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arXiv · 2009.08575

Prisoners, Rooms, and Lightswitches

Daniel M. Kane·Scott Duke Kominers
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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.

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Daniel M. Kane, Scott Duke Kominers. 2020-09-18. Prisoners, Rooms, and Lightswitches. https://arxiv.org/abs/2009.08575

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