Generalizing the Futurama Theorem

TL;DR

This paper provides a new proof and generalization of the Futurama Brain Swap Theorem, extending it to cyclic brain swaps among prime number of characters, with implications for understanding complex brain exchange systems.

Contribution

It introduces a new proof of the original theorem and generalizes it to cyclic swaps involving a prime number of participants, expanding the theoretical framework.

Findings

New proof of the Futurama Brain Swap Theorem

Generalization to cyclic brain swaps among prime p characters

Insights into the structure of brain exchange systems

Abstract

The 2010 episode of \emph{Futurama} titled \emph{The Prisoner of Benda} centers around a machine that swaps the brains of any two people who use it. The problem is, once two people use the machine to swap brains with each other, they cannot swap back. In this article, we present a new proof of this theorem and also a generalization of it to what would happen if, instead, the machine swapped cyclically the brains of $p$ characters, where $p$ is prime.