user31998: I am a big fan of solving mate in N problems and believe they help tremendously with i.e. visualization. The only issue I seem to face in certain problems are when they are very vague. For example this one from the famous Polgar book. How can one proceed to solve it? I know the answer now, but it is very vague. Especially when there is no zugzwang or check to start with.
[Title "Bela Schläffer, Magyar Sakkelet, 5/1952, Mate in 2"]
[fen "nK6/4N1p1/pk4r1/p1b5/p6p/2N5/p3RB1Q/1q6 w - - 1 1"]


user26382: "Hmm, the rook is the only piece holding the 6-th rank... So if only I could distract it, the queen or the rook would deliver mate. Oh, and the black queen is doing exactly the same but vertically. Both heavy pieces are completely tied by their defensive duties. Can I somehow bother them, especially with a fancy move? Qg1, of course." Then it's only left to ensure all the lines work.

user15049: First, usually it is not a move which gives check and not a move which takes a piece, because this is considered to be not elegant in modern studies. For example 1.Qd6+, 1.Qc7+, 1.Bxc5+, 1.Re6+, check with a knight on d5, c8 or a4, these are probably not the solutions. Ignore these moves and only come back to them when you did not find the solution using the following method:
Step 1 (this step can also be left away in mate in 2 studies): Check what would happen if it was black's move. Which moves would lead to a white mate in one? From this you can learn the weak spots of blacks pieces. E.g. here you can learn that a queen move which loses control over b2 may lead to 2.Rb2 mate. A rook move away from the sixth rank may lead to 2.Re6 or 2.Qd6 mate. His bishop is pinned, if he moves it, this leads to 2.Bxb or 2.Qxf2 mate. If he moves his rook to c6 this leads to 2.Ned5 mate, if he moves his queen to b5 it leads to 2.Ncd5 mate.
Step 2 Act as if you had two moves in a row. Which maneuvers would lead to mate? This would be 1.Kc8 and 2.Qb8 mate, 1.Qh1(g2) and 2.Nc8 mate, 1.Re5 and 2.Bxc5 mate, Qe5(g1) and 2.Bxc5 mate.
Step 3 Go through the list of maneuvers, found in step 2 and consider all black answers to the first move of the maneuver. The 'bad' black moves from step 1 may give you indicators which maneuver leads to success. Here we would find that the following attempts do not work: 1.Kc8 Rc6+!, 1.Qh1(g2) Qe4! 2.Rb2+ Qb4, 1.Re5 or Qe5 Qb4!.
This leads to the conclusion that 1.Qg1! has to be the move and by checking all possible black answers we can find that each leads to a mate in one, see the lines given in Brian Towers' answer and in RossMillikan's comments. Especially the fact that the Qg1 attacks the Rg6 makes the difference, 1...Qb4 now leads to 2.Qxg6 mate.
ChessBase also has a good article which gives further hints: How to solve chess problems.

user11144: First, you have to split into "gamelike 2#" and "artistic 2#". The approach is, at least in principle, completely different.

In a gamelike 2#, proceed as if you are looking at a normal game. That is, analyze checks, captures, strong threats first.
In artistic 2# like your example (and it's a major crime to reprint art without attribution :-); but as a problemist, I know how to find the source extremely fast, here it is) things are different. "Strong" moves are rare.

Nevertheless, as Brian's answer shows, the usual laws of tactics are not "out of order" in an artistic 2# (what you call "vague 2#"). I tried to solve it myself (as I am an extremely fast solver of artistic 2#). I began by choosing the key piece: Very obviously the N are tied to c6/b5 until Black blocks these squares, the B can do nothing, the R can't attack anything, leaves the wQ which is out of play anyway and the obvious key piece. I played Qg1 without thinking since that double "sacrifice" would be exactly the key I would have in one of my own compositions. (So, expect spectacular stuff in an artistic 2#...or subtle.) Only then I verified: Yes, Q and R are tied to the pin mates by the wR. I even overlooked Qb4. (The composer might have snookered you! So carefully check all defences.)

