Imagine there are only 5 prisoners, not 100, under the same rules. Before the 5th day, a prisoner CANNOT ask for freedom, as it is impossible that every other prisoner was called. So, after the 5th day, if a prisoner is called to the room and hasn't been called before, he can asume that either everyone was in the room before, or that there is someone still waiting to go. If the lamp is off, that means he was the last one and should ask for freedom.

But WHY did the other prisoner set the light off? Basically, what you said is: the last prisoner set the light off if he has been into the room before, but this can lead to an error. Let's asume only ONE prisoner is called for 5 straight days, and he sets the light off when he is called again the last day. So another prisoner gets called, it is the 6th day and the light is off, so he claims to be the last one, and is wrong.

I say it's mathematically impossible, DanielB says it can be done. Anyone else here might want to give an opinion, so we can decide once and for all if this one has a solution?