# A fair piece of the π

# BLOG: Heidelberg Laureate Forum

March the 14th, written 3.14 (for at least some people in the world – other date formats are available), is a very special day for mathematicians. It’s been celebrated for years as being an opportunity to recognise the brilliance of the wonderful circle constant, π (or, if you’re in a hurry, 3.14).

**Pi day**

Pi day, as it’s known, has become a great excuse for a party and all over the globe (an area of 4πr²) people celebrate by doing mathematical things and making terrible puns about Greek letters. You could join in by baking a pie – preferably one with some mathematical designs on the crust, but as long as it’s circular you can still claim π is in there somewhere; by reading this blog post from Pi day 2018 rounding up (pun intended) some interesting facts about π; or, you could use a physical method to approximate π – just for fun, swing a weight on the end of a pendulum and time its swing, then rearrange the formula to get an approximation for π (this YouTube video shows you how it’s done, and this Wired article goes into a bit more depth.

In fact, Pi day is such an important part of the mathematical calendar that since 2020, it’s been officially declared the **International Day of Mathematics**. With an official website at idm314.org, and coordinated by the International Mathematical Union you’re invited to participate in a worldwide celebration – with activities for students and the general public in schools, museums, libraries and other spaces. This year the website lists over 427 events, ranging from online talks and workshops to activities you can join in with from home.

Along with the main website, IDM also has an annual theme, which this year is Mathematics for a Better World. There’s a section of the website devoted to this theme, and it includes a lovely collection of examples of ways in which mathematics makes the world a better place every day – from allowing people to build technology to help others, through medicine, storm warnings and fighting crime, to creating fun experiences for people through digital music, and computer generated graphics in games.

One topic that stands out on the page, and something which definitely forms part of a better world for everyone, is that of fair sharing algorithms – a somewhat niche mathematical curiosity exploring ways to divide things up fairly, so that everyone gets what they consider to be a portion of equal value. This kind of mathematics has been around since… well, as long as people have been trying to share things – but the real work in the area has been mainly in the last century. In the 1940s, a group of Polish mathematicians formalised the problem and started work on algorithms to determine the best way to divide things fairly.

**Divide and Calculate**

It might seem obvious that if you’re sharing something like a plain cake, that’s the same all the way through and has no toppings or frosting, then the fairest way to share it between N people is by just cutting it into N pieces each with an equal volume (or weight, if you’re unsure it has consistent density). This does work when the resources you’re sharing are homogenous, and of equal value to everyone.

The interesting questions come when there are differences in the cake – maybe fruit inside that’s unevenly distributed, or frosting on the outside, or cherries arranged on the top. Obviously, it’s not worth mathematically studying such things and finding a way to assign them value – especially since their value might be subjective. I might not like cherries; or maybe I’m all about the frosting. The way to decide what’s fair is to come up with an algorithm in which each person is confident that the portion they get is what they would consider to be fair – so, they might not get exactly 1/N of the cake but if it’s got four cherries on it instead of three, they don’t mind.

The simplest fair division algorithm is when you’re sharing between two people, and it’s highly likely you’ll have used it at some point without even thinking about it too much. The process is that one person divides the resource (cake) in what they consider to be half, and the other one chooses which to take. This works, because as long as the cutter is convinced that their cut exactly halves the cake, they’d be happy with either piece since they consider both halves to have equal value – and the second person, who might not consider it to have been an exactly equal split, will be happy because they get to pick the one they consider better.

**The Three-Body Problem**

Like many mathematical problems, this one becomes significantly more complicated when you move from the case of two people to three (or more generally, N people). The first person cutting and the second choosing could leave the third person with an unfair (or, what to them seems unfair) portion.

Even though the overarching criterion here is fairness, there are different ways to mathematically define what we mean by ‘fair’ – for example, a method for fair-sharing can be deemed **proportional** if each of the N participants believes the piece they have represents at least 1/N of the value of the whole, by their own measure. It’s called **envy-free** if each believes that the piece they have is better than, or at least as good as, all the others, and there isn’t another piece they’d rather have taken.

Even though the overarching criterion here is fairness, there are different ways to mathematically define what we mean by ‘fair’ – for example, a method for fair-sharing can be deemed **proportional** if each of the N participants believes the piece they have represents at least 1/N of the value of the whole, by their own measure. It’s called **envy-free** if each believes that the piece they have is better than, or at least as good as, all the others, and there isn’t another piece they’d rather have taken.

An **equitable** division is one in which each person feels they got the same amount by their own measure (bearing in mind people with different priorities might favour different things). While these ideas all seem quite similar, they’re subtly different and some can be satisfied by a division that doesn’t satisfy all the others.

One simple cake-sharing algorithm is the **Dubins-Spanier Moving knife method**, which goes like this: one person picks up a knife, and moves it slowly from one end of a long, thin cake. Everyone who’s sharing the cake watches as the knife moves, and as soon as one of them thinks the piece to the left of the knife is a fair 1/N of the cake, they shout ‘stop’. They can then take this piece and leave. The process is then repeated with the remaining N-1 people, until all the cake is shared out.

Theoretically, this process should result in a fair division – since if anyone thinks the piece the person got was more than 1/N of the remaining cake, they would have shouted sooner than them, at the point they thought it was 1/N. This method guarantees proportionality, but it doesn’t guarantee an envy-free division – it’s possible that the last player might get a piece that others would prefer to their own (although presumably if they’ve left, they wouldn’t know, and might well have already eaten their bit of cake by the time this becomes apparent).

There are other, more complicated methods for fair sharing – one is outlined step-by-step in this video with Hannah Fry, which is included in the IDM resources on the topic. The resources also include a link to a lecture given as part of MoMath’s ‘Math Encounters’ series by the wonderful Francis Su, which includes an explanation (and live demonstration!) of the Dubins-Spanier method. There’s also a great worksheet of activities to try various algorithms for real.

Mathematical methods like this can be applied to sharing out any kind of finite resource – they can be used to allocate land, natural resources or money; or to divide up tasks to be done, like chores; you can even apply them to situations like how to divide the rent between housemates if the rooms they’re each renting are all different sizes. It’s a great example of how a complex real-world problem can be reduced to its simplest factors, and modelled using mathematics – for a better world.

Yes, mathematics has many practical applications not only in the world of science and technology, but also in the world of law and order and in the social sciences (and everywhere else?).

By construction, mathematics is not of this world. And that means exactly that it has many applications in this world, because whenever you want purity, you need something that is not of this (so impure) world.