Solving a problem from a Jan Prezewoznik book The following position is White to move:

[fen "1Q5r/1p2kppp/1q2b3/r1p3P1/p2pB3/P2P3P/1PP1P3/1KR2R2 w - - 0 1"]


I'm a newbie in chess and I want to know, how to solve this problem in a quick way.
The nature of the problem that this position presents is first this: to correctly assess the situation on the board, and in particular to recognize the immediate and impending problems White is facing. As this answer will show, there is a lot of concrete analyzing one can do from this position, but the point I want to stress is that just answering that first question (and with only simple calculating) goes a very long way toward narrowing down what moves are even reasonable to consider, and can ultimately lead to 1.Rxf7+ as mentioned in Dag's comment above. What follows is one rough path of thought that one might take when facing this position in a game, and some analysis follows below.

Obviously White's queen is attacked, and something must be done about that, so that already cuts down our options considerably. There is a free rook on offer via 1.Qxh8, so what happens if we just take it? The thing to note there is that after the direct 1...Rb5, all three of Black's pieces are bearing down on the white king, while the four white pieces are all unable to help defend. It turns out that to stave off mate there, White would need to shed a losing amount of material. Once one sees this 1...Rb5 threat at all, it's pretty clear that any of 1.Qh2/Qg3/Qf4 leave the queen similarly out of play such that White is unable to defend properly against 1...Rb5.

1.Qe5 is a different case, as it can potentially serve a defensive purpose (supporting Bd5 to nullify the black bishop) and has some bite to it by keeping the queen in the vicinity of the black king. In its favor, Black could no longer respond with 1...Rb5? without the tables turning (see analysis below). Since e5 was the only remaining safe square for the queen, we might feel satisfied with that choice.

But here we should do a little concrete calculating: instead of 1...Rb5?, the move 1...c4 unveils a new attack on the queen and threatens 1...c3; because of that, it's almost game over by that point. But 1.Qe5 seemed like the only place we could put the queen, so we might need to bite the bullet and answer 1...c4 best we can. There the options would be truly limited, and the only way to keep the game going at all from there would be avoiding ...c3 by stirring things up with the desperado sacrifice 2.Rxf7+. (As noted in the analysis, Black actually does need to take care in those complications, and so though this whole approach is objectively not the best from the starting position, it might be a decent practical play.)

With all that laid out, though, we could now at least ask ourselves, what if we try the same complicating idea from the start? The play will be similar to after 1.Qe5 c4 2.Rxf7+, and at this point the only way to decide between that path and 1.Rxf7+ is to bear down and try to calculate the consequences of each (as in the analysis tree below). But note that we're able to get down to these two basic options just by looking a couple of moves deep in combination with a general sense of danger (e.g. our pieces out of play while our opponent's pieces are all attacking). If all goes right in our calculations, we'll find that in fact 1.Rxf7+ gives at least a perpetual (which isn't trivial to work out in full from the start); but again, we're forced into the idea from more general considerations before even getting bogged down in variations.

[FEN "1Q5r/1p2kppp/1q2b3/r1p3P1/p2pB3/P2P3P/1PP1P3/1KR2R2 w - - 0 1"]

1.Rxf7+ 
    ( 1.Qe5 c4 2.Rxf7+ Kxf7 3.Rf1+ Ke8 
        ( 3...Ke7 $2 4.Qxg7+ Kd6 5.Qxh8 c3 6.Qf8+ Kc7 7.Qe7+ Bd7 8.Qb4 $10
        )
    4.Qb8+ Qd8 
        ( 4...Kd7 $2 5.Qxh8 c3 6.Qxg7+ Kd6 7.Qf8+ Kc7 8.Qe7+ Bd7 9.Qb4 $10
        )
    5.Qxb7 Qd6 {White has no perpetual, and ultimately the extra black 
    rook will tell against the extra white pawns.} )
1...Bxf7 
    ( 1...Kxf7 2.Rf1+ Ke7 3.Qxh8 Rb5 4.Qxg7+ Kd6 5.Qf8+ Kd7 6.Rf7+ Bxf7 7.
    Qxf7+ Kc8 8.Qe8+ Kc7 9.Qe7+ Kb8 10.Qe8+ Ka7 11.Ka1 Rxb2 12.Qxa4+ Kb8 
    13.Qe8+ Ka7 $10 )
2.Qe5+ Qe6 
    ( 2...Kd7 3.Bf5+ $16 )
    ( 2...Be6 3.Qxg7+ Kd6 4.Qxh8 Rb5 5.Qf8+ Kd7 6.Qg7+ $10 )
3.Qc7+ Ke8 4.Qb8+ 
    ( 4.Qxa5 Qa2# )
4...Kd7 5.Qxb7+ $10 *