I was given a copy of *Galois Knot Theory* by J. Maruyama for Christmas last year. It is one of the greatest presents I have ever received, and is a book that truly highlights everything that is beautiful about mathematics. Continue reading

# Category Archives: Front page feature

Featured on the front page slideshow

# Modelling blood

From understanding the effect of aneurysms and what causes strokes to simulating and constructing artificial organs, maths has a huge role to play in developing new medical treatments. But one key part of the human physiology is the study of blood. It’s fairly obvious that blood is key to life – if you bleed too much you die. It has been studied by many eminent figures, from Aristotle who believed blood was required to transport heat around the body to Poiseuille who derived derived a simplified model of mathematical flow in a pipe to describe flow through arteries. We now understand that blood carries oxygen and essential nutrients to our cells, and carries waste products such as urea away to be processed.

# Maths trumps review

On a blustery early spring afternoon, three of the Chalkdust team gathered to test out an intriguing new product: a mathematically-themed version of the classic “my-car-is-faster-than-yours” card game, trumps. If you’ve not played trumps before, the idea is simple. Each card in a set of trumps depicts a member of a certain group and statistics about that member. Players take it in turns to read out a statistic from the top card in their hand, and the one with the highest number wins all of the cards from that round. For example, a set might be all about wild animals, and each card will show a picture of the animal along with its weight, speed, life-span etc. Your aim is to collect all of your opponents cards by choosing which statistic you will do the best in.

So it’s a game involving sets, statistics and probability… seems only natural that mathematicians might want to get involved, right? Right! We recently got our hands on some maths trumps, a new twist on the game with six different sets of cards all themed around mathematics. Read on to hear what we thought about two of the sets, “2D shapes” and the mysteriously titled “Connections”.

Continue reading

# Our favourite (and not-so-favourite) Euler equations

In previous issues of Chalkdust, we shared with you a selection of our favourite “things” in maths, such as our favourite functions, shapes and sets. On the other hand, there are also some things we find annoying and very much dislike, such as bad notation, certain numbers, etc. For this special occasion (commemorating Euler’s birthday Euler a few weeks ago), we decided to spread some of our favourite, and not-so-favourite, examples of things named after Euler throughout issue 07.

We would also like to hear yours! Send them to us at ** contact@chalkdustmagazine.com**. Continue reading

# The new Chalkdust T-shirt

If you’ve been to one of our issue 07 launch events or you’ve been keeping an eye on Twitter, you may have spotted the new Chalkdust T-shirt. If you like it, you can order one here!

Whether or not you’d seen it before, you’re probably wondering what the pattern on the T-shirt means… Wonder no more, we’re about to reveal all in this blog post. If you’d like to try to work it out yourself then stop reading now; spoilers ahead.

The pattern on the new T-shirt is a position in John Conway’s Game of Life. Life is a cellular automaton that was invented by John Conway in 1970, and popularised soon afterwards by Martin Gardner.

In Life, cells on a square grid are either alive or dead. In this post and on the T-shirt, we use white for alive cells and black (or the colour of the T-shirt) for dead cells. Life begins at generation 0 with some cells alive and some dead. The aliveness of a cell in the following generations is determined by the following rules:

- Any live cell with four or more live neighbours dies of overcrowding.
- Any live cell with one or fewer live neighbours dies of loneliness.
- Any dead cell with exactly three live neighbours comes to life.

These three simple rules leads to some surprisingly complicated behaviour.

The pattern to the left is called a glider. This is because as the generations progress it glides across the grid. You can see what I mean in the GIF below.

The T-shirt shows a position in Life. But it’s not just any old position: if you go forward one generation, you get the following:

If you like that, you can buy a T-shirt with it printed on here! You can also use this tool to write any word/phrase you like in Life.

Of course, you could continue to look at what happens to the T-shirt’s pattern after more generations. Unfortunately, not much of interest happens:

# A few of Euler’s masterpieces

Leonard Euler wrote more mathematics than anyone in history. It is said that he was responsible for around a third of all the mathematics, physics and mechanical engineering research published in all of Europe between the years 1726-1800. Much of our modern notation is due to him. He left his mark on every subject he touched. In fact, there is a whole Wikipedia page dedicated to simply listing all the things named after him. Almost everything on the list has its own Wikipedia page. Instead of attempting the impossible by trying to summarise of all of his work, we will present a few personal favourites from the world of pure maths and hope that it encourages others to read further and find their own personal favourites.

