I almost missed the Cheryl’s Birthday Party internet phenomenon this week. An awkwardly written logic problem went viral, and the internet was abuzz with attempts to solve it. Check out the NYTimes treatment of the origial pnroblem, and the afterparty.

The problem is an example of a metalogic puzzle. Logic puzzles usually consist of organizing what you know and don’t know to suss conclusions out of ambiguous seeming clues. Metalogic puzzles are a different sort of beast: they require you to imagine what characters know and don’t know to solve problems. In a metalogic puzzle, a character not knowing something can be critical information.

Here’s some examples of my favorites.

#### The Three Hats.

Three princesses are taken prisoner by an ogre. For sport, the ogre puts them in a row from eldest to youngest, so that each sister can only see her younger sisters. Then he blindfolds them and says, “I have three white hats and two black hats. I will put one hat on each of your head. If any one of you can tell me the color of your own hat when I take off your blindfold, I’ll let all three of you go.”

The ogre takes the blindfold off the eldest princess. She looks ahead at the hats on her two younger sisters, thinks, and finally says, “I don’t know what color my hat is.”

The ogre takes the blindfold off the middle princess. She looks ahead at the hat on her younger sister, thinks, and says, “I don’t know what color my hat is.”

Then the ogre takes the blindfold off the youngest princess. She looks ahead into the woods–she can’t see anyone else’s hat. Then she thinks, and says, “I know what color my hat is.”

What color is her hat? How did she know?

I love this puzzle, because it doesn’t seem like there’s enough there to answer it. The secret lies in imagining yourself in the head of each princess as they imagine themselves in the head of their sisters. It requires a fantastic leap of imagination. I’m not going to solve it for you, but consider:

What does the eldest sister know about her sisters’ hats?

What does the middle sister know from the fact that the eldest sister doesn’t know her own hat? What does it tell us that even with this information she doesn’t know the color of her own hat?

I’ve recently come up with my own series of metalogic puzzles. I don’t think these are too hard if you can get past the fact that they seem so devoid of usable information to start.

#### The Known Unknown

Abby and Bill play a game, where they each think of a whole number and try to guess each other’s number by asking each other questions which must be answered truthfully.

*Game 1*

Abby and Bill each pick a number in the 1 to 30 range.

Abby: Is your number twice my number?

Bill: I don’t know. Is your number twice my number?

Abby: I don’t know. Is your number half my number?

Bill: I don’t know. Is your number half my number?

Abby: I don’t know.

Bill: I know your number.

What is Abby’s number?

*Game 2*

Abby and Bill each pick a number in the 1 to 40 range.

Abby: Is your number twice my number?

Bill: I don’t know. Is your number twice my number?

Abby: I don’t know. Is your number twice my number?

Bill: I don’t know. Is your number twice my number?

Abby: I don’t know. Is your number twice my number?

Bill: I don’t know. Did you think we chose the same number?

Abby: We didn’t.

What are their numbers?

Finally, here’s a classic that I’d say is a level more difficult, and seems even more impossible to solve than the previous ones. It’s definitely doable though–you just need to remember who knows what, and write it all down.

#### The Surveyor’s Dilemma

A surveyor stops at a mathematician’s house to take a census.

Surveyor: “How many children do you have, and how old are they?”

Mathematician: “I have 3 children. The product of their ages is 36.

S: “That’s not enough information.”

M: “The sum of their ages is the same as my house number.”

S: “That’s still not enough information.”

M: “My eldest child is learning the violin.”

S: “Now I have enough information.”

What are the ages of the mathematician’s children?