Can two bishops checkmate a lone king?
Yes, and unlike almost every other checkmate, any corner of the board will do. A king and two bishops always win against a lone king, as long as the bishops are on different colours. That is true of the bishops you start the game with, one on light squares and one on dark squares.
Two bishops on the same colour cannot mate. Neither can a king and one bishop, or a king and one knight. Those endings are drawn.
The good news is that the winning method is short. You need the two bishops and your king. The king does at least half the work.
What is the plan?
The plan has three steps, and you do them in order.
1. Bring your king towards the enemy king. Bishops alone cannot push a king anywhere.
2. Put the bishops on two diagonals next to each other. They form a wall that the enemy king cannot cross. Step the wall forward one square at a time.
3. Push the king into a corner and mate it there. Any corner works with two bishops. You do not have to pick a colour.
The common mistake we see from learners is starting with step two and forgetting step one. Move your king first.
How do the two bishops build a wall?
A diagonal is a slanting line of squares of one colour. One bishop controls one diagonal at a time. Two bishops on side-by-side diagonals control two lines of squares that touch, and a king cannot step through them.
In the study, look at the position after White plays the bishops to d5 and d6. Every square on the two diagonals is covered. The black king is stuck on one side.
Once the wall stands, you move it forward carefully. Move one bishop, keep the other still, and never let a gap open. If the black king attacks a bishop, move that bishop along its own diagonal and keep the wall.
How does the mate in the corner work?
The mating picture always looks the same. With the black king on a8:
- Your king stands on b6. It covers a7 and b7.
- One bishop covers b8, so the king cannot step sideways.
- The other bishop gives check on the a8 square along the long diagonal.
That is three attackers on three squares, plus the check. There is no gap.
Getting there needs one small trick called a waiting move. When the black king sits in the corner and it is your turn, you sometimes have nothing useful to do. Move a bishop backwards along its diagonal. Black must then step out of the corner, and you give check to force it back.
What is the stalemate trap?
Stalemate means the side to move has no legal move and is not in check. The game is an immediate draw. It is the only way to lose this ending.
The danger square is the corner. Picture the black king on a8, your king on b6, and one bishop covering b8. Your king already covers a7 and b7, so Black has no moves at all. If your next move is not check, the game is drawn.
So before every move, ask one question: after this move, does Black have a legal move? If the answer is no, your move must be a check.
How long do you have to finish?
The fifty-move rule says that if fifty moves pass for each side with no capture and no pawn move, either player may claim a draw. In this ending there are no pawns and nothing left to capture, so the count of fifty starts on the very first move.
Fifty moves sounds tight. It is not. Computers have checked every possible position of king and two bishops against a lone king. The longest one needs nineteen moves with perfect play.
If you follow the three steps and never lose the wall, you will finish in about twenty moves. Practise the method a few times and it becomes automatic.