## Geometry

There’s quite a lot going on in this picture but let’s just focus on the miraculous red line in the middle known as the *Euler line*.

What’s so miraculous about it? After reading what it is, perhaps you’ll agree that it’s very *existence* is a miracle. Start with three arbitrary points $A, B, C$ and draw the triangle $ABC$. Next, construct the three perpendicular bisectors of the edges of $ABC$. These are the green lines and they all meet at a single point which we label $O$. The gold lines are the *medians* of the triangle. They are the lines through the vertices and opposing midpoints and they also meet in a single point, which we label $G$. The blue lines are constructed by dropping a perpendicular from each vertex to its opposite edge of the triangle. Again, they meet in a single point which we call $H$. It turns out, and this is what Euler proved, that no mater how the original points $A,B,C$ are arranged, the points $O,G,H$ always line up in on straight line. What’s more, the distance $GH$ is always exactly twice that of $OG$.

## Analysis

Logarithms are introduced in school nowadays as being related to exponentiation by the formulas

$$y= a^x \text{ if and only if } x = \log_a(y).$$

It was Euler who first clearly perceived logarithms in this way. Before Euler, logarithms were used by scientists and engineers to simplify calculations by converting multiplication (which was hard) into addition (which was easier). Euler recognised the significance of logarithms as mathematically interesting functions in their own right, independently of their use in calculations. He observed that $a^\delta$ is very slightly larger than 1 when $\delta$ is very slightly larger than 0. In fact, for $\delta$ positive but very small, $a^\delta \approx 1 + k \delta$ for some proportionality constant $k$ which depends on $a$. He gives the numerical examples, $a=10$, $\delta = 0.000001$ for which $k = 2.3026$ and $a=5$, $\delta = 0.000001$ for which $k = 1.60944$, and found that the number $e = \sum_{n=1}^{\infty} \frac{1}{n!} = 2.7182818284\ldots$ is exactly that number with proportionality constant $k=1$. This number $e$ is appropriately called *Euler’s number*. The power series

$$e^x = 1+x+\frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots.$$

is also due to him. Many expressions become simpler because $k=1$ and this is why it is *natural* to take logarithms to the base $e$. He also discovered the power series

$$\log_e(1+x) = x-\frac{x^2}{2} + \frac{x^3}{3} – \frac{x^4}{4} + \frac{x^5}{5} – \cdots.$$

## Number theory

*“These works are recorded to have been completed in six days $\ldots$ because six is a perfect number – not because God required a protracted time, as if He could not at once create all things,$\ldots$ but because the perfection of the works was signified by the number six. For the number six is the first which is made up of its own parts, i.e., of its sixth, third and half, which are respectively one, two and three, and which make a total is six.”*

This is an excerpt from St Augustine’s City of God (Part XI Chapter 30) explaining that God created the world in six days because six is the first *perfect* number. A perfect number is a positive integer which is equal to the sum of all its proper divisors, so excluding the number itself. For example, 6 is perfect because its proper divisors are 1, 2 and 3, and 1+2+3 = 6. The next smallest perfect numbers are 28, 496 and 8128. Perfect numbers have been entertaining the imaginations of mathematicians and non-mathematicians alike for literally thousands of years. In fact, as far back as c. 300 BC, Euclid proved in book IX of his *Elements* that * if* $n = 2^p(2^p-1)$ where $p$ and $2^p-1$ are both prime numbers

*$n$ is a perfect number. A prime number of the form $2^p-1$ is known as a Mersenne prime. Although it had been conjectured previously, it wasn’t until Euler worked on the problem (around 2000 years later!) that someone finally succeeded in proving that*

**then****$n$ is an even perfect number**

*if**$n=2^p(2^p-1)$ where $p$ and $2^p-1$ are both prime numbers. This result, now called the “Euclid-Euler Theorem”, establishes a strikingly curious one-to-one correspondence between even perfect numbers and Mersenne primes.*

