A lone king cannot be mated in the centre - there is always somewhere to go. So every mating technique comes down to three steps:
The single exception is mate with two rooks: there your king can stand anywhere, the rooks manage on their own.
Each mate below opens on an interactive board with the position already set up, so you can move the pieces and test your technique.
The order to learn them is straightforward: two rooks, queen, and queen with rook are the base you should be able to deliver without thinking. Rook and king is the compulsory minimum for tournament play. Two bishops and bishop with knight are for those playing seriously: in practice they appear a handful of times in years.
The queen mate is the first one people learn and the only one where a beginner regularly draws instead of winning. The reason is stalemate.
What stalemate is
The opposing king has no legal move but is not in check. The game ends immediately in a draw, no matter how much material you have.
The queen is so powerful that it easily takes away every square at once. Hence the rule: before every queen move, check that the king still has at least one square. If it has none and there is no check, that is stalemate and half a point thrown away.
The safe technique: approach the enemy king with the queen a knight's move away, shrinking the box step by step, and only bring your own king up once the king is pinned to the edge.
Even with a queen, the mate cannot take forever. If fifty consecutive moves pass with no capture and no pawn move, either side may claim a draw.
In practice the limit almost never bites: the queen mate takes about a dozen moves and the rook mate is quick too. It only becomes a real problem with bishop and knight, where the technique runs to some thirty moves and any error costs time.
Note that the draw does not happen automatically - it has to be claimed. In an event with notation the players count the moves themselves from their scoresheets.
Some material combinations cannot deliver mate at all, however well you play. Such games are drawn immediately:
The practical lesson for the endgame: when trading down with a material edge, make sure you are not left with a single bishop or knight. In that situation an extra pawn is worth more than an extra minor piece.
Related reading - tactics for beginners, pawn endgames.
Mate with two rooks and mate with the queen: they are the simplest and the most frequent. Then rook and king, which is the compulsory minimum for tournament play. Leave bishop and knight for later - it appears a handful of times in years of practice.
The queen takes away too many squares at once. Before every queen move, check that the enemy king still has at least one legal square. If it has none and is not in check, the game is drawn on the spot.
No. King and bishop against king, like king and knight against king, is a theoretical draw. With two knights mate is possible only through a gross error by the defender and cannot be forced.
If fifty consecutive moves pass with no capture and no pawn move, either side may claim a draw. It is not automatic - a claim is needed. For queen and rook mates the limit is not a problem.
Almost always yes: a single long-range piece pins the king to the edge but only mates together with your king. The exceptions are mate with two rooks and with queen plus rook, where the two pieces manage alone.
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Improvement - tactics, pawn endgames, reading chess books.