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, The remainder theorem of my textbook wants me to find the set of values of $k$ such that $3x^4 + 4x^3 - 12x^2 + k = 0$ has four distinct real roots, and I don’t know where to start! - Quite Ugly Algebraic Relation: Targeting Its Constant Hi, QUARTIC, and thanks for your message! I …
I had cause to revisit an old @edsouthall tweet: It’s a nice puzzle. Try it yourself! Spoilers below the line. As a clarification: the square drawn must have the starting point as its top left corner. My approach Given that you’re at one corner of the square, you always have two moves that will keep …
Dear Uncle Colin, How on earth would I work out the 100th digit after the decimal point in $(1 + \sqrt{2})^{500}$? The would require an enormous calculation! - Perhaps A Simplification Could Answer Logically Hi, PASCAL, and thanks for your message! The answer to this lies in one of my favourite …
You know you’re in for a treat when @robeastaway asks a birthday question. I don’t know if the poll shows up embedded, but you’ll see that the answers are split pretty evenly between $\frac{1}{3}$ and $\frac{1}{2}$, each of which is about twice as popular as “100%”. You might wish to have a go at it …
From the 2008-9 British Mathematical Olympiad 1: Find all of the real values of $x$, $y$ and $z$ such that: $(x+1)yz = 12$ $(y+1)zx = 4$ $(z+1)xy = 4$. I find this sort of cyclical simultaneous equation extremely unintuitive: there are tricks and hacks I use, but it’s all very ad hoc and I don’t …
Dear Uncle Colin, Does $a > b$ imply $\frac{1}{b} > \frac{1}{a}$? - Inequality: Mightily Perplexing Logically, Yeah? Hi, IMPLY, and thanks for your message! On the face of it, that seems sensible, doesn’t it? It’s tempting to say something like $1 > \frac{b}{a}$, so $\frac{1}{b} > …
There’s a problem, in measuring (for example) wind direction that I hadn’t ever thought about until it was pointed out to me: taking averages is fraught with danger. For example, if you have three readings: one at a bearing of 010, one at 000 and one at 350, you might quite reasonably say “the …
Dear Uncle Colin, Apparently, $\sum_1^\infty \left(\frac{n^2}{2^n}\right) = 6$, which surprised me. Can you explain why? - Some Intuition Gone Missing, Aaargh! Hi, SIGMA, and thanks for your message! I’m not sure I can give you intuition, but I can explain where the result comes from. Let’s start …
For the first time, I’ve been using Goodreads to track what I read, and making sure to read a bit at bedtime almost every night. I figured it would be worth listing the books that I felt were five-star books, in case they help anyone else. In alphabetical order by author: How to Take Smart Notes, by …
Dear Uncle Colin, Why do you insist on giving smart-arse answers to simple puzzles on Twitter? - Malice Establishes Almost Nothing Hi, MEAN, and thanks for your message! First up, please don’t confuse my cheerful ragging of questions with malice; on the contrary, coming up with alternative …
A factorial puzzle I found at NRICH: Consider numbers of the form $u_n = 1! + 2! + 3! + \dots + n!$. How many such numbers are perfect squares? Write out a few terms The first thing I like to do, faced with a problem like this, is to write out a few terms to see if there are any patterns I might …
Dear Uncle Colin, The question says that $z_1$, $z_2$, $z_3$ and $z_4$ are distinct complex numbers representing the vertices of a quadrilateral ABCD, in order. Further, $z_1 - z_4 = z_2 - z_3$ and $\arg\left( \frac{z_4 - z_1}{z_2 - z_1}\right) = \piby 4$. The shape is supposed to be one of a …
I’m currently recommending Ratio by Michael Ruhlman to anyone who’ll listen - it’s a cookbook that teaches you how to cook, not just how to make things. Its one fault, though: it gives temperatures in degrees Fahrenheit. It’s not technically true that Fahrenheit picked 100º to be the temperature of …
Dear Uncle Colin, At a Mathscounts competition, a contestant was asked “how many six-digit positive integers are divisible by 1000 but not by 400?”. Within four seconds, they correctly answered 450 – how on earth could they do that so quickly? - Mathematical Evaluations Normally Take A Lot Longer, …
When I started this project, I realised I was going to run into trouble with some of the letters. The high-scoring Scrabble tiles ones in particular – although J would likely be OK, I figured I’d struggle a bit with Q and Z, and find nothing at all for X. I was mistaken. Today, we have a little bit …