**then**## Analytic number theory

*“The remarks I have decided to present here refer generally to that kind of series*

*which are absolutely different from the ones usually considered till now.”*

Euler wrote a lot of his work in Latin. The quotation above is the first sentence from a paper he wrote whose title translates into English as “Several Remarks on Infinite Series”. Theorem 7 of that paper is the following enigmatic identity

$$1 + \frac{1}{2} + \frac{1}{3}+ \frac{1}{4}+ \frac{1}{5}+ \frac{1}{6} + \cdots = \frac{2\cdot3\cdot5\cdot7\cdot11\cdot13\cdot17\cdot19\cdots}{1\cdot2\cdot4\cdot6\cdot10\cdot12\cdot16\cdot18\cdots} $$

where the numerator of the right hand side is the product of all the prime numbers and the denominator is the product of all the numbers 1 less than a prime. It is well known that the left hand side of this identity diverges to infinity. In fact, in a different work, Euler proved the stronger statement

$$\lim_{n \rightarrow \infty}\left(-\log n + \sum_{k=1}^{n}\frac{1}{k}\right) = \gamma$$

where $\gamma = 0.57721\ldots$ is the Euler–Mascheroni constant. It follows straight away from the fact that the left hand side diverges that there are infinitely many primes – because the right hand side cannot be a finite product. Euler didn’t stop there though. He used his product formula to prove the much more impressive result that

$$\frac{1}{2} + \frac{1}{3} +\frac{1}{5} + \frac{1}{7} +\frac{1}{11} +\frac{1}{13} + \frac{1}{17} + \cdots = \infty.$$

## Combinatorics

This next one is truly astonishing – both the statement and Euler’s proof. It concerns the number of different ways of expressing a positive integer as a sum of other positive integers. For example, there are 15 ways of expressing 7 like this and they are

\begin{align*}

&1+1+1+1+1+1+1+1, \:\: 1+1+1+1+1+2, \:\: 1+1+1+1+3, \\

&1+1+1+2+2, \:\: 1+1+1+4, \:\: 1+1+2+3, \:\: 1+2+2+2, \:\: 1+1+5, \\

&1+2+4, \:\: 1+3+3, \:\: 2+2+3, \:\: 1+6, \:\: 2+5, \:\: 3+4, \: \text{ and } \:7.

\end{align*}

The thing to notice is that there are exactly 5 ways where all the numbers are odd and also exactly 5 ways in which there are no repeats. This is no accident. In fact Euler proved that this always happens.

The number of ways of expressing a given number as a sum of **distinct** positive integers is the same as the number of ways of expressing it a sum of **odd** positive integers.

It’s hard to believe this at first since it seems like it has no right to be true, but it is. In order to better appreciate Euler’s ingenious proof, it is worth trying to imagine how helpless you would feel if you were asked to show this in an exam. Euler’s proof is as shocking as the statement.

He starts by noticing that the number of ways of writing $n$ as a sum of distinct positive integers is precisely the coefficient of $x^n$ in the expression $(1+x)(1+x^2)(1+x^3)(1+x^4)\cdots.$ Next, manipulate this infinite product to get

\begin{align*}

(1+x)(1+x^2)(1+x^3)(1+x^4)\cdots &= \frac{(1-x^2)(1-x^4)(1-x^6)(1-x^8)\cdots}{(1-x)(1-x^2)(1-x^3)(1-x^4)\cdots} \\

&=\frac{1}{(1-x)(1-x^3)(1-x^5)(1-x^7)\cdots}

\end{align*}

and expand using the formula for a geometric series to get that this is equal to

$$(1+x+x^2+\cdots )(1+x^3+x^6 + \cdots )(1+x^5+x^{10}+\cdots )(1+x^7+x^{14}+\cdots )\cdots.$$

Now finish by recognising the coefficient of $x^n$ in this last expression as being exactly the number of ways of writing $n$ as a sum of positive odd integers, where now we allow repeats.

