# A theory of trios.

Working on my book Math Games with Bad Drawings (by the way, for U.S. readers, two days left to preorder the deluxe package), I thought a lot about Set.

Maybe too much. I’ll let you judge.

Set is a classic pattern-hunting game. The cards have four traits: number, shape, color, and texture. Each trait comes in three flavors; for example, the shapes are squiggles, ovals, and diamonds, while the textures are filled, empty, and striped.

You lay out twelve cards, and then look for special trios called “sets,” where for every trait the three cards are either (a) all the same, or (b) all different.

For example, they might share three traits, and differ in one:

Or they might share two traits, and differ in the other two:

Or share a single trait, while differing in three:

Or differ in all four traits (the hardest kind of set to spot):

But watch out for deceptive not-quite-sets. If two cards share a trait while the third one differs, you’re out of luck.

The game was developed by geneticist Marsha Jean Falco in a moment of gorgeous serendipity. (I tell the story in the book.) Since then, it has become a favorite among mathematicians, who study its rich symmetries and peculiar generalizations to higher dimensions.

But personally, I’m most fascinated its potential as a Grand Unified Theory of Trios.

You see, pop culture is full of trios. Three friends will band together to fight noseless eugenicists.

Or to topple evil empires.

Or just to take on general crime and villainy.

To my eye, these trios resemble Sets. They share certain traits in common. On other traits, they all differ. But if two people share a special bond that the third does not… well, that’s poison for a friend group, isn’t it?

Maybe I’m shoehorning the world into my theory. Maybe fictional characters possess so many characteristics, and allow for so many interpretations, that you can always find a few shared and/or differing traits. Maybe this is all just an excuse to show off how amazingly good I have gotten at drawing scarves, lightsabers, and anthropomorphic chipmunks.

Or maybe Marsha Jean Falco gave the world something more than just a great card game: an organizing theory of friendship.

(Preorder the deluxe Math Games with Bad Drawings package through Kickstarter, or the still-thrilling ordinary version wherever you buy books!

## 4 thoughts on “A theory of trios.”

1. The applications for polyamory alone!

Okay, but seriously, I’m fascinated by this. It’s a falsifiable theory but not necessarily a provable one… and of course if you seemed to find a falsifiable example, how would you know it was really proof against, as opposed to “things haven’t just fallen apart yet?” In contrast, how would you know the examples that fit (the trio works together and everything is either different between all three or shared between all three) are not also about to fall apart due to… life? In short, you could never really prove this as a realistic way to view real world friendships, partnerships or other entanglements. Not in a scientifically rigorous way.

On the other hand, it feels true. I know that is an incredibly unrigorous, unmathematical, not-remotely-scientific way to put things, but you can still examine the emotion. Why does it feel true? Possibly because, if there is something shared by two of the three, the bond between the two is automatically strengthened, in a way that leaves the third out. This will inevitably lead to tensions and problems. In relationships, however, there is a possible solution that is not in the rules of the game, as explained here. If each pair has a bond, the balance is restored. So, for example, Han/Luke/Leia has a similarity of gender between Han and Luke that leaves Leia out, a family connection that Luke and Leia share that leaves Han out, and a romantic connection that Han and Leia share that leaves Luke out. These three balance out, and allow the trio to remain overall stable.

Of course, in real life, the traits are not going to have equivalent weight. If you have a trio A, B and C, and A and B share a traumatic past that C does not share, B and C share a passion for music that A does not, this will probably not be balanced by A and C both liking the same flavor of gum while B thinks gum is just gross and pointless. B will have two friends, A and C, but they won’t be a proper trio. This makes the algorithm impossible to apply, because the number of human traits can never be perfectly quantified and there can be changes, over the course of a person’s life, in which traits matter more, or continue to exist at all.

However, this can still be a good shorthand. I have been in situations where there was a desire for a B to make friends A and C all part of a companionable trio, and it never quite works unless A and C can share something as strongly as B and A and B and C share something. Conceptualizing it this way can, potentially, help resolve the problem, or at least help everyone involved understand the dynamic in a way that isn’t taken personally.

…no, I’m not going to apologize for overthinking this. This was fun!