bq. You are one of P recently arrested prisoners. The warden, a deranged computer scientist, makes the following announcement:

bq. You may meet together today and plan a strategy, but after today you will be in isolated cells and have no communication with one another.

You may meet together today and plan a strategy, but after today you will be in isolated cells and have no communication with one another.

I have set up a “switch room” which contains a light switch, which is either on or off. The switch is not connected to anything.
Every now and then, I will select one prisoner at random to enter the “switch room.” This prisoner may throw the switch (from on to off, or vice-versa), or may leave the switch unchanged. Nobody else will ever enter this room.
Each prisoner will visit the switch room arbitrarily often. More precisely, for any N, eventually each of you will visit the switch room at least N times.