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.
Suppose we have $ax^2 + bx + c = 0$. It’d be easier to complete the square if the $x$ term were even, so let’s double: $2ax^2 + 2bx + 2c = 0$ It’s also be nicer if the $x^2$ term were a square, so let’s multiply by $2a$: $4a^2x^2 + 4abx + 4ac = 0$ The first two terms are $(2ax)(2ax + 2b)$, which by …
Dear Uncle Colin, I have an exam question I don’t understand! There’s a toy truck of mass 5kg attached (by a rod) to another truck of mass 2kg on a slope at 10 degrees to the horizontal. The resistances to motion are 8N and 6N, respectively, and the whole thing is pulled up the slope by a string. …
One of the questions that occasionally crops up in the hive of scum and villainy I hang out in search of problems to solve is, “how do calculators work out trigonometric values?”. It’s not, typically, the way that the Mathematical Ninja would (find an angle nearby and adjust using heuristics), but …
Dear Uncle Colin, I need to figure out $\int \cos^3(2t) \sin^5(2t) \dt$ and I’m… just going round in circles. So to speak. What do you suggest? - Doing Integration’s Really A Chore Hi, DIRAC, and thanks for your message! It’s very easy to end up going around in circles on these - the trick is to …
A nice puzzle by way of @benjaminleis: In case you can’t read that, we need to find the sum of the digits in $N = 9 + 99 + 999 + 999\dots999$, where the last number consists of 321 consecutive nines. As usual, I’ll let you pause to think about it here and post spoilers below the line. It feels …
Dear Uncle Colin, I want to stretch my Year 12 Further Maths class - what extra-curricular topics would you recommend? Something To Really Engage Their Creative Reasoning Hello, STRETCH, and thanks for your message! How excellent to be looking beyond the curriculum for ways to engage and develop …
I’m in the process of clearing out old bookmarks, and stumbled on this puzzle from @jase_jwanner: I shall give you a moment to ponder these, and put my spoiler below the line. The first and last The first and last of these are, if you look at them the right way, completely obvious: the two numbers …
Dear Uncle Colin, I’ve figured out that $x^{x^{x^{\dots}}} = 2$ when $x = \sqrt{2}$, but I’m struggling to make sense of the function - it seems to have a vertical gradient when $x = e^{\frac{1}{e}}$, but it doesn’t seem to have what I think of as an asymptote there. What gives? - Puzzling …
“We’ve been through this a hundred times, sensei. I say something like ‘$10^{1.35}$. Hm, let me get my calculator’ and you torture me in some unspeakable way an blurt out the answer…” “22.4” “… thank you, especially for refraining from the torture bit.” “You’re welcome.” “Then, of course, you tell …
Dear Uncle Colin, On a recent revision course, my tutor couldn’t integrate $\int_0^{\piby2} \frac{\sin(2\theta)}{1+\cos(\theta)} \d \theta$. Can you? - Reasonable Expectation For University’s Nameless Don? Hi, REFUND! Far be it from me to disparage a fellow professional’s integration skills, …
I’m a big fan of the doodle. My lecture notes, even my schoolbooks, are covered with geometric patterns and impossible shapes and simple cartoons. Today’s entry in the Dictionary of Mathematical Eponymy started jumped out of Stanislaw Ulam’s notes while he was listening (according to Martin Gardner) …
Dear Uncle Colin I wonder: at what height is the volume of a cone above that height equal to the volume below? What about the surface area? Are there any cones where it’s the same height? Finally Researching Uniformly Splitting Things Up, Mate Hi, FRUSTUM, and thanks for your message! Let’s suppose …
I’m on a bit of a continued fractions jaunt at the moment, as they’re my current “oo! That’s a tool I haven’t played with enough, and that I think might be interesting!” One of their applications (surprisingly to me) is to answer the question “A survey firm says ‘82.43% of our respondents said [x].’ …
Dear Uncle Colin, Can you explain why $\frac{a+b}{c + d}$ is between $\frac{a}{c}$ and $\frac{b}{d}$? - Grinding Out Solid Proof Explaining Rationals Hi, GOSPER, and thanks for your message! As per usual, there are several methods to show this. I’m going to assume (since it hasn’t been stated) that …
Lately I’ve been playing around with continued fractions - I blame credit @evelynjlamb for pointing me at this post by @johndcook. I’ve done most of my learnin’ from here and from various Wikipedia pages, and I thought I’d revisit some of it for the benefit of others. In fact, I’ll start with the …
