You've just played the most brilliant game of chess you've ever played, and after several hours of concentrated thought you have reached the position at left. You have a king and a queen against your opponent's lone king. Would you know how to finish him off? Or would all your effort go for nothing as you floundered around the board chasing hopelessly after the enemy king? Clearly you need to know a few basic checkmates. There are four basic checkmates that every chess player needs to learn eventually: king and queen vs. king, king and rook vs. king, king and two bishops vs. king, and king and bishop and knight vs. king. I'll cover the first two of these only in these lessons.
In all of these basic checkmates, the enemy king must be forced to the edge of the board, and if you have only minor pieces at your disposal (i.e., two bishops or bishop and knight) then the enemy king must also be forced into the corner. The king and queen vs. king checkmate is the easiest, so let's begin with the position above. The correct procedure is one of progressively confining the enemy king to smaller and smaller sectors of the board using the combined actions of your king and queen.
1. Qb5 Ke6 The other fellow, of course, tries to avoid being confined and at all costs tries to stay away from the edge of the board. 2. Kf4 Kd6 3. Ke4 Ke6 4. Qc6+ The only thing you really have to be careful about at this point is accidently giving away your queen. I've seen even experienced players do it when they had only a few seconds left on the clock and were in a time scramble to get the enemy king checkmated! 4...Ke7 5. Kf5 Kf7 6. Qd7+ Kg8 7. Kg6 Kh8 Now white has a choice of five moves that administer mate. When the enemy king is finally backed into the corner is when you have to be the most careful. Now there is a danger of stalemate. If white were to play 8. Qf7?? here, the game would be drawn. 8. Qd8++ Also perfectly fine would be 8. Qg7++ or 8. Qh7++. I'll leave the other two up to you to find.
In this position, the queen has been replaced by the less powerful rook. This is going to be a bit more difficult, but only a bit. The basic procedure is the same: confine the enemy king to smaller and smaller sections of the board using the combined force of the king and rook. In this case, the king and rook need to work more closely with each other. The enemy king could not approach and intimidate your queen, because the queen radiates force in all directions. The enemy king can harass your rook, however, and your king must be there to give the rook support. You need to be somewhat efficient in bringing the game to a close at this point, because you have only fifty moves to do it. Remember, if a pawn is not moved or a piece is not captured for fifty moves, your opponent can declare the game drawn.
You need to practice these two mates. Believe me, when the situation arises in one of your games, and it will, you'll wish you had!
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