## Infinite series

Jakob Bernoulli’s 1689 *Tractatus de seriebus infinitis* was a state-of-the-art account of infinite series, as they were understood at the time. It included results like the fact that the harmonic series $\sum_{n=1}^{\infty}\frac{1}{n}$ diverges and explicitly evaluated a number of convergent series. For example, the geometric series $\sum_{n=1}^{\infty}a^n = \frac{1}{1-a}$ for $|a|<1$, the sum of the reciprocals of the triangular numbers,

$$1 + \frac{1}{3} + \frac{1}{6} +\frac{1}{10} + \frac{1}{15} + \cdots = \sum_{n=1}^{\infty}\frac{1}{n(n+1)} = 2,$$

and others like $\sum_{n=1}^{\infty}\frac{n^2}{2^n} = 6$ and $\sum_{n=1}^{\infty}\frac{n^3}{2^n} = 26$ were all known at the time. At some point Jakob decided to think about $\sum_{n=1}^{\infty}\frac{1}{n^2}.$ He knew that it converged but tried and failed, as did a number of others, to evaluate it explicitly. Concerning this sum, the *Tractatus* included the line

*“If anyone finds and communicates to us that which thus far has eluded our efforts, great will be our gratitude.”*

Euler rose to the challenge in spectacular fashion by showing that

$$1+\frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \frac{1}{5^2} + \frac{1}{6^2} + \cdots = \frac{\pi^2}{6}.$$

His original argument, although not entirely justified at the time, is undoubtedly the work of a genius. He reasoned as follows. Just like polynomials can be factored according to their roots, Euler factorised $\frac{\sin x}{x}$ according to it’s (infinitely many!) roots, which are $\pm n \pi$ for $n = 1, 2, 3, \ldots$, as

$$\frac{\sin x}{x} = \prod_{n=1}^{\infty}\left(1-\frac{x}{n\pi}\right)\left(1+\frac{x}{n\pi}\right) = \prod_{n=1}^{\infty}\left(1-\frac{x^2}{n^2\pi^2}\right).$$

The power series expansion

$$\frac{\sin x}{x} = 1 – \frac{x^2}{3!} + \frac{x^4}{5!} – \frac{x^6}{7!} – \cdots$$

was well known to Euler. To evaluate the sum in question, it just remains to expand the infinite product and compare coefficients of $x^2$ in these two representations of $\frac{\sin x}{x}.$

#### Further reading

To learn more about Euler’s mathematics the following resources are highly recommended.

- W. Dunham,
*Euler Master of Us All*. This is an excellent book that explains in much more detail some of things written about here. It nicely puts Euler’s contributions into context by outlining the state of knowledge before Euler and explaining what later work it inspired. - P. Nahin,
*Dr Euler’s Fabulous formula: Cures Many Mathematical Ills*. This whole book is dedicated to Euler’s formula $e^{i \theta } = \cos \theta + i \sin \theta$. It’s packed full of wonderful identities and important applications. - eulerarchive.maa.org is a website that describes itself as “A digital library dedicated to the work and life of Leonard Euler”. Amongst other things, it contains links to a huge number of his original papers, many of which have been translated into English.

# Let them share cake

There comes a point in every person’s life where they have to learn how to share fairly. Admittedly some people seem to ignore this point, sailing on through life gleefully seizing more than would be justified, but we’re willing to bet that the situation of having to divide up a resource (for example some food or a list of chores) into parts that everybody is happy with is pretty much universal.

If there are just two people who want to split the resource, then there is a simple method to ensure that it is divided fairly. This concept (called the “I cut, you choose” method) is so old that it’s even mentioned in the Bible. As the name suggests, the method involves one person splitting the resource into what they consider to be equal halves, and then the other person picking which (if any) of the pieces they think is worth more. The person who chooses is bound to be happy, and the person who cut can’t complain since they were supposed to divide the resource into equal pieces. The solution for two people, then, is so simple that it doesn’t seem like mathematics at all. However, the problem becomes significantly harder once you start to include more people — so difficult, in fact, that a completely ‘satisfactory’ answer for an arbitrary number of sharers was not found until 2016… Continue reading

# Magic behind the Fibonacci sequence

In mathematics, there are countless sequences such as arithmetic sequences, geometric sequences, and many more. The Fibonacci sequence is one of them, but it is different from other sequences in that it can be easily found in everyday life. Let’s take a look at patterns that can be discovered in Fibonacci numbers and how we can find them around us.

