Stick Figure Puzzle

Wait But Why holds a work retreat for all of its stick figures. Everyone’s excited about it—it’s a chance to get to know each other better, which is rare because they’re typically confined to their own drawings and not able to socialize.

Soon, though, things take a dark turn, as it is announced that Wait But Why can no longer employ all of them and needs to downsize.

The decision is that all stick figures with perfect head circles will keep their jobs, and those whose heads Tim messed up by not connecting the circle will be erased.

The thing is, stick figures can’t look in the mirror and there’s no way for them to tell what their own head looks like. They can see all the other stick figures’ heads, but not their own.

To avoid the awkwardness of having to tell the messed up drawings that they’re going to be erased, Tim and Andrew make the following statements and then spinelessly leave the retreat:

Statement 1: From here forward, there will be no mention of anyone’s head quality whatsoever, including any facial expressions or body language that could give someone information about what their head looks like.

Statement 2: If at any point you discover that your head is messed up, you are required to erase yourself that night at midnight.

Statement 3: All stick figures must be perfectly obedient to the rules, perfect at logic, and aware that all the others are perfect at logic as well.

Statement 4: It is announced that there are 100 stick figures present at the retreat and at least one stick figure present has a messed up head.

Disturbed, the stick figures go on with Day 1 of the retreat and try to enjoy themselves as best they can. That night at 10pm, they all head to their bedrooms and go to sleep.

The next day (Day 2), the stick figures come out for the day, and everyone is still there. Same story on Day 3. In fact, it’s the same story through Day 8.

But when the stick figures emerge from their bedrooms on the morning of Day 9, the only ones present have perfect heads. It seems that all stick figures with messed up heads erased themselves on the night of Day 8.

The question is: How many stick figures erased themselves on Day 8 and why did things play out this way?

This is based on a famous logic puzzle. Think long and hard about it before you look at the solution. There are two hints below to help you along the way if you’re stuck.

[expand title=”HINT 1″]How would things play out if only one of the stick figures present had a messed up head?[/expand]

[expand title=”HINT 2″]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?[/expand]

[expand title=”SOLUTION”]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?[/expand]

If the hint and solution links won’t expand on your device, view them here.

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More puzzles on Wait But Why:

The Three Jellybean Problem

The Puzzle of the Pirate Booty

The Two Envelopes Problem

The Infinite Checkerboard Quandary

And one very big puzzle:

What Makes You You?

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