I'm excited to share the following riddle. It was given to me more than two years ago and I finally solved it last summer (after not thinking about it for a long time). In my desperation, I tried to find a solution online, but couldn't even find the riddle anywhere else. I'm excited to see if somebody knows the riddle or if not, how you approach the solution.
Three mathematicians are in prison. Each of them is in a single cell and they are not able to communicate in any way. They are imprisoned for an arbitrary number of days.
Each cell has a single light bulb that is either on or off on a given day. The warden tells the mathematicians that the light system of the prison has three modes:
- neutral mode, where each lightbulb is independent of the others
- bright mode, where two bulbs turn on every day and the other turns off
- dark mode, where two bulbs turn off every day and the other turns on
(All distributions are not necessarily uniform.)
The prison starts in neutral mode. After an unknown but finite number of days, the warden will select either bright mode or dark mode, which is locked in permanently.
After countably infinitely many days have passed, the mathematicians are asked which one the warden picked. They may discuss strategy before going into the cells, but there will be no communication afterwards. They have unlimited capacities to communicate and remember strategies that they come up with. Two of the three need to guess correctly to escape; how can they ensure this? You may assume that the axiom of choice holds.