In a Fibonacci sequence, every number after the first two numbers is the sum of the two preceding ones.

0, 1, 1, 2, 3, 5, 8, 13,…

# Read Issue 07 now!

Our latest edition, Issue 07, is available now. Enjoy the articles online or scroll down to view the magazine as a PDF.

## Features

### Game, set, maths (no more tennis puns)

No more Katie Steckles.### Thinking outside outside the box

Rob Eastaway joins the dots.### Baking a Menger sponge sponge

Sam Hartburn bakes your favourite fractal### e to thee x

Zoe Griffiths on the life of e### Adventures in Chance-ylvania

High stakes gambling with Paula Rowińska### Counting palindromes

Alex Xela shows us the world of palindromic numbers, and calculates the chances of getting one### Routes: Edsger Dijkstra

In this edition of the series, we instead learn about 'routes' and Edsger Dijkstra's shortest path algorithm.### On the cover: Chladni figures of a square drum

Find out more about the patterns on the cover of Issue 07### Significant figures: Sir Christopher Zeeman

Biography of Sir Christopher Zeeman### Primes à la Euler

Infinitely many primes ending in 1, 3, 7 and 9 proved in typically Eulerian style.### Machine learning 101

Start your quest to conquer the planet with this introduction to the wonderful world of machine learning### Mike Jordan on machine learning

Undoubtedly the most influential voice on this hottest of hot topics.### Domino tiling & domineering

John Dore and Chris Woodcock join the dots### In conversation with Chris Budd

We chat to one of the UK's most qualified voices in mathematics communication

## Fun

### Dear Dirichlet, Issue 07

Pigs, popes and produce are among the topics of discussion in this issue's Dear Dirichlet advice column### Prize crossnumber, Issue 07

Win £100 of Maths Gear goodies by solving our famously fiendish crossnumber### Comic: The Inverse Homotopy, part 6

Part 6 of our mathematical comic's adventure### Top Ten: Mathematical celebration days

The definitive chart of the best dates### Conference bingo

Race to be the first to get five-in-a-row### How to make: a catastrophe machine

Make a catastrophe machine just like Zeeman's### Page 3 model: Frictional unemployment

Modelling unemployment using simple differential equations!### What’s hot and what’s not, issue 07

Fashion is fleeting, Chalkdust regulars are not.

## Printed versions

or

# Game, set, maths (no more tennis puns)

A while ago, my friends and I learned of a brilliant and simple game, played by comedians Alex Horne and Tim Key. We

discovered a clip of them playing the game backstage at a show, and were immediately hooked. It’s a naming game, requiring a vague knowledge of a bunch of famous people, and anyone can play. The game is called No More Women (reasons for which will become clear later) and I’m taking the opportunity both to share the rules of the game with you, and explore a little of the maths behind it.

#### No More Women

Each move in the game consists of two phases. First, you name a celebrity (for example, “Simon Singh”). Then, you exclude a category of people—and primarily, the category you give must include the person you’ve just named (for example, “no more authors”). Play then passes to the next person, who must name another celebrity—but crucially, they can’t name anyone who falls in a category that’s already been excluded. Play continues until someone’s stated celebrity is proven disallowed, or they can’t think of someone.

There are a few other rules—for example, you’re not allowed to disallow everyone. There must always be at least one person left in the remaining chunk of people that are allowed—for example, having heard the move described previously, you wouldn’t be allowed to say “Courteney Cox; no more people who aren’t authors”—as the two categories, ‘authors’ and ‘non-authors’, together make up the whole set of people.

If the people you’re playing with suspect that your most recent exclusion has made the next move impossible, they may challenge—and you’d have to name someone you can think of who’s still allowed. Challenging like this is a risky move, as you’re then not allowed to say the example just given, and you have to think of yet another one.

It’s also generally agreed that the people you’re naming should be famous people—as in, people everyone might reasonably be expected to have heard of. If you name someone that nobody has ever heard of, your fellow players can insist you think of someone else, or decide

whether a quick cheeky online search is allowed, to verify that they are indeed a tennis player (or whatever you’ve claimed them to be). It’s a matter of opinion whether someone who’s not a real celebrity but is known to everyone in the group playing should be allowed, although this does make it a bit more fun if you allow it.

