Can a bishop and knight force checkmate?
Yes, but only into two of the four corners, and picking the wrong one is how grandmasters have drawn this ending. A king, a bishop and a knight beat a lone king every time, with correct play.
The difficulty is the corner. You can only mate in a corner square that is the same colour as your bishop. With a light-squared bishop that means a8 or h1. With a dark-squared bishop it means a1 or h8.
If you push the king into the wrong corner, there is no mate there. You have to walk it along the edge to the right corner, and that is where most players get lost.
What is the plan?
There are three stages, in this order.
1. Push the enemy king to the edge of the board. Use all three pieces. This part is like any other mating technique.
2. Walk the king along the edge to the right corner. This is the hard part, and the knight's W manoeuvre does it.
3. Deliver mate in the corner. The bishop checks, the knight takes one square, and your king takes the rest.
Stage two is the one to learn properly. Stages one and three become easy once you have seen them two or three times.
What is the W manoeuvre?
The W manoeuvre is the path the knight walks while the enemy king is being pushed along the edge.
In the study the knight visits f7, then e5, then d7, then c5, then b7. Write those squares down and draw a line between them. The line goes up, down, up, down, and it makes the shape of the letter W.
Each stop does one job. The knight covers a dark square that the light-squared bishop can never reach. The bishop covers the light squares, your king covers the squares next to the enemy king, and together the three pieces leave no gap.
The knight never gives up its square until the bishop or the king has taken over that duty. That is the whole trick.
How does the mate look?
The mating picture is always the same. Here it is with a light-squared bishop and the black king on a8:
- Your king is on b6, covering a7 and b7.
- Your knight is on a6, covering b8.
- Your bishop gives check on the a8 square from c6 or from further down that diagonal.
Nothing is left. Every square around the enemy king is taken and the king is in check.
The same picture appears in the h1 corner, turned around. Your king on f2, your knight on g4 and your bishop on g2 give mate to a king on h1.
What goes wrong most often?
Three mistakes account for nearly every failure.
1. Chasing the king into the wrong corner. Check the colour of your bishop before you start pushing. If the king is heading for the wrong corner, use the bishop to close that side off.
2. Letting the king slip past the bishop. Keep the bishop on a diagonal that cuts the board, and only move it when your knight or king covers the gap.
3. Stalemate at the very end. When the enemy king is in the corner with no squares left, your move must be a check.
A fourth problem is the clock. This ending takes many moves, so learn the path rather than working it out from scratch each time.
How many moves does it take?
Computers have worked out every position of king, bishop and knight against a lone king. From the worst starting position the win takes thirty-three moves with perfect play.
That fits inside the fifty-move rule, which lets either player claim a draw after fifty moves with no capture and no pawn move. But perfect play already uses thirty-three of those fifty moves, so a few wrong moves will use up the rest.
So the honest advice is this. Learn the W manoeuvre until you can play it without thinking. If you have to work the method out at the board, you will probably run out of moves.