CONCEPTUAL NON SEQUITURS
This is a variant of the trial-and-error puzzle, a problem whose solution requires thinking so lateral that it's completely irrational. The term describes something along the lines of "put the sombrero on the bulldozer" or "sharpen the headphones with the banana." A few games try to get away with this by claiming that it's surrealism, but true surrealism is informed by some kind of underlying point; it's not just random weirdness. Chapter 12, "General Principles of Level Design," discusses conceptual non sequiturs at greater length.
A variant of this is the opposite-reaction puzzle, one whose solution turns out to be the exact opposite of what you'd expect. In the original Adventure, the player could drive away a menacing snake by releasing a little bird from its cage. Fortunately, at that point in Adventure, the player didn't have many options, so he usually found
the solution quickly. But unless you design an entire game on this principle, players may see it as just an annoying gimmick.
Illogical spaces were a classic challenge in text adventures. If you went north from room A, you got to room B, but if you went south from room B, you didn't necessarily go back to room A. Modern games use teleporters to provide a similar effect; the player may step out of a teleporter with no idea where it has taken her. In such a space, the player simply has to wander around taking notes until she can figure out the relationships among the various locations. Unless you offer some clues, this is another puzzle that can be solved only by trial and error.
Many adventure games include references to things outside the game world for comic effect, but those references shouldn't be part of a puzzle. A game that requires the player to know information from a source other than itself is unfair.
For example, Haunt offered puzzles that only players familiar with the movie Monty Python and the Holy Grail could solve. It didn't really matter because Haunt was a game made by a student for fun, but in a commercial game, such puzzles would be unreasonable unless you explicitly make it clear that the game requires the player to know trivial facts. If you want to make humorous references to popular TV shows, movies, and so on, do them in narrative events or in an NPC's conversation rather than as solutions to puzzles. Beware, though: Cultural references age quickly and will make the game seem dated after a few years.
You have to be even more careful when developing games for foreign markets because other countries don't always have the same idioms. For instance, the action, "Wear the lampshade on my head," could cause other characters in the game to assume that the player's avatar is drunk, which might be desirable in the context of the story. However, wearing a lampshade as a sign of drunkenness is an American cultural idiom that might not be understood in, say, Japan. Again, it's OK to make cultural references in your game; just be careful about requiring the user to understand them in order to win.