Also, the properties you give should be ones which people might reasonably be expected to know the truth value of for the majority of people you might name: while it does exclude a sensible proportion of people, “no more June birthdays” is harsh unless you are prepared to let people look up every single name then given to find when their birthday is—and that kind of thing leads to cheating. Impromptu discussion can also break out, for example around what exactly constitutes an author—does it have to be someone who makes a living writing books, or does a presenter who’s written an autobiography count as an author, since they have a published work?

The title of the game is obviously a nod to one massive move you could make in the game, excluding basically half of all people (although is it half of all famous people? Smash the patriarchy etc). As the game continues, the set of people you’re allowed to name gets

increasingly small, and the game becomes more difficult. The rate of increase in difficulty depends on how cruel your fellow players are with category choices—“no more brunettes” is much harsher than “no more people from Bristol”, but if that’s the only fact you know for sure about your given celeb, you’ll have to go with that.

Particularly mean moves include “no more currently living people” and “no more people whose first and second name begin with different letters”. The trivial move is to say, for example, “Pat Sharp; no more people who used to present the kids’ TV show Fun House”, excluding precisely one celebrity (this is valid, but boring).

This is an excellent game to play when you’re all, for example, sitting around on public transport, or somewhere you don’t have access to conventional game-playing equipment and/or tables. Part of the challenge is remembering all the categories that have been counted out already, although if you prefer you could write down the categories on a board or piece of paper everyone can see as they’re named, and be real sticklers. But is writing down the names in a list the best way?

#### No more lists

A more interesting way of writing down the moves could make the game more concrete and help us understand it mathematically. Since each move divides the space cleanly (assuming the definition of your category is precise enough), you could imagine the people all on a page, then draw a circle around a set of people and everyone inside the circle could be in the category, and vice versa. This leads us to a natural way to describe moves in the game—using Venn diagrams.

Everybody knows Venn diagrams—the overlapping circles of categories described by John Venn in 1880, now a staple of set theory and a lovely way to visualise sets of things with certain properties. In this game, by definition, every person falls neatly in or out of a set. For example, you could have “no more authors” which would exclude everyone inside the set ‘authors’ . In terms of Venn diagrams, the nicest way to display this would be to call the outside of the circle ‘authors’ and the inside ‘non-authors’, so the circle contains all the people still allowed (see diagram). By our rules, there must be at least one person outside the circle (in particular, the one you just named on your turn), and one person inside the circle (which you’d need to be able to name if challenged).

A second move (eg “no more brunettes”—agreement will have to be reached whether you count natural brunettes only) could then overlap with this, restricting the next move to those in the intersection of those two circles. You’d also need at least one person to be in the region you’d just excluded (the non-author brunette you just named) and at least one still allowed (one non-author non-brunette you can name if challenged). However, it’s not necessary for there to be a non-brunette author (no offence, JK Rowling), as that category has already been excluded, so we don’t care what happens there.

Each move of the game adds another circle to this diagram—it’s easy enough to draw a third circle to create a three-way Venn diagram. Let’s say, for example, someone says “no more politicians”; this would overlap with all four existing categories (brunette authors, in the outer region; brunette non-authors; non-brunette authors; and non-brunette non-authors, which is where allowable moves still lie). You’d need to have named a non-brunette non-author politician as your turn (we went with Charles Kennedy, confirmed redhead) and you should also be able to name a non-brunette non-author non-politician (such as Lupita Nyong’o, who was in *The Jungle Book*, but hasn’t published one *yet*).

A four-way Venn diagram using only circles is not possible—in the example shown on the left below, not all sets of intersections are present. For example, there would be nowhere to place Steven Tyler (a brunette non-author who’s tall and not a politician). However, a four-way diagram can be constructed using ellipses (shown on the right), and was designed by John Venn himself.

