100 perfect logicians

100 perfect logicians are told to sit in a circle in a room. Before they enter the room, they are told at least one person has a blue forehead. When you determine your forehead is blue, you need to leave the room when the lights turn off. Then the lights are turned back on, with those people who have deduced they have blue foreheads having left the room.

They enter the room in the dark. Then the lights turn on, they make deductions but cannot communicate with each other. Then the lights go off, and those that fully deduced that they have blue foreheads leave. The lights turn back on, they do more deducing, and the lights go off and the process repeats.

• Let’s say we have painted everyone’s forehead blue. What happens during the above process ?