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.
This post is inspired by a Futility Closet article. Do visit them and subscribe to their excellent podcast! Suppose you’re dealt a bridge hand1, and someone asks whether you have any aces; you check, and yes! you find an ace. What’s the probability you have more than one ace? This is a …
Dear Uncle Colin I have been asked to describe how $y = \frac{3x^2-1}{3x+2}$ behaves as $x$ goes to infinity. My first answers, “$y$ goes to infinity” and “$y$ approaches $x$”, were both wrong. Any ideas? - Both Options Reasonable, Erroneous Limits Hi, BOREL, and thanks for …
One of the more surprising results a mathematician comes across in a university course is that the infinite sum $S = 1 + \frac{1}{4} + \frac{1}{9} + … + \frac{1}{n^2} + …$ comes out as $\frac{\pi^2}{6}$. If $\pi^2$s are going to crop up in sums like that, they should be required to …
Dear Uncle Colin, What is $\frac{1}{\infty}$? - Calculating A Number, Though Outside Reals Hi, CANTOR, and thanks for your message! The short answer: it’s undefined. The longer answer: Infinity is not a number. It’s not something you’re allowed to divide by. The calculation …
“So,” said the Mathematical Ninja, “we meet again.” “In fairness,” said the student, “this is our regularly-scheduled appointment.” The Mathematical Ninja was unable to deny this. Instead, it was time for a demand: “Tell me the square root of …
Dear Uncle Colin, I’m aware of Binet’s formula for finding the $n$th Fibonacci number, $F_n = \frac{\phi^n - (-\phi)^{-n}}{\sqrt{5}}$, and wondered if there was an inverse version - to find $n$ given a Fibonacci number. -- Fibonacci Explicit Inverse, Getting Extremely Nervous But Am …
In an early draft of my forthcoming book, The Maths Behind, which will be available wherever good books are sold from September, I believe, I took the following unprovoked dig at Australia: “… it crashed into the ocean about 1,600 miles to the west of Perth, Australia. There’s …
Dear Uncle Colin, I’m a bit stumped by a logs question with a variable base: $\log_{\sqrt[3]{x+3}}(x^3 + 10x^2 + 31x + 30) = 9$. I know the basics of logarithms, but this is currently beyond me. -- Obtaining Underwhelming Grade, Having To Review Every Definition Hello, OUGHTRED, and thanks for …
In this month’s episode of Wrong, But Useful, we are joined by Special Guest Co-Host @jussumchick, who is Jo Sibley in real life. Colin’s audio is unusually hissy in this one, which is why it’s a little late; he apologises for both inconveniences. We discuss: Jo’s background …
For all the grief I give @reflectivemaths on Wrong, But Useful, he does occasionally ask an interesting question. In episode 45, he wondered how many packs of Lego cards one would need to acquire, on average, to complete the set of 140?1 A simpler case Suppose, instead of 140 cards, there was a …
Dear Uncle Colin, In a recent exam, I was invited to solve $12x^2 - 59x + 72=0$ without a calculator. Is that a reasonable thing to ask? Very Irate EdExcel-Taught Examinee Hi, VIETE, and I don’t blame you for being cross - in a non-calculator exam, I’m not sure that really tests the …
The brilliant @dragon_dodo sent me this puzzle: Evaluate $\int_0^1 \left(1-x^\frac{1}{7}\right)^3 - \left(1-x^\frac{1}{3}\right)^7 \d x$. I’m not going to give you the solution right now; that will come after I’ve rambled for a bit. After I’d solved the puzzle (see below), I …
Dear Uncle Colin, In a recent test, I stumbled across $9x^4 + \frac{1}{144x^4} + \frac{1}{2}$, which apparently factorises as $\left(3x^2 + \frac{1}{12x^2}\right)^2$. How on earth am I supposed to spot that?! - Feeling Almost Cheated, That’s Only Reasonable Hi, FACTOR, and thanks for your …
“You know how you’re always putting things like ‘just to keep @RealityMinus3 happy’ in your posts?” “Of course, sensei!” “Well… you remember that post about missing solutions in a trig problem?” “Ut-oh.” What follows is a guest …
Dear Uncle Colin, I’ve got a question that asks me to find the coefficient of $x^5$ in $(1+x)^5 + (1+x)^6 + (1+x)^7 + … + (1+x)^{100}$. I can easily work out the coefficient in each term (it’s just $\nCr{k}{5}$), but I can’t see an easy way to add them up. Any ideas? - …
