user12396: I have a collection of middle-game chess puzzles classified in levels. I can solve most puzzles in the first level after a few minutes, and some of the puzzles in level 2.
But for higher levels there is no hope I can solve any of them, even if I tried for a thousand years. Nevertheless, those are supposed to be interesting positions with hidden treasures, and I wonder if I can still learn something from them. What should I do? Play against a computer from those positions until finding the solutions? Look at the solutions and memorizing them?

user11245: Delivering instruction in how to solve tactical problems is much too complex a topic to address in a forum post, unfortunately. The main points are these:

You need to learn to recognize 1st-level tactics when they appear. Most players can do this.  1st-level tactics include things like forks, skewers, pins, etc. You will only be able to exploit this skill (by itself) if your opponent blunders.
You need to recognize situations that may lend themselves to a 1st-level tactic after a short sequence of 1 or more forcing moves. Forcing moves are mate threats, captures, checks, and threats to capture. I call these 2nd-level tactics, because they involve a setup before the 1st-level tactic appears. 

Such situations include possibilities such as being able to force an alignment of the opponent's queen and king that may then become vulnerable to a pin or skewer.

Finally, you need to recognize situations where no 1st-level tactic is available, and the opponent has replies that would frustrate a series of setup moves that would otherwise work.  It may still be possible to employ one or more other, supporting tactics to eliminate those replies, thus allowing the setup to work. Then, the 1st-level tactic will become possible.  

I call these 3rd-level tactics, because of the additional preparatory moves that must be played to eliminate the opponent's defensive options prior to (or during) the setup to the final, 1st-level tactic.

Once you have mastered some degree of 3rd-level tactics, you will be able to move onto 4th-level tactics. These involve multiple 2nd- and 3rd-level tactics in a sequence.

This is my personal approach to understanding combinations, and there is much more theory around it than I can relate here. I am writing a book to cover this gap in current teaching.
Until you can find such a book, I strongly recommend George Renko's Intensive Chess Tactics Course, now in its 2nd edition, from ChessBase.  Renko has spent many years codifying this classification of tactical themes (of which he has enumerated 112).
Please note he has made a claim of plagiarism against Convekta for their product Encyclopedia of Chess Combinations by NM Kalinichenko, which is an exact copy published several years after Renko's.  Snopes.com has confirmed this claim. For this reason I advise against purchasing the Convekta product.

user10233: This is a decent question because it tries to delve into the thinking process of solving a tactical puzzle. Of course, Tactical motifs and Positional motifs are important but you gain those by experience or study time, and I believe what the asker is looking for is a methodology he can apply to a difficult puzzle in hopes of solving it. So, with that in mind:
Note all captures, checks, and unprotected or insufficiently
protected pieces. Visualize each separately, alone. When you do this, you begin to connect some dots in the position, like "if I take here and he takes, that leaves this piece undefended...", etc., etc.
Note all potential double-attacks, probably the most common tactic in chess. 
Visualize all avenues of attack for every long-range piece - Bishops, Rooks, and Queens; this means mentally ignore any obstructing enemy or friendly pieces, i.e., Bd3 looks down the b1-h7 diagonal regardless if there are black or white pieces in the way. This paves the way for clearance sacrifice ideas, and discovered attacks.
Visualize all potential knight-forks, and all knight-forks after the move of the knight. 
Honestly, I would start there and avoid using the computer. It will not teach you anything you cannot figure out yourself - you just need to be persistent. 
I would also write it all down so you can drop the puzzle and come back to it later. You are gaining good habits by doing this, and it gives you valuable references.
Only then when you are completely spent on the puzzle do you pass it to Stockfish.
2c.

user11629: Chess puzzles (and to a large extent playing chess) are based around recognizing certain patterns or motifs which help you to quickly assess a position and to visualize potential lines without the need to calculate move by move in your head. 
Chesstempo has a good list of tactical motifs and positional motifs. It is important that when you solve problems (or when analyzing your games) that you realize which motives (usually there is more than one) apply to this problem. If you cannot solve a problem on your own it is fine to look up the solution or to get help from a computer, but make sure you understand the solution (i.e. match them to motifs/patterns). Again chesstempo is very useful for this as it shows you the motifs of each problem as "tags" (after you solved it or after you failed to solve it).
Once you have built a repertoire of patterns/motifs in your mind you should have a good basis to solve more complicated problems as well.
Another factor could be visualization when calculating long lines, but there is not much you can do about it other than concentrating, playing games, analyzing games, solving puzzles.... basically everything you should be doing anyway.

