# November, 2020

## A random number puzzle

A puzzle that crossed my path via @drmaciver: player 1 pushes the button twice, and multiplies the two outputs together to get a score (e.g. 0.45 x 0.9=0.4). then player 2 pushes the button once, and squares the result to get their score (e.g. 0.67 x 0.67 = 0.4489) the

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

## A proof without words

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. $\blacksquare$

## Ask Uncle Colin: Angles and roots

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 Headaches Evaluating This Angle Hi, THETA, and thanks for your message! My third thought

## The Mathematical Ninja and the Cube Root of 81

“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

## Ask Uncle Colin: A Calculator Error

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 is slightly unusual in that it

## Continued fractions and the square root of 3.

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

## Ask Uncle Colin: A Seemingly Undefined Integral

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

## Dictionary of Mathematical Eponymy: Wahba’s Problem

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: Given