How do you checkmate with a King and Queen versus King?
Is there a method that is faster than other methods in terms of moves?
Is there a foolproof method if you are in time trouble?
Grave Danger in King and Queen vs lone King
It's important for the winning side to avoid this common pitfall in this basic endgame.
The winning side should avoid placing their queen on the c2, c7, f2 or f7 squares if the opponent's King is in the nearby corner, because this leads to stalemate.
Below, I include some other types of stalemates that are not as well known.
The plan is to drive the king to one of the edge ranks or files.
After this is achieved, the checkmating positions are usually like this -
This method is the fastest and similar to the rectangle method involving checkmate with a rook. The trick is to confine the opponent's king in the smallest possible rectangle with the queen's first move (if not already confined or can be confined at a later move) and then bring the side's own king closer to the opponent's king.
In the position to the left below, White plays 1. Qc6 as the first move, making the rectangle smaller.
Once this has been achieved, White simply brings the King closer to the Black King to drive it into one of the edge ranks or files. The rook check trick discussed in the rectangle method for the rook applies here with the queen check.
[FEN "8/8/8/4k3/8/8/8/K6Q w - - 0 1"] 1. Qc6 Kf5 2. Kb2 Ke5 3. Kc3 Kf5 4. Kd4 Kf4 5. Qf6+ Kg4 6. Ke3 Kh5 7. Qg7 Kh4 8. Kf4 Kh3 9. Qg3#
Knight's Square Method
This method is mainly for beginners. Normally, experienced players will do better to use the rectangle method even in time trouble. However, it may at times help to use a combination of this and the rectangle method.
If you're a in time trouble, this a very fast (in terms of how quickly you can make the moves on the board without thinking much) and foolproof (stalemates are always to be watched out for) way to checkmate the opponent's king.
With this method, you can literally finish playing the move before the opponent's hand even reaches the clock (assuming you're not playing online).
Consider this position where the Black King is on e5.
The King can be checked by a White knight from each of the 8 squares d3, c4, c6, d7, f7, g6, g4 and f3. In the Knight's Square Method, the idea is to move the queen to one of these squares (preferably one away from your king to avoid collisions in time trouble!) and then simply "copy" the opposite king's move. Thus, in the position below, White first moves the queen to a "knight-square" 1. Qc6!
After that, the White Queen can "copy" the Black King's moves and remain in the knight's square, except, I REPEAT!!, except when the king moves to the corner (a1, a8, h1, or h8). For example, king moves right one square, queen moves right one square; king moves diagonally up one square, queen moves diagonally up one square. Once the Black King is driven to one of edge ranks or files, White brings the King to deliver the checkmate. This is demonstrated below.
[FEN "8/8/8/8/3k4/8/8/K6Q w - - 0 1"] 1. Qc6 Ke5 (1... Kd3 2. Qc5 Kd2 3. Qc4 Ke3 4. Qd5 Kf4 5. Qe6 Kf3 6. Qe5 Kg4 7. Qf6 Kg3 8. Qf5 Kg2 9. Qf4 Kg1 10. Qf3 Kh2 11. Qg4 Kh1 12. Kb2 Kh2 13. Kc2 Kh1 14. Kd2 Kh2 15. Ke2 Kh1 16. Kf2 Kh2 17. Qg2#) 2. Qd7 Kf6 (2... Kf4 3. Qe6 Kf3 4. Qe5 Kf2 5. Qe4 Kf1 6. Qe3 Kg2 7. Qf4 Kg1 8. Qf3 Kh2 9. Qg4 Kh1 10. Kb2 Kh2 11. Kc2 Kh1 12. Kd2 Kh2 13. Ke2 Kh1 14. Kf2 Kh2 15. Qg2#) 3. Qe8 Kg5 (3... Kg7 4. Qe6 Kf8 5. Qd7 Kg8 6. Kb2 Kf8 7. Kc3 Kg8 8. Kd4 Kf8 9. Ke5 Kg8 10. Kf6 Kf8 11. Qf7#) 4. Qf7 Kg4 5. Qf6 Kg3 6. Qf5 Kg2 7. Qf4 Kg1 8. Qf3 Kh2 9. Qg4 Kh1 10. Kb2 Kh2 11. Kc2 Kh1 12. Kd2 Kh2 13. Ke2 Kh1 14. Kf2 Kh2 15. Qg2#
Sometimes, one's own King can get in the way of the Queen in this method, as shown below.
[FEN "8/8/8/2k5/4K3/5Q2/8/8 w - - 0 1"] 1. Qb3 Kd6 2. Qc4 Ke7 3. Qd5 Kf6
In such cases, the King can simply be moved away and the same method continues.
[FEN "8/8/5k2/3Q4/4K3/8/8/8 w - - 0 4"] 1. Kf4 Kg6 2. Qe5 Kh6 3. Qf5 Kg7 4. Qe6 Kh7 5. Qf6 Kg8 6. Qe7 Kh8 7. Kg5 Kg8 8. Kg6 Kh8 9. Qg7# *
As mentioned before, the one pitfall to be avoided in this method is the stalemate that can occur when mistakenly occupying the knight's square when the king is in the corner.
[FEN "8/8/8/4Q3/8/8/6k1/K7 w - - 0 9"] 1. Qf4 Kh1 2. Qg3 $4 (2. Qg4 $1)
Lol, this question is not as silly as it sounds as once I was in time trouble and ended up stalemating my opponent by accident! Most players resign when their opponent is up a queen so such blunders can happen if you occasionally come across a player who is stubborn enough to play till the end. Anyway, after that incident I went back to studying the Queen and King vs. King endgame to avoid such embarrassments in the future!
So, this is how you do it:
1) Remember that stalemates are more easy with Q & K so it is important to check for a possible stalemate before every move.
2) Use your queen to slowly try to corner the opponent king until he is confined to one of the edge row or columns of the board.
[FEN "8/8/8/k2K4/8/1Q6/8/8 w - - 0 1"]
3) Bring the king in to the position shown in the diagram below:
[FEN "8/2K5/8/k7/8/1Q6/8/8 w - - 0 1"]
Remember that the king has to be in this position to deliver the checkmate.
4) Slowly nudge the opponent king towards the corner.
[FEN "8/k1K5/8/1Q6/8/8/8/8 w - - 0 1"]
5) When the opponent king is in this position, deliver the checkmate!
[FEN "8/k1K5/8/Q7/8/8/8/8 w - - 0 1"]
[FEN "8/8/8/3k4/8/7K/8/5Q2 w - - 0 1"] 1. Qf4 Ke6 2. Kg4 Kd5 3. Kf5 Kc5 4. Qd2 Kc4 5. Ke5 Kb3 6. Qc1 Kb4 7. Qc2 Kb5 8. Qc3 Kb6 9. Qc4 Kb7 10. Qc5 Kb8 11. Qc6 Ka7 12. Qb5 Ka8 13. Kd6 Ka7 14. Kc7 Ka8 15. Qa5# *
I don't know about the 'faster than other methods' bit, but it works!
The simplest answer is to restrict the enemy king's movements by "walling off", not attacking, with the queen, move by move, driving it to the edge and then bringing up the king to help administer the mate. You must remember to always leave the king at least one free square once it's in a corner to avoid stalemate. This is basically the same method used with the rook and king, although the king and rook must be used together in that case to drive the king to the edge and finally into a corner for the mate. All this may sound confusing in the abstract, but if you try it I'm sure it will become clear quickly enough. I don't think trying to memorize a "formula" beyond what I've just suggested will be of any more benefit.