The Mathematics of Murder

How to create the perfect murder mystery dinner party

This January, I wrote my first murder mystery. No, not a book, but a script for a dinner party where someone would be “offed,” and the assembled guests would try to figure out who did it. Larry and I went to one of these last December, the brainchild of Gautam Rao from the class on Improv I was taking, and it was an absolute blast. I wanted to create another such experience for the same crew and also take up the writing challenge. Having read untold murder mysteries while growing up, what would it be like to write one?

The first question was the setting. I decided it had to be a university, given that I’ve been a math professor for over four decades. So I invented the University of New Horizons, upstate New York, an institution which had multiple secrets festering under the surface. With inter-colleague rivalries bitter enough to plausibly lead to homicide.

In case you’re wondering, mathematicians are too self-absorbed in their own theoretical worlds to be good candidates for murderers, so this wasn’t based on my own experience. To distance myself further, I endowed New Horizons with a medical school (which my university doesn’t have), with the action unfolding in a new lab dedicated to finding whether Yusawa could cure pancreatic cancer.

Yusawa? That’s the name I gave to a made-up hallucinogenic plant used for religious purposes by a small Native American tribe in upstate NY (the “Mahoras,” I called them). The Native American connection was inspired by the endless land acknowledgements I’ve suffered through over the years at university functions (is it just my imagination, or are these toothless acknowledgements metastasizing?). Act I kicks off with a university dignitary paying tribute to the Mahoras for the land they’ve “contributed” to the university, and a tribal elder barging in to announce that acknowledgements be damned, the Mahoras are suing.

A common trope in murder mystery parties is for all the players to be connected to a central character who then gets killed. In my script, this person was Professor Arnold (“Arnie”), the head of the new pancreatic cancer lab. Someone who’d spent many years studying the medicinal (and self-administered hallucinogenic) properties of Yusawa.

In Act I, even though everyone from the university president to a graduate student assistant comes forward to spout high praise for Professor A during his new lab’s dedication, they each have an axe to grind against him. Arnie is a hardcore Yusawa addict, he’s seduced faculty wives and sexually harassed students, he’s deceived the Mahoras and shortchanged his staff members, he’s broken promises and misappropriated funds. The reason he has to be such a wheeler dealer, embodying so many negative qualities, is simply because I needed enough different motives for each of the other players to want to kill him.

Initially, I thought it would be ten of us, which meant I’d need nine motives – one for everyone except Arnie. As more Improv classmates said they’d come, I found the script became disproportionately more difficult to compose. Gautam had mentioned this effect as well, and it wasn’t just our imaginations – there was a mathematical reason behind it (see below). At one point, it seemed like I’d need a script for 15 – but in the end, we were 13. The perfect number for killing one of the assembled – I could have called my party “The Last Supper.”

Instead, I called it “Coq and Dagger” – Coq au Vin combined with Cloak and Dagger. The chicken with wine was the main course, but prepared with Indian spices – a recipe I concocted a long time ago in Mumbai, to use up a less-than-drinkable bottle of Indian red. Larry made Romanian cheese pie for the appetizer and his famous mocha chocolate brownies for dessert – an essential treat given that it would be the final meal for Professor Arnold. (No, the brownies weren’t laced.)

I died at the end of the first act (a Yusawa overdose injected into my right arm did me in). Sadly, the death was off camera, so I didn’t get to ham it up for my Improv friends. Yes, I played Arnie – I had to, so that I could resurrect myself as a police investigator for the next act. In this second role the trick was to guide the action along without intruding too much.

I’m happy to report that the group successfully identified the murderer. Many thanks to X, who only learnt about their homicidal streak when I secretly revealed it to them mid-game (can’t reveal who X was in case I reprise the game with a different cast). Kudos also to Sandy, who took the selfie, for her inspired performance – complete with a startling wig – as the jilted office manager driven to drink by her crush on Arnie.

Which brings me to the math part, that sheds light on the work it takes to incorporate more players. In terms of preparing the coq au vin, it’s straightforward – all one needs to do is throw in another chicken piece or two for each extra guest. In other words, if n is the number of participants, then the amount of cooking work is proportional to n. I’d get a straight line were I to plot this on a graph.

At first glance, creating extra motives might also seem to follow such a linear path. After all, if there are n participants, I need n motives. However, there’s a hidden cost involved. The motives all must be compatible, the script must be convincing and consistent. Each time I add an extra player, I need to remember there is already a network of connections, motives, prejudices, alliances, between the players previously included. I need to smoothly imbed the new player into this network. That means creating (or at least mulling) new connections and interactions with each of the previous participants.

If the game has just 2 players, then there’s only a single connection to consider. For 3 players, this increases to 3 connections, and with 4, we get 6 connections (the 4 sides and the 2 diagonals in the rightmost picture below, where the nodes represent players).

We can continue using geometric diagrams to count that for 5 players, we’ll have 10 connections, and this increases to 15 connections for 6 players and 21 for 7. (A nifty explanation for the general mathematical formula for n players is given in the note at the end.)

Plotting this count no longer gives a straight line, but a parabola. In other words, the work required for figuring out connections grows much faster than that for cooking (quadratically, instead of linearly). We’re so conditioned internally for things to increase linearly that we can’t help but notice when they fail to do so.

I’m already thinking of possible setting for the next murder mystery. Perhaps one of those remote artist colonies, which I’ve been to a few times while authoring my books. Writers, I feel, have the capacity for much more cutthroat behavior than university personnel. Plus, just think of the wigs and the quirks and the airs Sandy could put on!

NOTE: The number of connections between n players is surprisingly easy to figure out. Let’s take one of these n players. They can be connected with each of the other players, of which there are n - 1. In other words, we’d have n - 1 connections involving this player.

This means we have a total of n players in the game, each with n - 1 connections. So the total number of connections will be n times n - 1, i.e. n(n - 1).

But notice that each connection gets counted twice in the above formula (the connection between A and B gets counted while counting A’s connections, and also while counting B’s connections).

So the total number of distinct connections is half the number, i.e. n(n - 1)/2. That’s the formula!