It is also possible to construct Venn diagrams with five identical pieces—the one on the left, devised by Branko Grünbaum has rotational symmetry. The five-way Venn diagram shown next to it is called an Edwards–Venn diagram, and was devised by Anthony Edwards. They represent a way to create Venn diagrams with arbitrarily many regions, using sections of a sphere projected back down onto a plane, and were devised while designing a stained-glass window in memory of Venn. Starting with the left and top hemispheres, given by the two rectangles, and the front hemisphere represented by the circle, the next set is the shape made by the seam on a tennis ball (winding up and down around the equator) and further sets are made by doubling the number of oscillations on subsequent winding lines. They’re sometimes called cogwheel diagrams, due to the shape.

Technically any number of segments is possible, although the diagrams get much more complex as you go on. There’s a lovely interactive seven-way Venn diagram by Santiago Ortiz (on his website: moebio.com/research/sevensets—neither he nor I am able to determine the original author of this shape). Of course, this would still only get you seven moves into a game of No More Women, and representing a full game in diagram form would be challenging and likely uninformative.

#### No more Venn diagrams

Another way to interpret the game’s structure would be to use half-planes. If you could theoretically arrange everyone in the (so far useful) 2D plane of all celebrities—presumably a private jet—then a category excluded could be represented by a line cutting the plane in half, with all the allowed persons on one side and the disallowed persons on the other side.

For example, the line $x=2$ describes a vertical line on an $x y$-plane running through the point $2$ on the $x$-axis, and everyone to the left of the line might be an author and everyone to the right not an author. Then, a second line at $x + y = 1$ might cut diagonally across, with brunettes above and non-brunettes below. It reminds me slightly of the loci problems we used to play with at school—a way to visualise solutions to sets of linear equations, or to determine which bits of a field a tethered goat can reach (for some reason, it’s always a goat).

We can intersect arbitrarily many half-planes to define the space of allowed people. In fact, any convex set can be described as an intersection of half-planes. But this is probably not useful either—firstly, the sections will become arbitrarily tiny and hard to see,

much as in the Venn diagram case; secondly, you’re kind of imagining the people all standing in the room (or on a very, very big plane) and each time you define a new line you’d need everyone to be miraculously standing on the correct side of it. This is leading me to imagine celebrities looking upwards, then sprinting across so they’re on the correct side before a giant imaginary looming line comes crashing down, which is fun to picture, but not hugely helpful. And finally, the lines are defined in a pretty arbitrary way—the categories we’re using are not quantitative (unless you’re using a category like “no more under-25s”, in which case you can plot that on an axis), and otherwise there’s no natural way to assign an equation to a category, so it’s a bit unsatisfying.

#### No more diagrams

So I guess we’ll have to fall back on classic set theory. Developed by Cantor, while he was attempting to work out a way to compare the magnitude of different infinite sets, the theory of sets underpins a huge amount of mathematical rigour and thinking, and contains parallels with algebra and logic in ways that illustrate the beauty of mathematics in its purest form. But we just want to play a stupid game about famous people, so here goes.

In set notation, we might define:

$$\text{Authors} = \{ \text{Simon Singh}, \text{Stephen King}, \text{JK Rowling}, … \}$$

The set is specified by the list of things in brackets, so this set equals this collection of people. Then our game would consist of moves as follows:

$$\text{Simon Singh}\in\text{Authors}$$

$$\overline{\text{Authors}} \neq \varnothing$$

Here the $\in$ symbol means ‘is an element of the set’. We use a line above the name of the set to mean ‘the complement of this set’, or the set of all things in the universe that aren’t in this set. In the Venn diagrams we defined earlier, $\overline{\text{Authors}}$ would be the inside of the ‘no more authors’ circle, and the set $\text{Authors}$ would be everything outside this circle.

We’ve also specified, in the second line, that the set $\overline{\text{Authors}}$—the set of all non-authors—is not equal to $\varnothing$, where this symbol denotes the empty set. This means that set is not empty, because something exists in it.

Play continues:

$$\text{Courteney Cox}\in\text{Brunettes}\cap \overline{\text{Authors}}$$

$$\overline{\text{Brunettes}} \cap \overline{\text{Authors}} \neq \varnothing$$

Here we’ve used the $\cap$ symbol to define an intersection—this is the set of all things that occur in both the given sets—here, brunettes who are also not authors.

