Prisoners, rooms, and light switches (Q2227823)

Summary: We examine a new variant of the classic prisoners and light switches 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 \(s\) light switches; 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. \textit{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.