Took lot of times to understand why Queen is going where she has to go and nowhere else. In many lines a rook or queen sacrifice completes the job. Very interesting puzzle.
4 stars.

e7 starts mate in four which seems inevitable. My mind stuck to this solution and i couldn't see nothing else...

This took me a long time to work out. 4 stars for me. This was VERY clever.

For me 1.Rxf8+ Kxf8 2.Qxh8+ Rg8 3.Qf6+ Ke8 4.Qe7 (or d7) mate found in 2 seconds is SHORTER then 1.Qa8 ! (which do mate in 3 moves but takes at least 10 minutes to be found).
In fact black would resign after 1.Rxf8+ Kxf8 2.Qxh8+, so that White won in 2 moves.

iberville, It's a puzzle, Black would resign after he read "White to move, mate in 3", so that White wins in 0 moves.  

Yeah, I'd have played 1.Rxf8 in real life too. And yeah, I know that since it's a puzzle I woulda failed to find the best response.

BUT, in a real game, I bet that 1.Rxf8! would have resulted in immediate resignation, so in practice, it would have been the faster move (to be played) and the most efficient move, causing resignation in 1!

... about this sort of puzzle is that it took only a few seconds to figure out a mate in 4. If it were a real game, I would have looked no further.

It soon becomes apparent that a pawn or a rook move won't do. After a minute or two (...or maybe three) it seemed that it had to be a Q move to b2 or a1. But for the solution to lie in that direction, there had to be some reason for rejecting one in favour of the other. On what grounds? The Black bishop provides the answer: 1...Bxd6 puts 1.Qb2 out of court.

And so we were able to conclude that 1.Qa1! was the key.

People resign when in their view their opponents will win 'with reasonable play' as I've seen it expressed. Note: 'reasonable', not 'best' or even 'good' play.

The 'reasonableness' criterion will differ from person to person. I've known people to resign a game they weren't losing (occasionally even had a winning position!). That's a pretty low threshold of 'reasonableness'! Others tend to keep playing until in their view they have exhausted all possibilities of resistance.

If a mating combination turns up I might well play it out rather than resign. This I regard as a courtesy to my opponent, but only in that circumstance. Ordinarily it is considered a courtesy to resign when it is clear not only that one's opponent has a winning game, but also that one's defensive resources have been exhausted - no counter-play left. But to carry the game past that point should not be seen as a discourtesy, as there can be all sorts of reasons for doing so, apart from this sort of thing being a judgement call.

Could someone tell me why Qa5 would not work as well?