In this week's episode of PBS Infinite Series, I shared the following puzzle:

Consider a square in the xy-plane, and let A (an "assassin") and T (a "target") be two arbitrary-but-fixed points within the square. Suppose that the square behaves like a billiard table, so that any ray (a.k.a "shot") from the assassin will bounce off the sides of the square, with the angle of incidence equaling the angle of reflection. Puzzle: Is it possible to block any possible shot from A to T by placing a finite number of points in the square?

As I mention in the video, this is one of a number of billiard problems that folks studying dynamical systems have asked. It was mentioned in a 2014 lecture given by Maryam Mirzakhani, which you can watch in the video on the left! Maryam describes the puzzle but leaves the audience to think about the solution. And in that audience was category theorist Emily Riehl, who did indeed think about a solution on and off for a few weeks. (You can read Emily's reflection on her experience in this AMS tribute to Maryam.)

Fast forward to last fall, when I ran into Emily at a Women in Topology workshop at MSRI. During a bus ride up the hill to the MSRI campus, Emily suggested that the puzzle might be a good fit for Infinite Series. I couldn't agree more! The solution she shared with me is delightfully simple for what seems to be an incredibly complicated puzzle. It's the solution she believes Maryam intended the audience to find, and it's what I want to share with you today! In the video above, I set up some of the mathematics needed to work through the solution, so if you haven't watched it already, now would be a good time. In any case, I'll assume you're comfortable with the notions of flat tori, a tiling of the xy-plane with flat tori, and a lattice in the xy-plane.

Alright, let's get to it!

My sincerest thanks go to Emily Riehl for today's episode and blog post!