12396: <p>I have a collection of middle-game chess puzzles classified in levels. I can solve most puzzles in the first level after a few minutes, and some of the puzzles in level 2.</p>

<p>But for higher levels there is no hope I can solve any of them, even if I tried for a thousand years. Nevertheless, those are supposed to be interesting positions with hidden treasures, and I wonder if I can still learn something from them. What should I do? Play against a computer from those positions until finding the solutions? Look at the solutions and memorizing them?</p>
 11245: <p>Delivering instruction in how to solve tactical problems is much too complex a topic to address in a forum post, unfortunately. The main points are these:</p>

<ol>
<li><p>You need to learn to recognize 1st-level tactics when they appear. Most players can do this.  1st-level tactics include things like forks, skewers, pins, etc. You will only be able to exploit this skill (by itself) if your opponent blunders.</p></li>
<li><p>You need to recognize situations that may lend themselves to a 1st-level tactic after a short sequence of 1 or more forcing moves. Forcing moves are mate threats, captures, checks, and threats to capture. I call these 2nd-level tactics, because they involve a setup before the 1st-level tactic appears. </p></li>
</ol>

<p>Such situations include possibilities such as being able to force an alignment of the opponent's queen and king that may then become vulnerable to a pin or skewer.</p>

<ol start="3">
<li>Finally, you need to recognize situations where no 1st-level tactic is available, and the opponent has replies that would frustrate a series of setup moves that would otherwise work.  It may still be possible to employ one or more other, supporting tactics to eliminate those replies, thus allowing the setup to work. Then, the 1st-level tactic will become possible.  </li>
</ol>

<p>I call these 3rd-level tactics, because of the additional preparatory moves that must be played to eliminate the opponent's defensive options prior to (or during) the setup to the final, 1st-level tactic.</p>

<ol start="4">
<li>Once you have mastered some degree of 3rd-level tactics, you will be able to move onto 4th-level tactics. These involve <em>multiple</em> 2nd- and 3rd-level tactics in a sequence.</li>
</ol>

<p>This is my personal approach to understanding combinations, and there is much more theory around it than I can relate here. I am writing a book to cover this gap in current teaching.</p>

<p>Until you can find such a book, I strongly recommend George Renko's Intensive Chess Tactics Course, now in its 2nd edition, from ChessBase.  Renko has spent many years codifying this classification of tactical themes (of which he has enumerated 112).</p>

<p>Please note he has made a <a href="http://www.avlerchess.com/chess-misc/Chess_plagiarism_289873.html" rel="nofollow noreferrer">claim of plagiarism</a> against Convekta for their product Encyclopedia of Chess Combinations by NM Kalinichenko, which is an exact copy published several years after Renko's.  Snopes.com has confirmed this claim. For this reason I advise against purchasing the Convekta product.</p>
 10233: <p>This is a decent question because it tries to delve into the thinking process of solving a tactical puzzle. Of course, Tactical motifs and Positional motifs are important but you gain those by experience or study time, and I believe what the asker is looking for is a methodology he can apply to a difficult puzzle in hopes of solving it. So, with that in mind:</p>

<p>Note all captures, checks, and unprotected or insufficiently
protected pieces. Visualize each separately, alone. When you do this, you begin to connect some dots in the position, like "if I take here and he takes, that leaves this piece undefended...", etc., etc.</p>

<p>Note all potential double-attacks, probably the most common tactic in chess. </p>

<p>Visualize all avenues of attack for every long-range piece - Bishops, Rooks, and Queens; this means mentally <em>ignore</em> any obstructing enemy or friendly pieces, i.e., Bd3 looks down the b1-h7 diagonal regardless if there are black or white pieces in the way. This paves the way for <em>clearance</em> sacrifice ideas, and discovered attacks.</p>

<p>Visualize all potential knight-forks, and all knight-forks after the move of the knight. </p>

<p>Honestly, I would start there and avoid using the computer. It will not teach you anything you cannot figure out yourself - you just need to be persistent. </p>

<p>I would also <em>write it all down</em> so you can drop the puzzle and come back to it later. You are gaining good habits by doing this, and it gives you valuable references.</p>

<p>Only then when you are completely spent on the puzzle do you pass it to Stockfish.</p>

<p>2c.</p>
 11629: <p>Chess puzzles (and to a large extent playing chess) are based around recognizing certain patterns or motifs which help you to quickly assess a position and to visualize potential lines without the need to calculate move by move in your head. </p>

<p>Chesstempo has a good list of <a href="http://chesstempo.com/tactical-motifs.html">tactical motifs</a> and <a href="http://chesstempo.com/positional-motifs.html">positional motifs</a>. It is important that when you solve problems (or when analyzing your games) that you realize which motives (usually there is more than one) apply to this problem. If you cannot solve a problem on your own it is fine to look up the solution or to get help from a computer, but make sure you understand the solution (i.e. match them to motifs/patterns). Again chesstempo is very useful for this as it shows you the motifs of each problem as "tags" (after you solved it or after you failed to solve it).</p>

<p>Once you have built a repertoire of patterns/motifs in your mind you should have a good basis to solve more complicated problems as well.</p>

<p>Another factor could be visualization when calculating long lines, but there is not much you can do about it other than concentrating, playing games, analyzing games, solving puzzles.... basically everything you should be doing anyway.</p>
