How to checkmate with a queen: shrink the box
King and queen against a lone king is always a win, and the method takes a few minutes to learn. The queen does the herding on its own, your king comes over for the finish, and the only way to throw it away is stalemate.
The method, played out
Step 1: shut the king in a box
1.Qf4. The queen cuts the king off: it's shut in a box, and every queen move after this makes the box smaller. A simple way to keep shrinking it is to keep the queen a knight's move away from the king, so it can never come closer to the queen and has to step back.
You don't need checks. Checks often let the king out of the box.
Step 2: stop chasing, bring your king
Once the king is on the edge (here after 4.Qe7, with Black's king on h8), stop moving the queen. It has done its job. Now your own king walks over: 5.Kf2, 6.Kg3, 7.Kh4 and so on, while Black's king shuffles between g8 and h8.
The mistake: stalemate
With Black's king in the corner, Qf7?? is stalemate: the king has no legal move and isn't in check, so it's a draw. Before every queen move near the end, check that the other king still has a move.
Step 3: mate
With your king on h6, guarding g7 and h7, 10.Qe8 is mate. 10.Qg7 is mate too.
The idea to remember
Box the king in, push it to the edge, then stop and bring your king. And near the end, before every queen move, make sure the other king can still move.
Train it
This is one of the endgame techniques OptiChess teaches as a walkthrough: the moves play themselves until a moment that matters, then you find the move, checked against the tablebase. More endings are in the chess endgame trainer.
Try it. Real positions, set at your level, every answer checked against the board. No account needed to start.
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