2-1}\right)$ B: $(1,m)$ C: $(1-m,m)$ D: $\left( \frac{1-m^2}{1+m^2}, \frac{2m}{1+m^2}\right)$ …
Dear Uncle Colin, Suppose I’ve drawn a triangle with an angle of 30º and an opposite side of 5cm. Is there a simple way to estimate what the opposite side would be if the angle was, say, 40º? Some Kindof Estimated Trigonometric Calculation Help Hi, SKETCH, and thanks for your message! The short …
I feel like I spend a lot of time explaining the difference between $P(A | B)$, $P(B | A)$ and $P(A \cap B)$, so I figured it would be good to have an article I can point people at to explain properly. Let’s imagine we’re at an animal shelter where there are many animals of all descriptions. I’m …
Dear Uncle Colin, How can I find the turning points of a cubic (for example, $y = x^3 - x$) without calculus? - Calculus Unnecessary (But It’s Clever) Hi, CUBIC, and thanks for your message. This one gave me a proper ‘oo’ when I realised how it worked! The key insight here is to translate the cubic …
Many of the entries in the Dictionary of Mathematical Eponymy have been 20th century eponyms - that’s been a deliberate choice: partly because I wanted to tread ground that was relatively new for me, partly because I wanted to find at least a few things named after women and - a few notable …