Game, set, maths (no more tennis puns)

No more Katie Steckles.

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Image: Flickr user Spiralz, CC BY 2.0

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

“Simon Singh, no more authors.”
Richard Cooper, CC BY-SA 3.0

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.

“Courteney Cox, no more actors.”
Alan Light, CC BY 2.0

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?

“John Venn, no more logicians.”

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.

 

Two attempts to draw a four-way Venn diagram. In the attempt on the left, you can dream on if you think there’s a space for Steven Tyler.

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.

Two five-way diagrams: Grünbaum’s diagram (left) and an Edwards–Venn diagram (right)

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.

A goat acting out a maths puzzle.

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.

{Katie Steckles is a mathematician based in Manchester, who gives talks and workshops on different areas of maths. She finished her PhD in 2011, and since then has talked about maths in schools, at science festivals, on BBC radio and TV, at music festivals, at comedy shows and on the internet. She blogs at the Aperiodical, and has a YouTube channel.
Twitter  @stecks   YouTube Website  katiesteckles.co.uk    + More articles by Katie

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