Thus a general approach still works for both 2# types: Analyze the position. Assess the functions the pieces have to fulfil. (White attacking, Black defending.)
This comes especially handy in the artistic 2# because it must fulfil "economy": For the smothered mate, pawns around the white king are irrelevant for mating the black king. No composer would leave them on the board. Thus, sometimes white pawns are treacherous: maybe they just stand there to block a square of the key piece! (Clearly, useless for this problem...)
Sometimes, already the name of the composer is a giveaway (artistic 2# has different "schools") but this leads way too far...


user4418: With mate in 2 problems you basically have to try and look one move ahead and imagine a position where you are going to mate in 1. This will either involve mating the king on its current square or luring/forcing the king to another square and mating there.
Here there are some clues. The black king is currently in stalemate, i.e. has no moves, and black has no checks. As you would expect there are lots of "nearly" mate-in-1s. So, it looks like it is going to be mate the king where it is now rather than lure to another square.
The key then is to find a way of threatening a mate in 1 which can only be parried by allowing another mate in 1. Then it becomes clearer. If the bishop on c5 could be attacked again then Bxc5 would be mate. If the queen on b1 could be lured away from the b file then Rb2 would be mate. If a major piece (queen or rook) could take the queen on b1 then that would be mate. If the rook on g6 could be lured away from the 6th rank then Re6 would be mate. If the black queen wasn't defending the rook on g6 and white could take that rook with the queen then that would be mate.
Then the answer becomes clear:
[fen "nK6/4N1p1/pk4r1/p1b5/p6p/2N5/p3RB1Q/1q6 w - - 1 1"]

