# Chalkdust Issue 05 coming 7 March

We are pleased to announce that Chalkdust Issue 05 will be released on Tuesday 7 March. To celebrate this, we will be holding a free launch event on Tuesday 7 March in the Print Room Café at UCL from 7.15pm. We’d love to meet more of our readers, so please come along and invite your friends. If you can’t make the event but you’d still like to get your hands on some copies of Chalkdust, please see below for more information.

# To share, or not to share

The Montagues and the Capulets have never been friends and Juliet is quite aware of this. Even at her young age, she knows that her love for Romeo is an impossible dream her father will never accept. So, she designs a strategic plan. She will take a couple of sleeping pills, just enough to make her look like she is dead to trick everyone into thinking that she has passed away. Brilliant! If everything goes right, she will always be happy with Romeo… but if things go wrong… well, you never know. Continue reading

# Should you buy a Valentine’s day present?

Valentine’s day is just around the corner and you are still not sure whether or not you should buy your beloved one a present.  It’s a tough call. Should you spend money on buying your partner some chocolates and a teddy bear (that no one wants anyway), or will you risk it and bring only a charming smile to your romantic dinner? Continue reading

# Curiouser and curiouser!

As many of you will undoubtedly know, mathematics underpins much of our everyday life, in aspects such as love and warfare, to ancient creatures and Mean Girls: areas which have been previously explored in past articles.  But what makes mathematics so beautiful is that it allows us to solve problems, both simple and complex. Some of these problems may initially seem counter-intuitive, or not at all obvious. But by expressing them in mathematics, their true nature/solution can be revealed.

## Birthday problem

A famous one to play at parties! In a group of 23 people, there is approximately a 50% chance that two people will share the same birthday, and a 99.9% chance with 70 people. But, to get 100%, if we include pesky leap years, we need 367.

## Monty Hall problem

This (slightly paraphrased) problem is as follows: You’re on a game show, and you have a choice of three doors. Behind two doors are deadly scorpions, but behind the other door is a Chalkdust T-shirt! You pick a door, say No 1, and the host, who knows what’s behind the doors, opens another door, say No 2, containing an evil scorpion. He then proposes to you: “Do you want to pick door No 3?” How do you increase your chances of winning that awesome T-shirt?

## The curious tale of the accountant

This problem was actually initially presented to me only a week ago during a maths circle here at UCL. Throughout the year 2016, the accountant noticed that in any five consecutive months, his income was less than his expenses. But overall, his income was more than his expenses. How can this be? A small hint:

$12\mod5 \not \equiv 0$

## Rubik’s Cube

We all can remember spending countless hours trying to solve the pesky Rubik’s Cube, with most of us giving up in frustration and going on to solve simpler puzzles (or eat pizza). But in fact, all positions of the Rubik’s Cube can be solved in 20 positions (or less!).

And while I’m sure that that $\exists$ many more curious problems, half the fun of mathematics is discovering them yourself (and then sharing them)! If you have any curious problems to share tweet us @chalkdustmag and you might even be featured in future articles!

Question Marks: Flickr user Valerie Everett, CC BY-SA 2.0
Birthday cakes: Flickr user Felix, CC BY-SA 2.0
Money: Flickr user 401(K) 2012, CC BY-SA 2.0
Rubik’s Cube: Flickr user Sonny Abesamis, CC BY-SA 2.0
Rubik’s Cube is a registered trademark of Seven Towns, Ltd.

# Two years of Chalkdust!

Chalkdust has made it through its second year! Yep. 731 days ago (that’s right, 2016 was a leap year) we got together with the idea of creating something, we weren’t sure what, to share maths. That’s it. And now, one million minutes later, we are celebrating our second anniversary and are in the process of creating our fifth issue.

# The limit does not exist!

What’s more fetch than toaster strüdel, foot cream in a jar and a fertility vase from the Ndebele tribe? Arguably maths. More specifically, the mathematics of Mean Girls. If you haven’t seen Mean Girls, it’s a not-too-bad movie about teenage girls, teenage boys, teenage drama, and (a little bit of) maths.

Stop trying to make fetch happen. It’s not going to happen.

It’s not exactly October 3rd*, but its been 13 years since Mean Girls came out, and

• $13 \times 2=26$
• $26+2\times2017=4060$
• $4060-3010=1050$, and
• $1050\log(2)+\frac{\pi}{1423211345}$ is… just a number.

