B+N vs. N

After the cooked study in Tablebase 1 - Roycroft ½ alerted me to the hidden possibilities in the elementary endgame of Bishop and Knight vs. Knight, I decided to investigate further, using the position in the diagram as a starting point. The White King and Bishop cooperate to keep the enemy King confined to the corner, although it is in no immediate danger of being checkmated. If the Black King moves to c8, attacking the Bishop, the Bishop goes to a5, when it will take the White King four moves to attack it again on a6. This gives White plenty of tempi to maneuver against the Black Knight.

As for the Knights, it's easy to vary the position of one or the other to see the effect this has on the optimal solution. For example, in the diagrammed position, with White to move (WTM), White wins in 34 moves; with Black to move (BTM), White wins in 62, in spite of the fact that the Black Knight appears to be out of danger.

Whoever is on move, White wins

WTM: win in 34; BTM: win in 62
[FEN "k2B4/8/2K5/3N4/4n3/8/8/8 w - - 0 1"]

A useful heuristic is to recall that with Bishop and Knight versus bare King, the forced checkmate takes about 30 moves, worst case. That means that in the BTM solution above, a win in 62 reduces to approximately 30 moves to trap and capture the Knight, followed by 30 moves to execute the checkmate. With WTM in the diagram, the optimal solution (wins in 34) starts 1.Kb6, when a possible sequence is 1...Nf2 2.Bc7 Nd3 3.Bd6 Nb2 4.Ka6 Na4 5.Ba3 Kb8 6.Ka5, and the Knight is trapped.

Moving the White Knight from d5 to other squares produces different effects. With the Knight on f3 and WTM, the win will take 62 moves; with BTM, it will take a few moves longer. With the Knight on g2, WTM takes 67 moves, BTM 76 moves. On h2, WTM takes 66 moves, BTM 92 moves, probably bumping into the 50-move rule. With its Knight on h1, with or without the move White can't win, because its own wayward piece is dominated by the Black Knight.

Returning to the diagram, moving the Black Knight to other squares produces similar results. On f3, WTM is a win in 66 moves, while BTM is a draw after ...Nd4+. On g2, WTM is a win in 51 moves, BTM in 61 moves. On h1, WTM is a win in 28 moves, because Bh4 traps the Black Knight immediately; BTM loses in 56.

Fine, in 'Basic Chess Endings', analyzed two positions in this endgame. No. 274a showed how the weak side's Knight could be trapped, while No. 274b demonstrated a mating attack.

• 274a: [FEN "4n2k/8/8/4KN2/2B5/8/8/8 w - - 0 1"]; 1.Bf7 wins in 47 moves.
  
• 274b: [FEN "7k/8/5NKB/8/3n4/8/8/8 b - - 0 1"]; 1...Nf5 loses in eight moves after 2.Bf8.

Fine underestimated the difficulty of this endgame. In 274b, he stated that the Knight on d4 'could draw if it were on one of a number of other squares: c6, c4, d3, h3, and of course any square from which a capture is possible'. He did not have the tools to see that with the Knight on h3, Black loses in 41 moves after 1...Nf4+ 2.Kf7 Ne6 3.Nd5 etc. Even with our modern tablebase tools, this endgame is not trivial.