# July, 2018

## A probability puzzle

A nice prompt from @shahlock, some time ago: Math Prompt #apstats #mtbosTwo players A, B. A is 4-0 against B. How would you estimate probability A wins next match? Assume independence — M Shah (@shahlock) November 27, 2016 Stand back, everyone: I'm going to apply Bayes's Theorem. A prior Let's

## Ask Uncle Colin: Stuck on some trig

Dear Uncle Colin, I'm trying to solve $2\cos(3x)-3\sin(3x)=-1$ (for $0\le \theta \lt 90º$) but I keep getting stuck and/or confused! What do you recommend? - Losing Angles, Getting Ridiculous Answers, Nasty Geometric Equation Hi, LAGRANGE, and thank you for your message! There are a couple of ways to approach this:

## Can you find a centre and angle of rotation without any construction?

Some time ago, I had a message from someone who - somewhat oddly - wanted to find a centre of rotation (with an unknown angle) without constructing any bisectors. (Obviously, if it was a right-angle rotation, they could use the set-square trick; if it was a half-turn, the centre of

## Ask Uncle Colin: inverses and all sorts

Dear Uncle Colin, I'm stuck on a trigonometry question: find $\cos\br{\frac{1}{2}\arcsin\br{\frac{15}{17}}}$. Any bright ideas? - Any Rules Calculating Some Inverse Notation? Hi, ARCSIN, and thanks for your message! That's a nasty one! Let's start by thinking of a triangle with an angle of $\arcsin\br{\frac{15}{17}}$ - the opposite side is 15

## Wrong, But Useful: Episode 58

In this episode of Wrong, But Useful, we are joined by freelance mathematician @becky_k_warren, formerly of NRICH Becky likes sharing maths with people who "don't like maths" and the #beingmathematical twitter chat Number of the podcast: 157, which is the middle of a sexy prime triplet. Colin goes all Rees-Mogg

## Why the factor and remainder theorem work

So there I was, merrily teaching the factor and remainder theorems, and my student asked me one of my favourite questions: "I accept that the method works, but why does it?" (I like that kind of question because it makes me think on my feet in class, and that makes

Dear Uncle Colin, I had to find the $n$th term of a quadratic sequence (1, 6, 17, 34, 57). I remember my teacher saying something about a table, but I couldn't figure it out. Can you help? Struggles Expressing Quadratics Using Educator's Notation - Concrete Explanation? Hi, SEQUENCE, and thank

## Two coins, one fair, one biased

When the redoubtable @cuttheknotmath (Alexander Bogomolny) poses the following question: Two Coins: One Fair, one Biased https://t.co/Rz2zR3LRDj #FigureThat #math #probability pic.twitter.com/HHhnyGjhkq — Alexander Bogomolny (@CutTheKnotMath) March 5, 2018 ... you know there must be Something Up. Surely (the naive reader thinks) the one with two heads out of three is

## Ask Uncle Colin: A Coin Toss Conundrum

Dear Uncle Colin, Please can you settle an argument? I say, if you toss a coin three times, the probability of getting all heads is one in four, because the only possibilities are HHH, HHT, HTT and TTT. My friend says it's one in eight, being $\frac{1}{2}\times \frac{1}{2} \times \frac{1}{2}$.

## A calculator puzzle

"Your calculator has broken, leaving you with only the buttons for $\sin$, $\cos$, $\tan$ and their inverses, the equals button and the 0 that starts on the screen. Show that you can still produce any positive rational number." When this showed up on Reddit, I knew I was in for