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How do I improve at tactics

My score is all over the place. 1200-1500. What can I do to improve? I'm going to continue to do tactics, and I'm watching videos on YouTube, but what else?

I think I need help performing calculations when there is a square attacked and defended by 2 or 3 pieces on each side. And sometimes when a piece has been taken and the puzzle continues and i'm expected to move the piece to safety it's not clear to me why the given square is the one I should have chosen. I appreciate that this is the point of good chess play, but I feel I need some help getting there.
Before taking up on puzzles, I suggest to learn the basics of combinations like pin, skewer, fork etc. With that knowledge it is easier to understand more complicated tactics.
yes I think so too, and also playing 'easy' wins such as rook+king vs king, but try to find the most exact solution.

and then repeat...
Yes, firstly learn the names and patterns of the tactical motifs. You can find the complete list on chesstempo website.

A simple way to see if a piece is under-protected is to count the number of attackers and defenders. If say black's knight is attacked by 3 different attackers, black needs at least 3 defenders to keep material balance. If black has fewer defenders, white will win material. However be careful as this is many a times an oversimplified view since there are intermezzo moves (do research on this I shan't do the work for you).

Regarding about moving the piece to safety after capturing a piece, I believe you might be referring to those tactics involving unsound sacrifices. It is unsound because you may take the sacrificed piece and move that piece you took with to defend the threat that was posed. There might be only 1 correct square to put your piece because of the incoming threat which could lead to checkmate or huge material losses if placed incorrectly. Usually this involves protecting the square/piece that your enemy threatens to attack. If you use lichess puzzles, it is easy to know why your move is wrong since you can always toggle analysis and see the best reply from the opponent after your move. However if you are still unsure why your move is wrong in the case of complex puzzles, you can always pose your question to the wider community here.

For me, I record down the puzzles that I have gotten wrong by saving the link and redo them periodically. Always analyse puzzles that you have gotten wrong or you will be wasting your time.

Good luck and try not to spend more than half your time learning chess on tactics. Endgame and middlegame strategy is of importance too. Basic opening ideas as well.
Does "solving puzzles" (referred to as tactics trainer) directly lead to improving skills in playing a game of chess?

I argue no. Indirectly, over the course of viewing/studying puzzles, patterns emerge that become recognizable. When playing a game, tactics learned may be remembered and a best move is played, but puzzle like positions occur maybe once in every game.

The problem is, when playing a game, each move to be played is not thought of as a "puzzle" where 1 or 2 specific moves decide the result. The thought process of choosing which move to make when playing games is quite different than the one used to "solve a puzzle". A puzzle like position happens in games at a critical position, but in fast time controls as blitz, time does not permit a "study" of the position.

Just my opinion. I think many incorrectly believe by increasing their tactics rating their game play rating should be directly affected as also improving. My assessment in arguing no is based on most people here are playing 10 minutes or less. For longer time controls and OTB play where each move is analyzed deeper, tactics trainer obviously will provide a greater benefit. But with making mistakes, finding yourself in bad positions, tactics training is of little help.

As to how to improve at learning tactics? I suggest researching how people "learn". There are many good simple tests that reveal how each persons learning process works and how they are best suited. An example: Often people think they learn best by "visualization" methods, but in fact another approach better suits them. I see in profiles people completing dozens of tactics training puzzles daily. Maybe quality over quantity will prove of greater benefit. As in the saying "reading just 1 good chess book and understanding it, is far better than reading a 100 and not understanding any of it."

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