You say: "Do the same puzzles over and over? I'm worried that once I've memorised a puzzle, then I'm not doing so much calculation as memory retrieval."
Memory retrieval is exactly what you should be doing! The brain improves recall not by repeated input, but by repeated output. This is why reviewing notes is a terrible way to study for a test, and why memorizing and recreating your notes from scratch is what top performers do.
The concept that should be driving your improvement is chunking. Chunking is the process whereby you are able to immediately recognize more complicated concepts without thinking. You still calculate during a live game, but instead of spending 60 seconds working out a tactic, you should learn to see that tactic immediately so that you can spend that 60 seconds on something important. If you cram tactical patterns into your brain, and learn to see them in less than 1 second, while your opponent is spending 30-60 seconds reinventing the wheel, you are going to crush your opponent. On the flip side, if you are trying to calculate everything at the board, you are going to lose. You improve when you are away from the board. Once you are at the board, you only play. You only have the skills you developed away from the board.
An example of chunking is when you learned to read. First you learned letters, then small words, then bigger words, then phrases, and sentences. Now when you read, you do not look at some writing and say, "there is an H, there is an I", nor do you even look at it and say, "there is the word Hi". You see the word "Hi" and you immediately understand the meaning without thinking.
Learning to walk is another example. As an adult, you do not think, "move the left foot, catch your balance, now move your right foot". No, you walk down the street without thinking. If a car is going to hit you, you run as fast as you can without thinking. Your body and mind learned how to handle these more complex situations and they know what to do without thinking. The same happens with chess improvement.
For chess, you want to reach the point where a simple tactic presents itself on the board, and it pops off the board at you without thinking. Once you can do this, you can move on to more complex tactics and combinations that require deeper calculation, but until you can recognize simple tactics without thinking, your mind is moving far too slow to attempt deeper calculation during a live game. As long as you are trying to do difficult tactics problems involving 5-move combinations, before you can see simple tactics instantly, you are never going to improve.
The approach I have seen recommended by masters (NM Dan Hesiman, GM Rashid Ziatdinov, among many others), is to get a simple set of tactics problems such as John Bain's Chess Tactics For Students, and you do the same tactics problems over and over. Your goal is not to figure out the tactic on your own. The goal is to get as many patterns into your brain as you can, so that when you see a similar pattern during a live game, it is immediately clear to you without thinking. Doing tactics problems is not about achievement and getting a good percentage correct. It is only about cramming patterns into your brain. That means, if you do not see the answer quickly, look at the answer and move on.
The plan is:
- Do simple tactics problems over and over, until you can fly through all of them without making a mistake.
- If you don't see the answer after 30-60 seconds, look at the answer and move on. Come back to the problems you missed later.
- Stick to one tactical theme at a time. Once you have mastered one tactical idea, only then start another.
- GM Ziatdinov recommended this approach: Number your list of tactics problems. Start doing problems 1-10 until you can do them all perfectly. Then do problems 11-20 to perfection. Then repeat problems 1-20 to perfection. Now do problems 21-30, then repeat 1-30. Repeat this until you can do all problems perfectly. He had a list of over 3000 tactics problems on his website, and he said if you followed this approach you would have the tactical ability of a grandmaster.