What we’ll be looking at in this article is the last 10 minutes of the movie, where Cady and the Mathletes from North Shore compete in a maths state championship. After 87 minutes of play it’s a tie, and so the two teams enter the sudden death round, where the teams choose a member of the opposition (happening to both be female) to battle for the trophy.

T. Pak as in ‘Trang Pak the grotsky little byotch’? No, just her brother.

In one corner, Cady Heron, AKA ‘Africa girl’, AKA the ‘used-to-be maths geek but not so much anymore after she met the Plastics but is kind of a geek again because her attempted sabotage of Regina George backfired and descended into chaos’. She is also the protagonist of the movie.

In the other, is Caroline Krafft, the female member of Marymount Prep, who “seriously needed to pluck her eyebrows, whose skirt looks like it was picked out by a blind Sunday school teacher, and had some 99-cent lipgloss on her snaggletooth”. This was when I realised, making fun of Caroline Krafft wasn’t going make me any better at writing this article (or maths for that matter).

The two were asked to solve the following;

Caroline Krafft hastily states the incorrect answer, $-1$. If Cady manages to state the correct one then the North Shore mathletes ultimately win.

In the midst of all this Cady then has a miraculous flashback to the week that Aaron got his hair cut, and as she takes a moment to see straight pass Aaron’s face and onto the board behind, then has an epiphany (the kind we would like to have in our exams sometimes) and quickly realises the function diverges.

Aaron Samuels is now a spin class instructor, so he won’t be expected to know what a factorial is anymore. Thank God.

So we don’t exactly know how Cady did this in under 10 seconds, but we can try to figure out the question for ourselves.

So the question reads$$\lim_{x\to 0} \frac{\ln(1-x)-\sin x} {1-\cos^2 x}.$$

We’ll consider L’Hôpital’s Rule, seeing as its the most straightforward way of going about it.

If we substitute $x$ with 0 we will find$$\frac{\ln(1)-\sin(0)} {1-\cos^2(0)} = \frac{0}{0}.$$
This satisfies the criteria for being able to apply L’Hôpital’s Rule, which is as follows,$$\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}.$$

Let’s apply it to our function, starting by considering the limit from the right;

\begin{align}
\lim_{x\to 0+} \frac{\frac{-1}{1-x}-\cos x} {\sin 2x} &= \lim_{x\to 0+} \frac{-x \cos x + \cos x + 1}{ (x-1) \sin 2x} \\
&= \lim_{x\to 0+} [-x \cos x + \cos x + 1] \lim_{x\to 0+}\left[\frac{1}{x-1}\right] \lim_{x\to 0+}\left[\frac{1}{\sin 2x}\right].
\end{align}
Now,\begin{align}
\lim_{x\to 0+} \left[-x\cos x +\cos x +1 \right] &=2, \\
\lim_{x\to 0+} \left[\frac{1}{\sin 2x } \right] &= \infty, \\
\lim_{x\to 0+} \left[\frac{1}{x-1} \right] &= -1,
\end{align}
so by properties of infinity we get
$$\lim_{x\to 0+} [-x \cos x + \cos x + 1] \lim_{x\to 0+}\left[\frac{1}{x-1}\right] \lim_{x\to 0+}\left[\frac{1}{\sin 2x}\right] = 2 \cdot \infty \cdot -1 = -\infty.$$

Repeating this method for the limit from the left, we will obtain the following,
$$\lim_{x\to 0-} \frac{-x \cos x + \cos x + 1}{ (x-1) \sin 2x} = +\infty.$$

Just checking.

And so since the limit from the right and the left differ, the whole thing diverges. Yay.

Another approach would be to look at the Taylor series approximations for the separate terms at $x$ near 0. We won’t do that though since it’s not as mathsy, but it might have been the quicker way to go, on the assumption that Cady really knows her Taylor series expansions. But

“Cause the next time you see her she’ll be like, Ohh Kevin G!”

CC-BY-SA: Mathletes, Matty McRib. Mean Girls, Flickr. Screenshots copyright of Paramount Pictures.

# Review: Mathematical socks

This Christmas, I received mathematical socks. A great gift, you might think. But is it good maths or fake maths? Can you wear them and be taken mathematically seriously? Thus I have undertaken this important review.

## Unboxing

The socks come beautifully packaged and folded, tied together with a fancy red label, which gives a nifty standing suggestion.

Beautifully packed mathematical socks