Black to move. Can Black save this endgame? If yes, how? 
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If the white queen landed on f7, it would be stalemate. So I think, black should play to this end.
I would try to check the white king along the diagonal a2-f7 (in some cases, checks from black squares can be necessary but black can resume its strategy based on the white squares).
Eventually, a check on f7 can show white pieces that it is impossible to progress. (This is the case, for example, from the start position, if white king escapes from check to h5).
I was considering 1..Kh7 (to stop the W King from coming to h5-h6), but now I see that Romanoski has an excellent point. So I endorse his analysis that Black should keep checking on light squares, ensuring that as soon as the W K lands on h5, he can check on f7 and draw.
My question is, is there is any ‘general rule’ that Susan is looking for, which allows us to draw these conclusions (about when perpetual checks can be avoided and when not), without calculating the variations (which are infinite in no. in Q-P endings). I was curious because in the epic Topalov-Anand games (San Luis, 1st Rd), Topalov agreed to Anand’s final draw offer rather quickly. i thought he ‘knew’ from some general rules that perpetual checks couldn’t be avoided (which wasn’t obvious to me at all)
That f7 plan sounds difficult, the threat of it might help keep the perpetual alive and curtail the white Kings advance to h6
1… Qf7 doesn’t work, because White surely isn’t going to take the queen.
However, keeping that Qf7 idea in mind, I don’t see how White can avoid a perpetual.
NO I do not like the diagonal. it messes up. not simple. it might work but here is some thinking. we need thinking.
working backwards. K winds up at g4 gets checked and goes to h5. then we must check on f7. but white does not take so he goes back to g4. but the important point is that black must absolutely be able to check on f7 when white goes to h5. and he can only do that from e6 or d7.
so when white goes to g4 black must get to either e6 or d7.
continuing backwards. before white gets to g4. he must be on g3 or f3. and when we check it must allow going to d7 or e6.
now with the white king on g3. the white queen can be on f2 or f3 or f4. and we need to check with access to d7 or e6.
so if the black queen is running up and down the “d” column. then it can check the king on g3 if the queen is on those 3 spots and still get to d7.
if the queen is running up and down the e file. then it can check if the queen is on f2, 3 or 4. so that looks good also.
It looks like black can run the queen up either the d file or the e file and still get to e6 or d7 if the king goes to g3. of course one can also stop the king from going to h5 with checks at d1 or e2.
basically black stops all white progress. eventually we hit the 50 move rule and or the 3 times repitition rule.
so the first move for black is Qe1+ now on queen blocks we can still check with repitition. so eventually white moves the king to g4. now we can go to d1 or e2 on the diagonal. or up and down to offer the move into h5 with queen going to f7.
I say give white a hard time and keep offering him a draw and hope for the best here.
I dont like this. if the king goes to g4 and I check on e2 then the king might go to f4 and then white is forced to check on d3. and if the black queen blocks on e4. the white queen will be in the wrong place for the f7 check and we will probably lose. so with the check on e1 king to g4 black needs to skip over a file to d1 check. that keeps the white queen on the f file. since if white goes to f5. then black checks Qd7 and the white queen can not change files.
oh this gets complex. now we have to worry about the white king on f5, and its related squares to make checks.
I just had a brilliant idea. lets send this puzzle to Pal Benko and ask him to plot out all the moves. he is so organized. he will love this one.
tommy
if this endgame is not “topalov-anand” , it is very similar. I remember that it was a draw… but I really don’t know how!
if black plays Qf7, then white can play Qf6+, getting the queens off and winning in a hurry. he can advance the pawn and have the h-pawn to avoid stalemate at the queening square.
True Falcon of New West
1..Qb3+ 2.Qf3 Qf7 3.Qa8+ Kg7 4.Kg4 Qc4+ 5.Kh5 Qf7+ leads to a draw , the key fact here is that white can not push the g or h pawns without conceding check infinitum or losing a pawn and the opposition in a trivial pawn ending.
Other variations leads to either perpetual checks or a similar ending.
sheaf
1… Qf7 loses because of
2. Qf6+ and
1… Qb3+
2. Qf3 Qf7
3. Qf6+
leads to the same fiasco.
Where to put interrogation marks?
6 years after this puzzle was published, I’ve checked the 6-men tablebases and was stunned.
Not only Black can hold this endgame, but he has many moves to do so, even Qe1+ or Qe8 do the trick.
I knew that there were Rook gpawn and hpawn vs Rook drawn endgames, but never I conceived Queen gpawn and hpawn vs Queen drawn positions. Clearly shocking!