Can white win this queen and pawn endgame?
From this position put Black queen on the best defensive square, and then it will be White's to move.
[FEN "8/1PQ3pk/6pp/8/8/8/8/3K4 w - - 0 1"]
It might look like this:
[FEN "8/1PQ3pk/6pp/8/q7/8/8/3K4 w - - 0 1"]
My computer gives +6 for White, but it doesn't seem to be able to find a winning strategy.
The 7-piece tablebase says that the position without the pawn on h6 is draw, so I believe this position is also a draw (by perpetual check).
A typical winning attempt in queen+pawn vs queen endgames is to find a position where white can block a check with his queen while at the same time giving check. This is not possible here since the black king is protected by pawns.
Another option is to hide behind the pawn somehow. A somewhat ideal situation would be if white could move the king to a8, and to have the white queen on the a-file (e.g. on a5). In this position there is no check on the a-file and if black checked on the 8th row, white could respond by queening the pawn. However even in this ideal position black has the single (but sufficient) resource of pinning the pawn along the h1-a8 diagonal.
So yes this position with the b-pawn is draw and you can almost freely arrange white king and queen and it still stays a draw.
If it was a c-pawn though it would be more difficult for black to hold this.
Black seems to be able to get away with a draw by perpetual check. The usual way for White to escape that, is to advance the pawn, or interpose his/her own queen while giving check (thereby forcing the exchange of queens). The latter is impossible because black's king is well protected. The former is impossible as well, as long as black remembers to give check from the a-file if the white king is on a8.
Your computer probably doesn't see a threefold repetition (indeed, it's possible for White to stave this off for more than 50 moves) and thinks the b7 pawn is worth almost 6 pawns because it is on the verge of queening.
No table base necessary... black gives check every move, done :P The only position I can think of where white can promote is:
[Variant "From Position"]
[FEN "K7/QP4pk/6pp/8/8/4q3/8/8 b - - 37 19"]
1... Qe8+ 2. b8=Q Qe4+ 3. Qbb7 Qe8+ 4. Qab8 Qa4+ 5. Q7a7 Qe4+ 6. Qbb7 Qe8+
which is also a draw thus. But you can't even reach that position.
I know that this is three years late, but Stockfish with partial 7-piece tablebases immediately says 0.00. Now, at a depth of 71, it still says 0.00. I think it is, indeed, a draw.
Of course, there is no harm in trying, and the plan in such endings is to try to walk the king toward your own pawn. Clearly, it should not work based on the computer assessment, but still, I would try for a while until my opponent could prove a perpetual.
Black can put a piece on a square where it checks White's king, and then perpetually check. There isn't enough cover for White to stop the checks.
