Please help me draft a systematic approach to puzzle solving.

Please (constructively) criticize my initial draft so that I can improve it. Copy/paste it and print it out if you find it helpful.

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Systematic Puzzle Solving:

1. Material. Important in order to know whether to look for stalement or not.
2. General Analysis of Position.
   1. Are there threats that must be dealt with? Important in order to know whether you are looking for a way to defend or win material/checkmate/stalement.
3. Candidate Moves.
   1. Consider forcing moves first.
      1. Checks?
      2. Captures?
4. Calculate Specific Lines (based on the candidate moves chosen).
   1. Evaluate position at the end of each line.

Additionally, keep in mind what you are looking for:

1. Double attack
2. Pin
3. Skewer
4. X-Ray
5. Discovered attack
6. Removing the guard
7. Mate
8. Stalemate
9. Trap

Tips and Considerations:

1. Quality over quantity—Take your time, understand, and remember/memorize the pattern.
2. Consider different move orders.
3. Don’t forget to look for traps.
4. Don’t forget to look at special moves: castling and en passant.
5. Practice specific tactics (e.g., pins, double attacks, etc).
6. When you find what you believe is the correct move, take a second to consider if there is an even better move (e.g., wins more material).
7. Calculate lines all the way through.
8. When you miss a tactic (after taking your time), write down which pattern it was. Is a particular type of tactic your weakness?

Well, I think you should always just look at the position without thinking so hard, just look around. 

The reason you can solve easy puzzles in 2 seconds is because you see the answer through intuition or pattern recognition. 

The issue with each of your categories is that you need intuition to guide you. For instance, how do you know what the opponent's threats are? How do you calculate when there are so many candidate moves, and more possible responses? How do you know when to stop calculating? How do you know when there's a trap?

It's really all about intuition. It explains why Hikaru can find tactics in 1+0 bullet that a lot of players can't find given 30 minutes to analyze the position.

Eventually I'm planning on launching a similar project, but it is way too huge a project to tackle in a single thread. Pattern recognition is a major contributor to narrowing the search, and you haven't even addressed that yet. Also the question of forward search versus backward search, which phase of the game it's in, a repertoire of tricks, algorithm representation, and more are important considerations, so those make this an absolutely huge project.

If it's a hard puzzle, yeah, first I look for whether there's a material difference, and then a general evaluation.

One thing you didn't include, which I think is helpful, is listing the opponent's biggest threats. I generally look for all checks and 1 move threats that can attack a queen or rook (no matter how suicidal or crazy the move might look).

Then I look for all my one move checks, captures, and threats.

If it's a mate puzzle I like to remind myself exactly how many squares around the enemy king I have covered. Sometimes the pattern of free squares helps give you ideas. I.e. if it's a diagonal then you stat looking for ways to get a knight or bishop close.

But yeah, after that it's sort of free form. A mix of pure calculation, pattern recognition, fantasy positions, and maybe other stuff. I only follow a strict routine if I'm feeling tired as if I'll probably miss simple stuff, or if I know the puzzle is harder than what I can normally solve.

I sometimes count pieces, maybe subconsciously I always do. It's obviously an important thing. However, a lot of tactics I get are perpetual check / forcing a repetition. That's why I don't think too much when I do the tactics, unless I feel it's a really technical line (like an endgame study or something). I just play the tactic that feels right. Otherwise you take too long.

Obviously you shouldn't use this approach in a real game, playing tactics just because you feel like it.

Try solving this puzzle!

Qxe8# :c

This position where white has no king and every legal move wins is a bit too bizarre.