$x$ Plays Pokemon, for Almost-Every $x$
C. Evans Hedges
TL;DR
The paper investigates whether a disjunctive real number $x$ will eventually force a win in any finite-state game by modeling play as a labeled directed graph $G=(V,E)$ with a sink vertex and using a synchronizing word $W$ of length at most $|V|^2$ to guarantee victory from any start state; $x$, by containing $W$ in its base-$b$ expansion, thus yields a winning sequence. It provides a constructive proof that such a synchronizing word exists and is obtainable by inductively combining per-vertex winning words, while also noting that finding the shortest such word is NP-hard. The paper derives an explicit upper bound on the required number of button pushes, $|V|^2 b^{|V|^2}$, illustrating the combinatorial explosion, and discusses a probabilistic extension showing that with full-support RNG the win occurs with probability $1$. By connecting folklore automata-theory results to a playful Pokémon context, the work highlights a fundamental gap between existential guarantees and practical feasibility, especially given astronomical bounds like $2^{10^{14}}$ for concrete games such as Pokémon Sapphire.
Abstract
This paper provides a brief write-up showing that for any finite state game, a disjunctive number $x$ will eventually win that game. The proof techniques here are well known and this result follows immediately from folklore results in graph theory and cellular automata. This short paper primarily serves as an expositional piece to collect this proof with the fun context of $π$ Plays Pokémon serving as motivation.