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La mia analisi: Il materiale

La lezione del GM Schwartzman So, was this a nice one or not?  It must be, cause I loved it! By the way, as you will see in a second, there are two solutions, but both lead to the same idea.  So, if you found either one of the moves, you are in business!
Anyway, let's see what the deal is with this position.  Let's start with the bad news:  we are a rook down!  Should that worry us?  I would certainly say so...  But before we despair, let's look at what we do have:   black's king in a really tough spot!
So, the situation at hand is quite clear:  we can either manage to perpetual or checkmate black, or we're dead meat.  If this isn't enough of a pressure, how about the fact that black is threatening to take on e3, and with check...   Obviously, if that happens, only black can win this game, no doubt about it.
So, at first glance, it looks like the only way to win this game, is to do it rapidly, with checks.  After all, if we do give black even one move, his queen will suddenly start to make her presence felt. So, if we take this as a given, then the only moves we can look at are 1.f4, 1.h4, and the bishop sacrifices on f6 and  h6.  
The latter options quickly disappear, as it there is really no way to   checkmate if we allow the black king to return back home.  1.f4, on the other hand, looks fairly attractive, since it sends the king to the only square available: h4, which should be good, as it is even closer to our home base . 
Unfortunately, the problem begins when you look for a way to actually finish him off.  Interestingly enough, the black king practically uses our pawns for protection!  And if our bishop can't get to f6, that leaves only the queen as a checkmating tool.  But, as it so happens, the queen is also in charge of defending the pawn on g4.  The meaning?  Simple:  if we move it to e1, in order to put the king in check, the king simply captures our pawn.  From there on, despite the many possibilities, there is really no way to checkmate: black's king heads to f5 and to e4, where it finds itself in a very safe position, thanks to the numerous black pieces in the proximity.
So, what does this leave us with?  1.h4!

Black's choice is not ideal:  he has only one - to take on h4 with 1...Kh4.   Now, here is where the real trick to this puzzle lies.  If you have not spent enough time looking at this position, you might have discarded it quickly as hopeless.  After all, it doesn't really seem like there is much difference from the position we just looked at.  The king is still on h4, and while the pawn on f3 allows us to check on e1 without losing our g4 pawn, there is still the question of what to do after black's king returns to g5.  As you can see for yourself, there is no clear path to checkmate.
So, then what is the solution?  Well, there is one subtle difference between the two positions we have been looking at:  remember how I said that black's king was using our pawns for his protection?  Well, in this line, there is then one less pawn to protect the king:  the one on h2.  Does that make a difference?   Oh , yeah!  You see, this means that the 'h' file has opened up, and guess what a queen there would do to black...
The only remaining question is how to get the queen there.  Since we can't do it with check, we can also not move our queen, because it would allow black to capture on e3.  So, it shouldn't take long to realize that the only move that satisfies our strict requirements is 2.Kg2!

But what a move this is!  Despite the fact that there is no check, black can not find any reasonable way to defend the mating threat, which is Qh1-h6.   Because the white king has moved, taking on e3 doesn't help either...  Which only leaves 2...Kg5 (2...Qg4 would manage to prevent the mate, but would give white winning material advantage). 
Now, it's only a matter of moves:  3.f4 Kh4 4.Qh1 Kg4 5.Qh3 checkmate!

A rather impressive performance by white, don't you agree?  Also, based on the same idea, white could have started with 1.Kg2!! which would have led to the same result, because black is still unable to stop h2-h4 on the next move.     So, both 1.h4 and 1.Kg2 are great moves that win this game for white...