Dear Uncle Colin, I need to solve $5^{2x+2} +16\cdot 15^x - 9^{x+1} = 0$ but I’ve hit a dead end! Can you help? - Puzzling Out Wild Exponent Relations Hi, POWER, and thanks for your message! From the working you sent, it looks like you’ve picked a good strategy: you’ve let $X = 5^x$ and $Y = 3^x$ to …
A nice observation from Futility Closet: Draw two circles of any size and bracket them with tangents, as shown. The chords in blue will always be equal. I’m hardly going to let that pass by without a proof, now, am I? Spoilers below the line. My proof Definitions Let the distance between the centres …
Dear Uncle Colin, I know a 3D circle passes through $A(4,-4,5)$, $B(0,4,1)$ and $C(0,0,5)$ and I need to find its centre and radius. I could do it in 2D, but I’m a bit stuck here! Can’t Interpret Radius/Centre, Looked Everywhere Hi, CIRCLE, and thanks for your message! There are, as usual, several …
I get a lot of my problems-to-solve from Reddit, since if someone’s posted it there, there are probably thousands of people with the same difficulty. This one isn’t from Reddit, but from @frauenfelder, one of the high-heidyins at BoingBoing. Out of the 25 homework problems, there was one that she …
Dear Uncle Colin, I have the graphs of $y=\sin(x)$ and $y=\cos(x)$ for $0 < x < 2\pi$. They cross in two places, and I need to find the area enclosed. I’ve figured out that they cross at $\piby 4$ and $\frac{5}{4}\pi$, but after that I’m stuck! - Probably A Simple Calculation, Absolutely Lost …
I recently had the chance to employ this one, but didn’t manage to: it turns out that four three-to-six-year-olds are not especially interested in putting down markers and following rules, they just want to run around the maize maze and say “maize maze” and make “amazing” jokes1 What is Tremaux’s …
Dear Uncle Colin, My textbook gives me an arithmetic sequence that starts $3 + 8 + 13 + \dots$, and asks me to find where the sum is 1575. I’ve got it down to $\frac{-1 \pm \sqrt{63,001}}{10}$, but I don’t know how to work out that square root without a calculator! - Not Terribly Experienced …
The student’s shoulder twitched slightly as he said “So I need to work out $\log_2(10)$…” and the crash of the cane against the table reminded him that the calculator was off-limits. “I think you can estimate that yourself,” said the Mathematical Ninja. “Uh… ok. There’s a change of base formula, …
Dear Uncle Colin, How would you go about factorising $6x^2 - xy - y^2 + 7x - y + 2$? - Argh! Getting Nowhere. Expression Simplification Impossible. Hi, AGNESI, and thanks for your message! I have, historically, not been a fan of these. However, I’ve recently come across a method that makes a bit of …
A nice challenge puzzle via Reddit: Find $\sum_{n=1}^{\infty} \frac{n2^n}{(n+2)!}$ There was a video attached to it that I didn’t watch, something about telescoping sums, but the moment I saw this, I thought: generating functions! Why would I think such a thing? The thing that jumped out at me was …
Dear Uncle Colin, A while back, you shared an easy way to factorise nasty quadratics. Why does it work? Dutifully Indulging Students’ Curiousity & Reasoning In Maths. I’m Not A Nasty Teacher! Hi, DISCRIMINANT, and thanks for your message! Let’s start by recapping the method in the post, go …
When Jake’s father (Bradley Whitford) comes to town, Jake is excited to see him, but Charles is wary of his intentions; Holt challenges Amy, Terry, Gina and Rosa with a brain teaser in exchange for Beyonce tickets. Brooklyn 99, S02 E18, Captain Peralta “Surely it isn’t that hard?” I have no interest …
Dear Uncle Colin, I’m trying to work out $0.03 \div 0.1$. The answer is apparently 0.3, but I don’t understand why a division would make something bigger! Division Isn’t Very Intuitive, Decimals Especially Hi, DIVIDE, and thanks for your message! There are several ways to tackle this. I’ll start …
I am not a number theorist. I mean… well scratch that. I am not an especially knowledgeable number theorist1 but I do enjoy number theory when it’s around my level. The Sieve of Sundaram is about my level. What is the Sieve of Sundaram? OK, let’s start with the Sieve of Eratosthenes, which is …
Dear Uncle Colin, I need to find the area between the graphs of $y=\sin(2x)$ and $y=\cos^2(x)$ in the interval $0 \le x \lt 2\pi$. I’ve found four solutions, but I think that means I need to do five separate integrals! Is there an easier way? Trigonometric Expressions: Double Integration Of Ugly …