So Beedo wrote a nice post about puzzles in his Mad God megadungeon.
I got to thinking about how puzzles fit into a game. Beedo's posts are generally like that. I figured I need to keep something in mind when I put in puzzles - just like when I put in traps, monsters, etc. It's a continuum between two extremes - purely player-centered and purely character-centered.
On one end of the spectrum, a puzzle is purely player centered. A player-centered puzzle is one that depends on the knowledge of the players. Chess match puzzles that depend on the players knowing chess moves are like this. Factual puzzles (like the golem number puzzle in S2 White Plume Mountain), logic puzzles, riddles, jokes missing punchlines - all of those things generally purely test the player, not the characters. It doesn't matter if your wizard is a genius if you don't know how to recognize the mathematical progression pattern in that door combo, or whatever.
On the far end of the spectrum is the character-centered puzzle. A character-centered puzzle is pretty much like any other skill check, combat "to hit" roll, damage roll, etc. - do you roll well enough to accomplish your goal of getting past it? You roll the dice, you compare your IQ to the Mind Flayers' puzzle box's required IQ score, whatever. No amount of player skill impacts the results any more than player skill impacts how that d6 lands when you roll damage - discounting cheating, of course.
Of course, you can mix these anywhere along the spectrum. You can place a minimum required character aspect, or require a character-based roll, in order to solve a problem that is otherwise player-knowledge required. The players might need to figure out what to do, but their characters must execute in-game tasks successfully to manage it. (Actually, that also describes combat in almost every game I've played.) Or you can have a puzzle that only needs to be identified as such before a character roll solves the whole thing.
Let's try a few examples.
Purely Player Knowledge: A statue-moving puzzle. Put the statues in alphabetical order by name, and a magic door opens. No character ability matters.
Mainly Player Knowledge: A statue that utters a complex math puzzle, which requires the players to do the math to succeed. However, the statue only speaks to those with a minimum level of Magery and/or Thaumatology skill (or minim levels in Magic-User, say). Others are just not worthy of the challenge, and the statue will ignore their words even if they speak the answer.
Minimal Player Knowledge: A chasm blocks your way and has a magical gust of wind that blows when you try to cross it. Player knowledge can identify the problem and the solution, but character skills such as Climbing skill or a Dispel Magic spell are needed to resolve it.
Pure Character Skill: A magic door will only open when a complex magical puzzle lock is opened. All it takes is an IQ-based lockpicking roll to open it.
There are plenty of wayposts on the continuum, from either direction. But I think it's worth remembering that pretty much all game systems abstract some things to player ability, and some to character ability. Puzzles are just another aspect of it. You need to consider when you put it down if it's primarily a character challenge or a player challenge.
Of course, remember that players treat puzzles the way Alexander treated the Gordian Knot - they might find an option you weren't expecting . . .