Prisoner dilemma

[borisp]

Ok, so this isn't THE prisoner dilemma, but it is a problem in which prisoners face a dilemma. I got this from my friend, who was asked this problem in some job interview.

You and 30 other prisoners (the # doesn't matter) are being held in a common jail cell. You know that shortly you will all be taken to solitary confinement, at which point you will no longer be able to communicate, but you are free to discuss strategy now.

You also know that after you are separated, the guards will sporadically take individuals to a room with two switches, labeled "switch A" and "switch B," and that each switch is either in the "up" or "down" position, and that right now they are both down. When you are taken to this room, you have to switch exactly one of the switches, and then you go back to solitary.

You don't know the order that the guards will take you; you might visit the room 10 times in a row while everyone else sits in their cell, or you might sit in your cell while the guards cycle through everyone else 1000 times, etc. (I.e., assume the guards act randomly in choosing who visits the room next.)

Your task as a group is to at some point have someone declare factually that every prisoner has visited the room. For the sake of argument, say that if someone guesses that everyone has visited, and that person is wrong, then you all are put to death, whereas if that person is correct, then you all go free.

So what is your group's strategy?