1. Qg1 Bxf2 (1...Rxg1 2. Re6#) (1...Qxg1 2. Rb2#) (1...Qb4 2. Qxg6#) 2. Qxf2#


31998: <p>I am a big fan of solving mate in N problems and believe they help tremendously with i.e. visualization. The only issue I seem to face in certain problems are when they are very vague. For example this one from the famous Polgar book. How can one proceed to solve it? I know the answer now, but it is very vague. Especially when there is no zugzwang or check to start with.</p>
<pre><code>[Title &quot;Bela Schläffer, Magyar Sakkelet, 5/1952, Mate in 2&quot;]
[fen &quot;nK6/4N1p1/pk4r1/p1b5/p6p/2N5/p3RB1Q/1q6 w - - 1 1&quot;]
</code></pre>
 26382: <p>&quot;Hmm, the rook is the only piece holding the 6-th rank... So if only I could distract it, the queen or the rook would deliver mate. Oh, and the black queen is doing exactly the same but vertically. Both heavy pieces are completely tied by their defensive duties. Can I somehow bother them, especially with a fancy move? Qg1, of course.&quot; Then it's only left to ensure all the lines work.</p>
 15049: <p>First, usually it is not a move which gives check and not a move which takes a piece, because this is considered to be not elegant in modern studies. For example 1.Qd6+, 1.Qc7+, 1.Bxc5+, 1.Re6+, check with a knight on d5, c8 or a4, these are probably not the solutions. Ignore these moves and only come back to them when you did not find the solution using the following method:</p>
<p><strong>Step 1</strong> (this step can also be left away in mate in 2 studies): Check what would happen if it was black's move. Which moves would lead to a white mate in one? From this you can learn the weak spots of blacks pieces. E.g. here you can learn that a queen move which loses control over b2 may lead to 2.Rb2 mate. A rook move away from the sixth rank may lead to 2.Re6 or 2.Qd6 mate. His bishop is pinned, if he moves it, this leads to 2.Bxb or 2.Qxf2 mate. If he moves his rook to c6 this leads to 2.Ned5 mate, if he moves his queen to b5 it leads to 2.Ncd5 mate.</p>
<p><strong>Step 2</strong> Act as if you had two moves in a row. Which maneuvers would lead to mate? This would be 1.Kc8 and 2.Qb8 mate, 1.Qh1(g2) and 2.Nc8 mate, 1.Re5 and 2.Bxc5 mate, Qe5(g1) and 2.Bxc5 mate.</p>
<p><strong>Step 3</strong> Go through the list of maneuvers, found in step 2 and consider all black answers to the first move of the maneuver. The 'bad' black moves from step 1 may give you indicators which maneuver leads to success. Here we would find that the following attempts do not work: 1.Kc8 Rc6+!, 1.Qh1(g2) Qe4! 2.Rb2+ Qb4, 1.Re5 or Qe5 Qb4!.</p>
<p>This leads to the conclusion that 1.Qg1! has to be the move and by checking all possible black answers we can find that each leads to a mate in one, see the lines given in Brian Towers' answer and in RossMillikan's comments. Especially the fact that the Qg1 attacks the Rg6 makes the difference, 1...Qb4 now leads to 2.Qxg6 mate.</p>
<p>ChessBase also has a good article which gives further hints: <a href="https://en.chessbase.com/post/how-to-solve-chess-problems" rel="nofollow noreferrer">How to solve chess problems</a>.</p>
 11144: <p>First, you have to split into &quot;gamelike 2#&quot; and &quot;artistic 2#&quot;. The approach is, at least in principle, completely different.</p>
<ul>
<li>In a gamelike 2#, proceed as if you are looking at a normal game. That is, analyze checks, captures, strong threats first.</li>
<li>In artistic 2# like your example (and it's a major crime to reprint art without attribution :-); but as a problemist, I know how to find the source extremely fast, <a href="http://www.schach-udo.de/albrecht/albrecht2/?rec=84219" rel="noreferrer">here it is</a>) things are different. &quot;Strong&quot; moves are rare.</li>
</ul>
<p>Nevertheless, as Brian's answer shows, the usual laws of tactics are not &quot;out of order&quot; in an artistic 2# (what you call &quot;vague 2#&quot;). I tried to solve it myself (as I am an extremely fast solver of artistic 2#). I began by choosing the key piece: Very obviously the N are tied to c6/b5 until Black blocks these squares, the B can do nothing, the R can't attack anything, leaves the wQ which is out of play anyway and the obvious key piece. I played Qg1 without thinking since that double &quot;sacrifice&quot; would be exactly the key I would have in one of my own compositions. (So, expect spectacular stuff in an artistic 2#...or subtle.) Only then I verified: Yes, Q and R are tied to the pin mates by the wR. I even overlooked Qb4. (The composer might have snookered you! So carefully check all defences.)</p>
<ul>
<li>Thus a general approach still works for both 2# types: Analyze the position. Assess the functions the pieces have to fulfil. (White attacking, Black defending.)</li>
<li>This comes especially handy in the artistic 2# because it must fulfil &quot;economy&quot;: For the smothered mate, pawns around the white king are irrelevant for mating the black king. No composer would leave them on the board. Thus, sometimes white pawns are treacherous: maybe they just stand there to block a square of the key piece! (Clearly, useless for this problem...)</li>
<li>Sometimes, already the name of the composer is a giveaway (artistic 2# has different &quot;schools&quot;) but this leads way too far...</li>
</ul>
 4418: <p>With mate in 2 problems you basically have to try and look one move ahead and imagine a position where you are going to mate in 1. This will either involve mating the king on its current square or luring/forcing the king to another square and mating there.</p>
<p>Here there are some clues. The black king is currently in stalemate, i.e. has no moves, and black has no checks. As you would expect there are lots of &quot;nearly&quot; mate-in-1s. So, it looks like it is going to be mate the king where it is now rather than lure to another square.</p>
<p>The key then is to find a way of threatening a mate in 1 which can only be parried by allowing another mate in 1. Then it becomes clearer. If the bishop on c5 could be attacked again then Bxc5 would be mate. If the queen on b1 could be lured away from the b file then Rb2 would be mate. If a major piece (queen or rook) could take the queen on b1 then that would be mate. If the rook on g6 could be lured away from the 6th rank then Re6 would be mate. If the black queen wasn't defending the rook on g6 and white could take that rook with the queen then that would be mate.</p>
<p>Then the answer becomes clear:</p>
<pre><code>[fen &quot;nK6/4N1p1/pk4r1/p1b5/p6p/2N5/p3RB1Q/1q6 w - - 1 1&quot;]

1. Qg1 Bxf2 (1...Rxg1 2. Re6#) (1...Qxg1 2. Rb2#) (1...Qb4 2. Qxg6#) 2. Qxf2#
</code></pre>
