Puzzles where perfect logicians are told different pieces of information, and the reader must figure out something based on their words and actions.
The setup of meta-knowledge puzzles consist of several perfect logicians, each informed about different aspects of some secret piece of information. They then either have a conversation, or perform a series of actions, based on what they know. The goal is for the reader is either to determine what the secret is, or to figure out how these perfect logicians will behave.
A common theme in these puzzles is knowledge about knowledge. To determine how the parties in the puzzle act, you have to think about what the parties know, and their knowledge about what everyone else knows, and their knowledge about their knowledge about what they know, and so on. A useful concept is the idea of "common knowledge," which consists of what is known by everyone, and known to be known by everyone, and so on to all higher orders of knowledge.
Some famous examples are:
Blue Eyed Islanders: There are several people wearing hats who know everyone's hat but their own. They must leave the island on the first day that they figure out their eye color. An "oracle" visits the island, and tells the islanders something about the eye colors she sees.
Sum and Product: There are two unknown numbers. One person is told their sum, and the other is told their product. They then have a conversation, and the reader must figure out the tow numbers from this conversation.
Cheryl's Birthday: Cheryl tells Alice and Bob that her birthday is in a certain list of dates. She tells Alice the month, and Bob the day. After a conversation, you must figure out the birthday.