Bishop and Knight vs King
Bishop and knight against a lone king is a forced win — at most thirty-three moves from the worst legal position — and it is the hardest thing a chess player is ever required to do with no counterplay on the board. The mate only works in the two corners of the bishop's color; the defender therefore runs for a wrong-colored corner, and White must drag him along the edge to the right one with a precise knight route known as the W-maneuver. Grandmasters have failed to convert this endgame with the fifty-move rule ticking. It is learnable — but only as a system, never move by move.
The Right-Corner Rule
Start with the fact that shapes everything. A checkmate needs every square around the king covered, and in a corner the bishop can only contribute on its own color. With a light-squared bishop the mates exist in the light corners, a8 and h1; the dark corners, a1 and h8, are safe houses. In the safe corner the defender's king sits on a square your bishop can never touch, and the knight and king alone cannot weave a mating net around it — one square always leaks.
So the endgame has three phases: drive the king to any edge (easy — king and bishop herd, knight cuts squares); let him run to the wrong-color corner, which he will; then perform the forced march along the edge from the wrong corner to the right one. Phase three is the whole exam, and the knight's path during it — f7, e5, d7, c5, b7 against a king running from h8 toward a8 — traces a giant W across the board. That W is not a mnemonic somebody invented; it is the only geometry by which the knight can keep sealing the escape squares in rhythm with the king and bishop.
The W-Maneuver: Evicting the King from the Wrong Corner
Black has reached the safe corner h8 (light-squared bishop, so h8 is wrong for him to fear but right for him to hide in). White has the ideal setup: king g6, bishop e4, knight e5. Watch the machine run — Black to move.
1...Kg8 2.Kf6 Kh8 3.Bf5! A quiet move with a precise job: the bishop takes up the diagonal from which it will control g8-adjacent light squares, and White is not afraid of the shuffle — 3...Kg8 4.Nf7 (first station of the W) seals h8 forever. 4...Kf8 5.Bh7! takes g8 itself, and the king must start the journey: 5...Ke8 6.Ne5 — second station, heading for d7 — 6...Kd8 7.Ke6 Kc7 8.Nd7! Third station: the knight covers b8, the next safe-looking square, always one step ahead of the king. Now Black tries the dash for freedom: 8...Kb7, aiming for a6 and the open board. 9.Bd3! The bishop, which has been idling, swings across to cover the a6–b5 escape diagonal from the other wing. 9...Kc6 10.Be2 — re-securing the same diagonal with tempo to spare — and the dash is over: the king must turn back with 10...Kc7, into the net, with the W half-drawn and the a8 corner waiting.
Two principles are visible in every move of that line. The knight zigzags between the colors, always sealing the edge squares the bishop cannot reach; the bishop controls the long escape diagonal and donates tempo moves whenever the position needs one. When students ask what to memorize, my answer is: not moves — the division of labor.
The Finish in the Right Corner
The drive's second half, from the position after Black's king has been turned back. White: king d6, bishop f3, knight d7; Black king c8, to move.
1...Kd8 2.Bh5! — the bishop pins the king to the eighth rank by covering e8; the back door is shut. 2...Kc8 3.Nc5 (fourth station) 3...Kd8 4.Nb7+ — fifth station, the W complete, delivered as a check that throws the king toward the corner: 4...Kc8 5.Kc6 Kb8 6.Kb6 Ka8 7.Kc7! The king takes b8 away; Black is reduced to a7 and a8. 7...Ka7 8.Be2! The bishop's last tempo move — Black must step to the corner. 8...Ka8 9.Nd6 Ka7 10.Nc8+ Ka8 11.Bf3#. The light bishop closes the h1–a8 diagonal it has owned all game, the knight covers a7, the king covers b7 and b8. From the diagram, mate in ten, tablebase-exact, and not one move of it was optional.
If the pattern of that finish looks familiar, it should: check into the corner, waiting move, mate on the long diagonal — the same cadence as the two bishops corner dance, with the knight playing the second bishop's role at one-third the speed.
The Thirty-Three Moves and the Clock
Tablebase facts: the worst legal position — king a1, bishop e1, knight a8, defender on d1 — is mate in thirty-three; a typical game position where you have just traded into the ending runs twenty to twenty-eight. Now set that against the arithmetic of the fifty-move rule. The margin is real but not generous, and it evaporates fast if you drive the king to the edge, blunder the geometry once, and let him run from one safe zone to another. Two disciplines protect you over the board: know phase three cold (the phases one and two almost play themselves — herd with king and bishop, keep the knight close, never stalemate a king that still has the whole edge), and count moves from the last capture so you know your budget honestly.
Should you learn it at all? Some coaches skip it, reasoning that it appears once in a career. I teach it, for a reason that has nothing to do with frequency: no other exercise forces three pieces of different geometry to cooperate square by square for thirty moves. Players who master the W-maneuver stop hanging squares in ordinary positions — the endgame is a coordination gym. And when it does appear — usually after a desperate defensive sacrifice at move sixty — it decides a game you spent five hours winning.
Frequently Asked Questions
In which corner does the bishop and knight mate happen?
Only in the corners the bishop's color controls: a8 and h1 for a light-squared bishop, a1 and h8 for a dark-squared one. In the other two corners no forced mate exists, which is why the defender runs there and why the forced drive to the right corner is the endgame's core skill.
What is the W-maneuver?
It is the knight's route while White drags the king along the edge from the wrong corner to the right one — against a king running from h8 to a8, the knight plays f7, e5, d7, c5, b7, tracing a W. Each station seals the next escape square in rhythm with the king and bishop.
How many moves does the bishop and knight checkmate take?
At most thirty-three from the worst position by tablebase count, and typically twenty to twenty-eight from realistic ones. That fits inside the fifty-move rule with margin only if the technique is known — hesitation or one wrong turn in the drive can genuinely run out the clock.
Is bishop and knight vs king always a win?
Yes, from every legal position where the pieces are not immediately lost — the tablebase confirms it. The practical results say otherwise only because humans fail the technique: the win is never in question, the fifty-move budget sometimes is.
Why can the defender not just stay in the wrong corner?
Because knight and king together evict him: the knight seals the corner square itself (as with Nf7 against a king on h8) while the king blocks the exit squares beside it. He is never mated in the wrong corner — he is escorted out of it, and the bishop closes each square behind him during the march.