Okay. So they choose "23" to be the counter. 23 alone recognizes when everyone else has gone. That means that a consistent process must exist that means nothing to the others but everything to 23.
Person 1 walks in. The lightbulb is off. He leaves it off.
Person 2 through 99 leaves the lightbulb off.
23 sees the lightbulb off. He turns it on. That means that the counter has been there and has left a mark.
Now the counter has to know when 99 other people have been chosen.
Person 24 (or whatever number) walks in and sees the lightbulb on. He can't turn it off because it will reset....no it won't.
Everyone leaves the bulb off except 23. When he arrives, he turns it on. The next person, 47, remembers that the bulb was on when he arrived. He turns it off. It continues until 23 is chosen again. He turns the bulb on. The next person, 83, turns it off. 47 sees the bulb off and leaves it off, remembering that he has gone. If 47 is chosen after 23 again, he leaves the bulb on for the next person.
This continues until 23 walks in and sees the bulb on. Everyone ahead of him has left it on for the next person. He will be the last person and will announce that all 100 have been to the lightbulb. And he will be correct.