Dear Uncle Colin, I’ve been shown a proof that goes like this: To show: $a^n = 1$ for all non-negative integers $n$ and for all non-zero real numbers $a$. Proceed by induction. Base case: $a^0 = 1$ by definition, so the base case holds. Inductive step: Suppose $a^j = 1$ for all integers $j$ such …
A puzzle that crossed my path via @drmaciver: You might want to have a play with it before I share my solution. Spoilers below the line. I mentally went around the houses a bit, before hitting the key question, which is: When does player 1 win? Suppose Player 1 - Aloysius - is given numbers $x$ and …
Dear Uncle Colin, I’m told there are two circles that touch the x-axis at the origin and are also tangent to the line $4x-3y+24=0$, but I can’t find their equations. Any ideas? - A Geometrically Nasty Example Seems Impossible Hi, AGNESI, and thanks for your message! I’m going to start with the …
Via nRICH: A circle touches the lines OA extended, OB extended and AB where OA and OB are perpendicular. Show that the diameter of the circle is equal to the perimeter of the triangle. $\blacksquare$
Dear Uncle Colin, In my non-calculator paper, I’m told $\cos(\theta) = \sqrt{\frac{1}{2}+ \frac{1}{2\sqrt{2}}}$ and that $\sin(\theta) = -\left(\sqrt{\frac{1}{2}-\frac{1}{2\sqrt{2}}}\right)$. Given that $0 \le \theta \lt 2\pi$, find $\theta$. I’ve no idea how to approach it! - Trigonometric …
“I would have to assume the teacher means $\sqrt[4]{81}$ instead.” “That’s as may be. But $4\ln(3)$ is 4.4 (less one part in 800). A third of that is $1.4\dot 6$, less one part in 800, call it 1.465.” “So you’d do $e$ to the power of that?” “Indeed! $\ln(4)$ is 1.4 less 1%, so 1.386 or so. We’re …
Dear Uncle Colin, I have to work out $\cot\left( \frac{3}{2}\pi \right)$. Wolfram Alpha says it’s 0, but when I work out $\frac{1}{\tan\left(\frac{3}{2}\pi\right)}$, my calculator shows an error. What’s going on? - Troublesome Angle, No? Hi, TAN, and thanks for your message! The cotangent function …
I’m a Big Fan of both @standupmaths and @sparksmaths, two mathematicians who fight the good fight. I was interested to see Ben tackling the square root of 3 using the ‘long division’ method. It’s a method I’ve tried hard to love. It’s a method I just can’t bring myself to do or recommend. …
Dear Uncle Colin, I need to evaluate $\int_0^{\piby2} \frac{1}{1+\sin(x)}\dx$ but I end up with $\infty - \infty$ and that’s no good! How should I be doing it? - Big Integral, Not Exactly Trivial Hi, BINET, and thanks for your message! This is a fun problem! I can think of several possible …
While my thesis has the word ‘topology’ in its title, at heart I’m a vectors-in-3D person. Give me matrices, not manifolds! So today’s entry in the Dictionary of Mathematical Eponymy is one that brings me joy. What is Wahba’s Problem? The mathematical statement of Wahba’s Problem is as follows: …
Dear Uncle Colin, How would you solve $5^{2x+2} + 16\cdot15^x -9^{x+1} = 0$? Doesn’t It Seem Genuinely Unpleasant? Insoluble? Strange, Even? Hi, Disguise, and thanks for your message! It turns out that this does have solutions! The trick is to ask “what would make it less unpleasant?” - and given …
I could probably have framed this post as an “Ask Uncle Colin”, but it feels somehow different, so I’m going to do it as a Monday post. My blog, my rules. On Twitter, @sharanjit asked: I had a point in my education where maths went from being easy to suddenly being hard (University). Did you have …
Dear Uncle Colin, Why is $\cos(36º)$ equal to $\frac{\phi}{2}$? I find trigonometry difficult and, uh, let’s say I have some demons to banish. Puzzled Educator Noticed That A Golden Ratio Appears Magically Hi, PENTAGRAM, and thanks for your message! I would imagine there are several ways to …
If you’ve read this blog for a while, you’ll know I’m a fan of @cshearer41’s puzzles (her book, Geometry Puzzles in Felt Tip, is available wherever etc). At a recent MathsJam, one jumped out of Chalkdust at us: (Image from Issue 10 of Chalkdust, a magazine for the mathematically curious.) It’s a …
Dear Uncle Colin, How many zeros are there on the end of $100!$? I worked it out to be 21, but the answer sheet says it’s 23 – and my calculator just gives an error message. What do you think? - Maybe A Tutor Has Exact, Reasoned Response? Hi, MATHERR, and thanks for your message! I think the correct …