A group of 100 criminal mathematicians (corrupt accountants, sleazy statisticians, etc) arrive together in a special prison. They aren't told how long they'll be there, but they're told how to get out.
On the day of their arrival, each prisoner has a colored spot tattooed in the middle of his back. The tattoo is either black or red. He isn't told which color he has, but he is told that all 100 prisoners have such a spot and both colors are represented; that is, all prisoners having black or all prisoner having red is not an option.
He's then put in an isolated room, where he will spend the rest of his prison term in solitary confinement. The room is filled with physical exercise machines, along with the usual amenities.
One wall of the room contains 99 recessed video monitors and the opposite wall a small video camera. On the morning of the last day of each year served, these will become active. Each prisoner will strip to the waist in front of the camera and face the video screens, where he will find displayed the bare backs of the other 99 prisoners. He's given 1 minute to note the colors of the spots on the backs displayed. The cameras then become inactive and the screens go dark for another year.
His task then, which he is well qualified to perform, is to determine the color of his own spot using logic only. Once he knows for a fact what his color is and can prove it logically, he informs the warden and is released from prison that very day.
At the end of the following year served (and every year served), the prisoners will again strip to the waist, back facing the camera, front facing the monitor screens, which will be displaying the tatooed backs of the other prisoners. The monitor screens of any prisoners released the previous year will be blank. Release is the only reason a screen will ever appear blank. Death is not allowed.
Cheating of any kind, which includes guessing, and facing or hand signaling into the video camera, is punished horribly, though not in such a way that the video camera would see the results the following year. So cheating is simply not done. Discovering the color of one's spot is to be accomplished by logic alone.
Assume that all the prisoners have the goal of getting out of prison as soon as possible.
. . .
So how does this play out? Does anyone ever discover his spot color and go free? If so, when?
Does everyone eventually go free? If so, when does the last one leave?
Explain the logic needed to solve the puzzle.