Prisoners, Rooms, and Lightswitches
Abstract
We examine a new variant of the classic prisoners and lightswitches puzzle: A warden leads his prisoners in and out of 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 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, , 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 is surprisingly small. The analysis gives rise to a number of puzzling open questions, as well.
Cite
@article{arxiv.2009.08575,
title = {Prisoners, Rooms, and Lightswitches},
author = {Daniel M. Kane and Scott Duke Kominers},
journal= {arXiv preprint arXiv:2009.08575},
year = {2020}
}