Along with the $\cup$ symbol for a union (the set of all things in either or both sets), this is one of the two main operations you can do with sets, and they correspond respectively to the AND and OR operators in Boolean logic, in a rough sense. On the Venn diagram, the intersection is the part of those two sets which overlaps—here, Brunettes is the outside of one circle, and $\overline{\text{Authors}}$ is the inside of another circle, so this is the portion of the non-Authors circle that doesn’t overlap with the non-Brunettes circle.

Again, the requirement of being able to name an example means that the set of non-brunette non-authors now has to definitely not be empty, and must contain something.

$$\text{Ed Miliband}\in\text{Politicians}\cap \overline{\text{Brunettes}} \cap \overline{\text{Authors}}$$

$$\overline{\text{Politicians}} \cap \overline{\text{Brunettes}} \cap \overline{\text{Authors}} \neq \varnothing$$

And so on. Sets can intersect arbitrarily, and while this doesn’t necessarily give us a nice visual way to imagine the celebrities, it

does give us a formal structure. Properties of set intersections can be considered as they apply to the game—for example, set

intersection is commutative:

$$A \cap B = B \cap A.$$

This means it’s independent of ordering, which feels obvious—brown-haired women are women with brown hair, duh—but in mathematics you have to be careful whether the order in which you do things matters (multiplying by three then adding four is different to adding four then multiplying by three, and often on Facebook only a real genius can work out the correct answer to this math problem, so keep your wits about you).

Intersection of sets is also associative:

$$A \cap B \cap C = (A \cap B) \cap C = A \cap (B \cap C).$$

Intersecting three sets is the same as first intersecting two of them, then intersecting the result with the third one. Because the action of intersecting sets is associative, it doesn’t matter which order you do this in, you’ll get the same result. Someone who’s a brunette author and also a philanthropist could also be considered to be an author/philanthropist with brown hair.

These two properties together mean that if you’re playing the game, and the categories defined are given in a different order—say, you’re playing against two others and they each give a specific category—they could occur in either order and still leave you with the same challenge afterwards. Non-brunette non-authors are just as hard to think of, and just as numerous, as non-author non-brunettes. It will, however, affect the choice of named celebrity each of the two other players is allowed to give.

There are other properties of sets you could think about in terms of gameplay—for example:

$$\overline{A \cup B} = \overline{A} \cap \overline{B}.$$

This says, the complement of the union of two sets is the same as the intersection of the complements. This is shown in Venn form to the right,

and it essentially means that if you consider the complement of each set (the things outside it) individually, the things they have in common will be the same as just the things that are in the complement of the union of these two sets—consider them together as an overlapping shape, and look outside of that.

In the context of the game, you can consider this to mean that if you’re looking for someone who’s not a brunette AND not an author (because of two successive turns that have occurred in the game), you need to think about the set of people who are either a brunette OR an author, and look for someone not in that set. Again, this feels obvious when you think about it in the context of naming celebrities, but the set theory confirms it.

Maybe formalising the game in this way will help you to get your head around the ideas of sets and set theory, if you’ve only recently encountered it. As someone who studied it—*cough*—years ago, it’s become part of my way of picturing any kind of problem like this, and a natural language with which to describe intersecting sets. I’ve been trained through years of maths to take any fun thing and abstract it into some symbols on a page. Yay!

#### No more No More Women

Of course, the mathematician’s real job is abstraction then generalisation—so this game could be played using any well-known set of things that can be categorised according to properties, and we have piloted among ourselves a much more niche version of the game, provisionally titled No More Integers

Played on the set of all numbers, starting with the complex plane, it’s basically a free-for-all and in general we’ve found it becomes very difficult very quickly, not just to name a number that hasn’t been excluded, but to think of a category that works and doesn’t just make the game totally boring and impossible. If non-integers are excluded early on, and then numbers with given factors start getting thrown out, it can get a bit like the coding/drinking/coding-while-drinking game Fizz Buzz.

It’s also totally possible for someone to say something that’s wrong/has already been excluded but nobody notices because their brains are all fried from factorising numbers in their head. It might take a few plays through before you all learn the best way not to end up in a mass brawl with whomever excluded the primes, but hopefully you can find some enjoyment in it. Or just play the version with celebrities—it’s a lot of fun.