# Enemies, Friends and Frenemies: Distance, Categorisation and Fun.

As Mario Puzo and Francis Ford Coppola wrote in “The Godfather Part II”:

(I bet you thought that was Sun Tzu, the author of “The Art of War”. So did I but this movie is the first use.)

I was thinking about this the other day and it occurred to me that this is actually a simple modelling problem. Can I build a model which will show the space around me and where I would expect to find friends and enemies? Of course, you might be wondering “why would you do this?” Well, mostly because it’s a little bit silly and it’s a way of thinking that has some fun attached to it. When I ask students to build models of the real world, where they think about how they would represent all of the important aspects of the problem and how they would simulate the important behaviours and actions seen with it, I often give them mathematical or engineering applications. So why not something a little more whimsical?

From looking at the quote, we would assume that there is some distance around us (let’s call it a circle) where we find everyone when they come up to talk to us, friend or foe, and let’s also assume that the elements “close” and “closer” refer to how close we let them get in conversation. (Other interpretations would have us living in a neighbourhood of people who hate us, while we have to drive to a different street to sit down for dinner with people who like us.) So all of our friends and enemies are in this circle, but enemies will be closer. That looks like this:

I have more friends than enemies because I’m popular!

So now we have a visual model of what is going on and, if we wanted to, we could build a simple program that says something like “if you’re in this zone, then you’re an enemy, but if you’re in that zone then you’re a friend” where we define the zones in terms of nested circular regions. But, as we know, friend always has your back and enemies stab you in the back, so now we need to add something to that “ME” in the middle – a notion of which way I’m facing – and make sure that I can always see my enemies. Let’s make the direction I’m looking an arrow. (If I could draw better, I’d put glasses on the front. If you’re doing this in the classroom, an actual 3D dummy head shows position really well.) That looks like this:

Same numbers but now I can keep an eye on those enemies!

Now our program has to keep track of which way we’re facing and then it checks the zones, on the understanding that either we’re going to arrange things to turn around if an enemy is behind us, or we can somehow get our enemies to move (possibly by asking nicely). This kind of exercise can easily be carried out by students and it raises all sorts of questions. Do I need all of my enemies to be closer than my friends or is it ok if the closest person to me is an enemy? What happens if my enemies are spread out in a triangle around me? Is they won’t move, do I need to keep rotating to keep an eye on them or is it ok if I stand so that they get as much of my back as they can? What is an acceptable solution to this problem? You might be surprised how much variation students will suggest in possible solutions, as they tell you what makes perfect sense to them for this problem.

When we do this kind of thing with real problems, we are trying to specify the problem to a degree that we remove all of the unasked questions that would otherwise make the problem ambiguous. Of course, even the best specification can stumble if you introduce new information. Some of you will have heard of the term ‘frenemy’, which apparently:

can refer to either an enemy pretending to be a friend or someone who really is a friend but is also a rival (from Wikipedia and around since 1953, amazingly!)

What happens if frenemies come into the mix? Well, in either case, we probably want to treat them like an enemy. If they’re an enemy pretending to be a friend, and we know this, then we don’t turn our back on them and, even in academia, it’s never all that wise to turn your back on a rival, either. (Duelling citations at dawn can be messy.) In terms of our simple model, we can deal with extending the model because we clearly understand what the important aspects are of this very simple situation. It would get trickier if frenemies weren’t clearly enemies and we would have to add more rules to our model to deal with this new group.

This can be played out with students of a variety of ages, across a variety of curricula, with materials as simple as a board, a marker and some checkers. Yet this is a powerful way to explain modelsspecification and improvement, without having to write a single line of actual computer code or talk about mathematics or bridges! I hope you found it useful.