Look, Knave

Abstract

We examine a recursive sequence in which $s_n$ is a literal description of what the binary expansion of the previous term $s_{n-1}$ is not. By adapting a technique of Conway, we determine limiting behaviour of $\{s_n\}$ and dynamics of a related self-map of $2^{\mathbb{N}}$. Our main result is the existence and uniqueness of a pair of binary sequences, each the compliment-description of the other. We also take every opportunity to make puns.