How to solve mate-in-N problems Here is a mate-in-4 I have been trying to solve:

[FEN "2r4r/b5pp/q3pp2/8/4QBk1/8/3R1PPK/4R3 w - - 0 1"]


(white mates in 4).  I have been looking at it for about half an hour but not solved it. My question is: what can I do to improve my ability to solve problems like this?

(please don't just give the solution, I can easily find that by computer, but you can't use computers in a game).



Here is what I have tried so far.

Enumerating attacks and checks.


White checks: Qf5+, Qg6+, Qf3+, Qxe6+, B-anywhere+, f3+ 
White captures: Qxh7
Black checks: none
Black captures: Bxf2


Enumerating other ideas:


Black can play Bxf2 or Qf1 which will cause havoc for a move , so probably the solution is checks on every move; at least the first two
White has to be careful not to let BK run to g6 as then it can move to f7 escaping mate in 4
In several variations below, Kh5 escapes with ...g5 or other pawn cover, so I am looking for tricks involving sacrificing rook to break that pawn cover.


Tree of analysis (have omitted some sub-variations that mate for brevity)

Be3+ Kh5; Rd5+ g5; ???
Bc7+ (or other B moves) Kh5; Rd5+ g5; ????
                             g4+ Kh4; ????
Qf3+ Kf5; Qh5+ (otherwise Kg6) g5; g4+ Kxf4; ????
Qf5+ or Qg6+ or Qxh7 don't go anywhere
Qe6+ Kxf4; Re4+ Kg5; Rg4+ Kh6; ????
f3+ Kh5; g4+ Kh4; ????
    Rd5+ g5; g4+ Kh6; ????

Rd3 Qxd3; Qxd3 Kxf4; no mate
Re3 Bxe3; Bxe3 can't be better than 1.Be3
Rd5 exd5 Be3+: doesnt work
g3: doesn't seem to do anything, Black can wait


Now I have looked at all the checks and some other interesting moves, none of them work, and so to solve the problem online I'd be reduced to just guessing random moves (and then probably being stumped on the resulting mate-in-3 for a while, as if I could solve that easily then I would have realized the first move by now).

But in a real game that is not possible; so where do I go from here?

Is there always a "key idea" to look for (and if so, how do I look) or sometimes do you just have to be able to brute force every possible move no matter how unlikely? I can't even think of any more candidates I haven't checked.
Assuming this is a chess problem, then you want to start by looking at things that show up as problem themes, more than as game themes.  There are a few salient features of the position:


White's queen-bishop battery; it seems very likely that discovered checks from the bishop are likely to be a mating resource.
The white bishop being on a line with the king, thus leaving it vulnerable to a pin from either the Ba7 or the Qa6.
The open h-file, and in particular the possibility of a king-rook battery to deliver mate.
More remotely, the possibilities for interference between the black pieces (e.g., bishop/rook interference on c5).


Given these, I would investigate the possibility of a move like Rh1; this offers up the threat of 2. Be3+ Kh5  3. Kg1 mate and forces black to defend immediately. The most natural defenses are via the aforementioned pinning moves, 1...Bb8 and 1...Qd6, so I'd start looking there.

--

Assuming, OTOH, that the problem is a tactics puzzle more than a 'chess problem' problem, then I'll just point out that your analysis missed one particular bishop check, and you should consider the question of where the bishop can go that covers key squares in the vicinity of the king; not just for a mating net, but also as possible 'drop points' for the queen.