# Some Mathematical Gems from John Conway

John Horton Conway died on April 11, 2020, at the age of 82, from complications related to COVID-19. See this obituary from Princeton University for an overview of Conway’s life and contributions to mathematics. Many readers of this blog will already be familiar with the Game of Life, surreal numbers, the Doomsday algorithm, monstrous moonshine, Sprouts, and the 15 theorem, to name just a few of Conway’s contributions to mathematics. In any case, much has already been written about all of these topics and I cannot do justice to them in a short blog post like this. So instead, I’ll focus on describing a handful of Conway’s somewhat lesser-known mathematical gems.

# COVID-19 Q&A

My friend Joshua Jay, who is one of the world’s top magicians, emails me from time to time with math questions. Sometimes they’re about card tricks, sometimes other things. Last night he sent me an excellent question about COVID-19, and I imagine that many others have wondered about this too. So I thought I’d share my response, in case it’s helpful to anyone.

JJ: Since the government is predicting between 100k – 240k deaths from COVID-19, let’s for argument’s sake split the difference and call it 170k projected deaths. They’re ALSO telling us they believe the deaths will “peak” something like April 20th. Am I wrong in assuming, then, that if we assume 170k total deaths, and the halfway point is a mere two weeks away, then they’re projecting 85k deaths before (and after) April 20th?

When I start to think about the idea of of 85k deaths between now and April 20th, and we’ve only experienced 5k so far, it means that 80k people are projected to die in the next two weeks. Surely that can’t be correct, or else it would be dominating the news cycle, right?

I’m not asking whether you think those projections are accurate… I’m just trying to wrap my head around the relationship between total projected deaths (whatever it is) and the projected peak of the curve.

# The Drunkard’s Walk and Catalan Numbers

In this post, I’d like to present an amusing and off-the-beaten-path solution to the classical “Drunkard’s Walk” problem which at the same time derives the well-known generating function for the Catalan numbers.  This solution evolved from a suggestion by my former undergraduate student Stefan Froehlich during a discussion in my Math 4802 (Mathematical Problem Solving) class at Georgia Tech in Fall 2007.

In case you’re not familiar with it, here’s the problem (in a formulation I learned from this book by Frederick Mosteller):

A drunken man standing one step from the edge of a cliff takes a sequence of random steps either away from or toward the cliff. At any step, his probability of taking a step away from the cliff is $p$ (and therefore his probability of taking a step toward the cliff is $1-p$). What is the probability that the man will eventually fall off the cliff?

What is the probability that two randomly chosen integers have no prime factors in common?  In honor of Pi Day, I’d like to explain the surprising answer: $6/\pi^2$.
The hero of this story is Leonhard Euler, who worked out this astonishing connection between prime numbers and $\pi$ through a series of brilliant insights.  In the spirit of Euler, I will be rather cavalier about issues of convergence and rigor here, focusing on the key underlying ideas.