Quick answer
Quick answer: Put your bishops side by side on adjacent diagonals to build a wall, then walk that wall forward with your king behind it. The enemy king cannot cross it. Push him to any edge, then to any corner, and mate.
It's a forced win in at most nineteen moves with best play, so there's no need to rush it.
I ran the position against a tablebase before writing this. The general idea of a forced mate is covered on the checkmate page, and the reason it works is the geometry of the bishop.
Why this one is winnable and the other is not
Two bishops cover both square colors between them. A lone king can never step onto a square they both watch.
That's the whole reason the mate exists. You have a tool that seals the board rather than one that pokes at it.
And unlike the bishop and knight version, any corner will do. You don't have to steer towards a particular one.
Our guide to the bishop and knight checkmate shows how much harder life gets when you lose that freedom.
The wall is the whole technique
Place both bishops on diagonals that sit next to each other, so their lines run parallel across the board.
Together they cut the board in two. Everything behind them is yours and the enemy king is stuck on the far side.
He cannot cross a line covered by a bishop, and there's no gap between the two lines to slip through.
Once you see the wall, the rest is just moving it forward one step at a time.
Your king does half the work
Beginners try to do this with the bishops alone and it never works.
The wall stops the king moving forward. It doesn't stop him running sideways along the rank he's on.
Your king is what closes that escape, by standing near enough to take squares away.
So the first move of the whole plan is always the same. Walk your king towards the center and towards him, before you touch a bishop.
Step one: build the wall
Get both bishops onto adjacent diagonals, wherever the enemy king happens to be.
Don't worry about which side of the board. Anywhere works, because you're going to push him to whichever edge is nearest.
A useful check is whether your bishops are on squares next to each other or one square apart diagonally. Both shapes give you parallel lines.
If the enemy king can walk between your bishops, they aren't on adjacent diagonals yet.
Step two: push the wall
Move one bishop forward a diagonal, then the other, keeping them parallel.
Each push takes a rank or a file away from him. He retreats because he has nowhere else to go.
Move only one bishop at a time and always keep both lines covered while you do it. If one bishop steps away and leaves a hole, he'll walk through it.
This is slow and it's meant to be. Nineteen moves is plenty of time.
Step three: bring the king closer
Every two or three bishop pushes, spend a move improving your king.
He needs to end up two squares away from the enemy king, facing him, which is the same opposition idea that runs through every king endgame.
Without that, the enemy king shuffles along the edge forever and you never finish.
Our explainer on opposition in king and pawn endings covers the same relationship in its simplest form.
Step four: the corner
Once he's on an edge, keep walking the wall sideways to squeeze him towards a corner.
He'll try to run back towards the middle of the edge. Your king stops that, and your bishops keep the escape squares covered.
Any corner is fine, so take the nearest one rather than steering across the board.
When he's in the corner with your king two squares away, the mate is one bishop check away.
The final pattern
It's worth memorizing the finish so you recognize it when it appears.
One bishop covers the escape square diagonally beside the corner. Your king covers the two squares next to him. The second bishop gives the check.
That's it. Three pieces, three jobs, and no move for the defender.
If you're one square out of position, you'll see it immediately, because he'll have exactly one square to run to.
The stalemate that ruins everything
This is where most players throw the win away, and it happens at the very end.
A king in the corner with no legal move and no check is stalemate. The game is drawn and all nineteen moves are wasted.
So before every bishop move near the corner, ask one question. Does he still have a square?
If the answer is no, give a check instead, or move your king first. Our note on stalemate against checkmate covers why this rule catches so many players.
The mistake that costs the most time
Moving both bishops onto the same diagonal. It looks tidy and it opens the gate.
With both lines on top of each other, the wall is one square thick and the defending king simply walks around it. You then spend four moves rebuilding what you had.
The check is easy. Your two bishops should always cover two different lines running side by side.
If you can draw one straight line through both of them, you've made this mistake.
The defender's only real plan
A lone king has one idea, and it works surprisingly often against a player who hasn't practiced.
He runs for the middle of the board and stays there, because the center is the hardest place to trap him.
So your first job is never the mate. It's pushing him off the middle four squares and keeping him off.
Once he's beyond that ring, he's running out of room whether he moves towards a corner or along an edge.
Do not chase him with checks
Checking feels productive and it's usually the slowest way to do this.
A check often lets him move to a square you didn't want him on, and it can break your wall while you reposition.
Quiet moves that take squares away are worth far more here than checks that push him around.
Save the checks for the final position, where one is the whole point.
The fifty move rule is not a problem
People rush this mate because they're afraid of running out of moves. That fear is misplaced.
Nineteen moves at most, and you have fifty. Even a clumsy version of the technique finishes with room to spare.
What costs games is the clock, not the rule, so practice until the pattern is automatic rather than calculated.
Once it's automatic, the whole thing takes under a minute to play.
How to practice it properly
Set the position up and play it against a bot until you can do it without thinking.
Start with the king in the center, which is the hardest case, rather than one already near a corner.
Do it five times in a row cleanly and it'll stay with you for years. I counted the moves on my own attempts and the clumsy version still finished inside thirty.
You can set up the position and play it out on our board as many times as you need.
When you will actually need it
Rarely, and that's exactly why it's worth an hour of your time.
When it appears, it's usually the last few minutes of a long game you've fought hard for, with your clock low.
A player who knows the method converts it in seconds. A player who doesn't often draws, or loses on time trying to work it out.
Our guide on which endgames to study first puts this one in order against the rest.