Math in… the Six Degrees of Kevin Bacon

You might have played Six Degrees before. Given an actor, try to find a chain of actors and common films that lead to Kevin Bacon.

For example, let’s take actress Emma Watson:

Since two movies separate Watson and Bacon, Watson’s Bacon number is 2.

The challenge is to find the shortest chain (six films or fewer). A chain from Emma Watson to Kevin Bacon doesn’t require more than two films. Since they haven’t starred in a film together, we can’t link them with one film, so a two-film chain is the shortest possible. Here’s another one:

Can you think of any other two-film chains?

We can think of this sort of game as being played on a multigraph, where the vertices are actors and there is an edge between two actors for each film they’ve costarred in. The goal is to find a shortest path from one vertex to another.

Looking at the Academy Award nominees for Best Actor, here’s a graph with just those five actors and their shared projects:

Adrien Brody has worked with everyone but Colman Domingo, while none of the other actors have worked with each other. If we want to be able to link Domingo to the other four, we need more actors.

Let’s add Anthony Mackie:

It turns out that adding a strategic sixth actor created short paths between any pair!

What might a graph for the Best Actress nominees look like? Definitely more complicated, since none of them have been costars and some pairs are three films apart.

Can you create a multigraph linking Cynthia Erivo, Karla Sofía Gascón, Mikey Madison, Demi Moore, and Fernanda Torres?