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Do you just grind them, or do you save some of the chess tactics that you miss?
Do you ever annotate your chess tactics?
Do you try to remember certain patterns and main ideas? Do you categorize your tactics by forks, pins, zugzwang, etc.?
How do you recommend we train tactics puzzles?
There is an article about tactics training at ChessTempo.com
http://www.apronus.com/chess/trainingtactics.htm
It deals with some of the issues raised in the question.
Training chess tactics should be done by theme. This means solve a set of tactics that are based on a similar theme, like Double Attack, Discovered Check, Absolute Pin, Relative Pin, Skewers, etc. This impresses upon the mind the ideas of that particular kind of tactic and gets your brain looking for it as a matter of course. After, you will find yourself 'seeing' these tactics (i.e., imagining) in your games or at least considering them as a means to an end in your positions. This is the creativity you cultivate in your chess game.
Well this is the sort of thing that changes from one person to another: there's no unique/best way of going about it. Almost everything you say are in general good recommendations. At least from my personal experience, most important is to find a routine for solving puzzles on a regular basis. Each puzzle contains at least a key idea that you can take away, mainly in terms of the structure and the weakness in it to be exploited in the tactic. Doing these on a regular basis fills up your bag of ideas as they often push you to think outside the box, but be patient, it takes time for them to residue and to lurk into your play in a natural way.
Another important aspect of puzzle solving is the fact that it is also by definition a calculation exercise, either before or after you've spotted the key idea. So it is strongly recommended to patiently1 solve puzzles without an analysis board, instead solve them entirely in your head, even if it requires a depth of 4-5 moves, and sometimes a bit more. This will improve both your speed and precision of calculation.
Finally, in terms of organization, almost all sources of chess tactics nowadays, be they books or websites2 offer categorized puzzles, in terms of checkmates, pins, double attacks, endgames, and so on, or options to do them all mixed in. So just find a rhythm for doing as many of them as possible, whenever you feel fresh.
1: Most importantly, don't try the first idea you see, once you spot a potential solution, check further to see if you cannot find a better one. Most puzzles often have only one solution (or one accepted one), so you have to think optimally in each step of the way. In other words there's definitely grinding involved :)
2: Just to name a few sources worth considering: chess24, chess.com and lichess.org all offer a wide range of tactics trainers, with small differences between them.
In his award winning book, Pump up you Rating, Axel Smith advocates something he calls the "Woodpecker Method". This is named after Hans Tikkanen (Tikkanen is Finnish for woodpecker) who first used the method to obtain 3 GM norms in 7 weeks.
There are a number of important principles in the method.
First you solve large numbers (larger numbers (1000s) for professional players, smaller numbers (100s) for amateurs) of tactical problems sorted by motif. When you have completed them you have a short break (a day or two) and then work through them again. You do this 4 or 5 times in a similar way to spaced repetition but without the same longer spacing between repetitions.
When you have done this you repeat with mixed motifs, so you don't automatically know what type of thing you are looking for and maybe include problems with no solution.
Second, the way you solve them is disciplined and has some rules. These are -
1) You do them against the clock, either something like 15 minutes per problem or 45 minutes for a set of 3. If your flag falls and you haven't written down at least a move for each problem you were supposed to solve then you "lose".
2) Even if you can't solve the problem you must choose a move, as you would have to do in a real game, and like in a real game you solve them in order. You are not allowed to go back.
3) Award yourself points. Smith gives a couple of scoring systems.
4) Calculate the variations through to the end. This is important when you do the repetitions. Don't just note the solution move.
The use of the clock and the points system is important to help you measure progress. You can use these to set targets which you must reach before going on to a new set of exercises.
The source for the tactical exercises is not important. What is important is that the solutions are given with extensive annotations so that you can check your calculations of logical defences and promising tries that don't quite work.
I never return to old tactics unless it is masterpiece. I don't sort them by motives, just calculating the best move and go on. I think mainly the calculation itself is important and if there is some new motive, you just remember it. I solve hard positions, taking time. Some strong players like plenty of easy positions, even repeatedly solving same positions. To me this looks like a waste of time.
Look for checks, captures, and threats.
Look for any undefended pieces.
Visualize where a tactic is possible. Can I pin the knight by moving my bishop here, and will it not leave my position vulnerable? Or, can I force the queen to move to this square, so I can checkmate/produce a royal fork/get him to resign/etc.
I like @phonon's answer; but still I guess some sub-questions are not answered.
Depends on the difference between your chess level and chess level of the audience the puzzle is intended to. If the puzzles are from books (esp. course-like structured series), authors will typically mention the level of the puzzles. For example, in winning chess series, Seirawan recommends time to be spent on puzzle and so on. The same is true for other series such as Yusupov's. If the puzzle is not a study, you should try to grinding and if you can't solve it in 'reasonable time', save it for later. If it is a study, one may be grinding against stone; but still that can be instructive (or not). What is 'reasonable time' also varies. To make the learning process more smooth and enjoyable, I would recommend doing puzzles that are just above your level of play.
2.Do you ever annotate your chess tactics?
Annotation doesn't matter much. But, analyzing all important variations can not be overemphasized. Writing down seems to help it.
IMO, you should set the problem on board and solve rather than solve looking at a figure. That would be a better practice. (I have seen master players pointing out this)
3.Do you try to remember certain patterns and main ideas? Do you categorize your tactics by forks, pins, zugzwang, etc.?
In fact, no need to try to remember things. It is more like getting familiar with it. Practicing with particular patterns or particular themes helps for sure.
IMO, you should mix it up. If you know for sure that you are to use a particular patter only, you can easily figure it out. People (incl. professionals) say that it imprints the pattern in your mind. May be, it does. But it kills all the fun. I would suggest noting down the puzzles and note down some positions from random games (need not be master games) and mix it up and try to solve the position as if you are facing the position in a game over the board.
4.How do you recommend we train tactics puzzles?
This question is answered very carefully by @photon. I don't have much to add; I am only expressing an opinion.
Also go through the recommendations of Seirawan and Yusupov they give on how they expect their reader to try out the problems in their book. You can not go wrong with it. (One such recommendation is fight not only against the problem, but also time as pointed out by @Brian Towers)
Find a set of puzzles that fits your level best (not too easy not too hard; may be a small percentage of them challenging). Just to reiterate, solving over the board is a better practice whether the source is a book or not (eg:a software or website).
In my opinion it depends on how well you can understand and memorize them. The next time you want to play chess try different methods, such as the methods you know before and the methods which are suggested by others and try to work out which of them is easier and more comfortable for you. It's not one method for all. Try to find what is suitable for you.
Be successful
