There are N prisoners and N cells in a prison (though the prisoners do not know the value of N). The cells are all identical from the inside, you cannot tell them apart. The cells are arranged in a circle, and in each cell there is a switch. The switch controls a light in the next cell clockwise around the circle. However, the power supply for these lights is usually off, and it only comes on at midnight each night for a tenth of a second (a tenth of a second not being long enough for a prisoner to see the state of their light and make a decision about their switch).

Each day, the warden will take the prisoners out of their cells and clean out the cells, setting the light switches back to "off". The warden will then place the prisoners back into the cells as he sees fit (the prisoners do not see eachother during this process). At any time, any prisoner may announce "I know how many prisoners there are." If they are then able to state the value of N, the prisoners win and are released, if they are incorrect they lose and all the prisoners are killed.

You are one of the N prisoners. Before the game, you may send out a single email outlining the strategy the prisoners must follow. Your email cannot refer to any prisoner in particular, as you know nothing about their identities. Find a strategy that guarantees the prisoners escape against the warden acting as an adversary.

Man, those wardens are nuts. Anyway, the light switch thing can be equivalently expressed as "every night each prisoner must simultaneously transmit a '0' or a '1' to the next prisoner clockwise from them". Your strategy may contain two types of prisoners, "you" and "everybody else".

The puzzle is pretty hard, I was only able to make a small amount of progress before giving up and just looking at the solution. There are a variety of degrees of solutions one can arrive at, there is a fairly easy strategy that works with probability 1-e for any e>0, and there is a better strategy that works with probability 1. In both of these cases, it is necessary to assume the prisoners have a private random number generator at their disposal to actually generate said probabilities. The full solution, however, is a guaranteed escape, and in fact is guaranteed that for each N there is a K such that "if the number of prisoners is no greater than N, we will escape before K days have passed". Though, K is a really really huge number compared to N.

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