How would things play out if only one of the stick figures present had a messed up head?
How about if there were two with a messed up head? On Day 1, what would each of those two think when they looked around at all the other stick figures?
Eight stick figures erased themselves on Day 8. Here’s why:
Start by thinking about the case in which there were only one stick figure with a messed up head. He would look around the room on Day 1, see no one with a messed up head, and realize that he must have a messed up head (since Statement 4 states that at least one stick figure has a messed up head). Since he’s perfectly obedient (as per Statement 3), he would erase himself that night.
Now imagine that there’s not one, but two stick figures with a messed up head—let’s call them Guy A and Guy B. On Day 1, both Guy A and Guy B would look around the room and see one stick figure with a messed up head and the other 98 with a perfect head. Guy A assumes that Guy B will see no one else with a messed up head, know he himself must have one, and erase himself that night.
But when Day 2 rolls around and Guy A sees Guy B emerge that morning, fully intact, he knows that Guy B must have gone to bed without the knowledge that his head was messed up. The only way that could have happened, since Guy B knows that at least one person has a messed up head, is if Guy B saw someone else with a messed up head. Since Guy A saw perfect heads on the other 98 stick figures, he concludes that he must also have a messed up head.
Guy B goes through the same thought process, so on Day 2, both of them know that they have messed up heads and both erase themselves that night at midnight.
Taking it a step further, if there are three stick figures with messed up heads (Guy A, B, and C), on Day 1, everyone can see three messed up heads, except the three who have them, who only see two messed up heads. So Guy A looks at Guy B and Guy C and thinks, “Those two don’t realize they have messed up heads and only see the other’s messed up head—so they will expect each other to erase themselves on the night of Day 1. But when Day 2 rolls around, they’ll see each other and realize the truth, that they have a messed up head too—and they’ll both erase themselves on the night of Day 2.” But when Guy A wakes up on the morning of Day 3 and sees both Guy B and Guy C alive and well, it hits him that he himself must also have a messed up head—which is the only explanation for Guy B and Guy C going to bed on Day 2 still not knowing if they had a messed up head.
Guy B and Guy C have this exact same revelation on Day 3, and that night, all three of them erase themselves.
Continuing this pattern, if Day 8 is the day the erasings finally happen, there would have to be eight stick figures with messed up heads. Each of them would have spent the retreat seeing seven other messed up heads and each would have gone to bed on Day 7 assuming that that night would be the night of the erasings. When all the messed up heads emerged on Day 8, each one of them then realized that they must have a messed up head too.
Make sense? How’d you do?