The estimable @colinthemathmo suggests a method for estimating the radius of the earth, which he credits to a sundial expert friend named Mike: Stand on a wall, perhaps two metres high, and wait for sunrise. When you see the sun just peak above the horizon, start the stopwatch, and jump off the wall …
Dear Uncle Colin, I’m pretty good with quadratic inequalities and pretty good with absolute values, but when I get the two together, I get confused. For example, I struggled with the set of values satisfying $x^2 -\left| 5x-3\right| < 2 + x$. Can you help? - Nasty Absolute Value …
It’s good to see @srcav back in the twitter and blogging fold - he’s been missed! As part of his comeback, he shared this lovely geometry puzzle: Assuming the situation is symmetrical (which it needs to be to get a sensible solution), there are - as usual - several ways to solve it. Like …
Dear Uncle Colin, When I solve $2\tan(2x)-2\cot(x)=0$ (for $0 \le x \le 2\pi$) by keeping everything in terms of $\tan$, I get four solutions; if I use sines and cosines, I get six (which Desmos agrees with). What am I missing? - Trigonometric Answers Not Generated - Expecting ‘Nother Two Hi, …
In this month’s edition of Wrong, But Useful, @reflectivemaths and I are joined by special guest co-host @dragon_dodo, who is Dominika Vasilkova in real life. We discuss: What maths appeals to a physicist. Dominika’s number of the podcast: $0.110001000000000000000001…$, …
Zeke and Monty play a game. They repeatedly toss a coin until either the sequence tail-tail-head (TTH) or the sequence tail-head-head (THH) appears. If TTH shows up first, Zeke wins; if THH shows up first, Monty wins. What is the probability that Zeke wins? My first reaction to this question was, …
Dear Uncle Colin, I was asked to find the tangent to the curve $r=\frac{8}{\theta}$ at the point where $\theta = \frac{\pi}{2}$. I worked out $\dydx = \frac{ \frac{8 \left(\theta \cos(\theta)-\sin(\theta)\right)}{\theta^2}}{\frac{-8\left(\theta \sin(\theta)+\cos(\theta)\right)}{\theta^2} }$, which …
As the student was wont to do, he idly muttered “So, that’s $\cos(10º)$…” The calculator, as calculators are wont to do when the Mathematical Ninja is around, suddenly went up in smoke. “0.985,” with a heavy implication of ‘you don’t need a calculator …
Dear Uncle Colin, If you know all of the factors of $n$, can you use that to find all of the factors of $n^2$? For example, I know that 6 has factors 1, 2, 3 and 6. Its square, 36, has the same factors, as well as 4, 9, 12, 18 and 36, but I don’t see an easy way to find them all – just …
I’m a big advocate of error logs: notebooks in which students analyse their mistakes. I recommend a three-column approach: in the first, write the question, in the second, what went wrong, and in the last, how to do it correctly. Oddly, that’s the format for this post, too. The question …
Dear Uncle Colin, I got stuck on this sector question, which asks for the radius of circle $P$, which touches sector $ABC$ as shown. I’m given that $ABC$ is a sector of a circle with centre $A$ with radius 12cm, and that angle $BAC$ is $\frac{\pi}{3}$. My answer was 3.8cm, but apparently it …
Another horde of zombies lumbered into view. “What are they saying?” asked the first, readying the shotgun as he’d done a hundred times before. “Something about the calculator exam,” said the second. “It’s hard to make out.” He pulled some spare shells …
Dear Uncle Colin, I’m struggling with a STEP question. Any ideas? Given: $q^2 - pr = -3k$ $r^2 - qp = -k$ $p^2 - rq = k$ Find p, q and r in terms of k. - Simultaneous Triple Equation Problem Hi, STEP, and thanks for your question! This is an absolute biter that took me several attempts to get …
An implicit differentiation question dealt with $y^4 - 2x^2 + 8xy^2 + 9 = 0$. Differentiating it is easy enough for a competent A-level student - but what does the curve look like? That requires a bit more thought. My usual approach to sketching a function uses a structure I call DATAS: Domain: …
Dear Uncle Colin, How would you find $\sqrt[4]{923521}$ without a calculator? -- Some Quite Recherché Technique Hi, SQRT! I have a few possible techniques here. The first is “do some clever stuff with logarithms”, the second is “do some clever stuff with known squares” and …