Prisoners, Rooms, and Lightswitches

TL;DR

This paper investigates a variant of the prisoners and lightswitches puzzle, determining the minimum number of switches needed per room for prisoners to guarantee escape under different knowledge conditions.

Contribution

It introduces a new variant of the classic puzzle and provides a solution for the minimum switches needed when prisoners know the initial configuration.

Findings

If prisoners do not know the initial configuration, escape is impossible.

If prisoners know the initial configuration, the minimum number of switches per room is surprisingly small.

Abstract

We examine a new variant of the classic prisoners and lightswitches puzzle: A warden leads his $n$ prisoners in and out of $r$ rooms, one at a time, in some order, with each prisoner eventually visiting every room an arbitrarily large number of times. The rooms are indistinguishable, except that each one has $s$ lightswitches; the prisoners win their freedom if at some point a prisoner can correctly declare that each prisoner has been in every room at least once. What is the minimum number of switches per room, $s$, such that the prisoners can manage this? We show that if the prisoners do not know the switches' starting configuration, then they have no chance of escape -- but if the prisoners do know the starting configuration, then the minimum sufficient $s$ is surprisingly small. The analysis gives rise to a number of puzzling open questions, as well.