Alphazero vs Stockfish Is the particular ending really a draw? Someone may try will FinalGen , please?
[Title &quot;White to move&quot;]
[fen &quot;8/1B2k3/p7/P6p/1n1p2pP/6P1/4KP2/8 w - - 0 1&quot;]
It's likely a draw. After going through a few lines (which strengthens the transposition table), Stockfish 11 at around depth 42 gave me 0.74 in the starting position. Especially in such a simplified endgame, this shouldn't be enough for White to claim a win. I also set it to White to move in the starting position.
Some variations I analyzed with Stockfish:
a) 1.Kd2 Kd6 2.Be4 Nc6 3.Bg6 Nxa5 4.Bxh5 Nc4+. Depth 38 gives 0.13, a clear draw.
b) 1.f3 gxf3+ 2.Kxf3 (2.Bxf3 Ke6 is 0.13, at depth 42) 2...Kd6 3.Kf4 Nc6: 0.13, depth 40.
c) 1.Be4 is the most challenging. Black should play 1...Kf6, and here White has a few tries:
c1) 2.Kd1 must be met with 2...Ke7!. After 3.Bg6 (3.Kc1 Kd6 4.Bg6 Ke7 5.Bxh5 Nd3+ is drawn: 0.25, depth 42) 3...Nc6 4.Bxh5 Ne5!, Black should hold: 0.38, depth 48. A more challenging move is 3.f3, but after 3...gxf3 4.Bxf3 Ke6 5.Bxh5 Nc6 6.Bg4+ Ke5 7.h5 (7.Bc8 Nxa5 8.Bxa6 Kf5 is theoretically drawn in the 7 piece tablebase) 7...Nxa5 8.h6 Kf6 9.Bh5 Nc4 10.g4 Ne3+ 11.Ke2 Nxg4, Black draws. Note that this 10...Ne3+ is extremely important - if Black couldn't play it with check, then White wins. This is why in variation c2 below, where White plays 2.Ke1 instead of 2.Kd1, 2...Ke7 will end up losing: this 10...Ne3+ resource no longer works.
c2) 2.Ke1 must instead be met with 2...Ke6!. Now 3.f3 can be met with 3...gxf3 4.Bxf3 Nd3+ 5.Kf1 Nc1 6.Bxh5 Nb3, which should be drawn, as Black is about to capture the a5-pawn. The 4...Nd3+ idea works nicely as its with tempo - in variation c1, this wouldn't have been the case as the White king was on d1. Another third move White can try here is 3.Bg6, and after 3...Ke7 4.Bxh5 Nd3+ 5.Ke2 Ne5, Black should draw - Stockfish gives 1.08 at depth 56, which shouldn't be quite enough to win. If White forces matters with 6.f3, then 6...gxf3+ 7.Bxf3 d3+ 8.Ke3 d2 9.Be2 Nc4+ 10.Kd3 Nxa5 11.Kxd2 is drawn in the 7 piece tablebase, as long as Black plays 11...Nc6 here.
c3) 2.f3 gxf3+ 3.Kxf3 Ke5 4.Bb7 Kf5 looks like it should be drawn.
c4) 2.Kd2 Ke7 3.Bg6 Nc6 4.Bxh5 Ne5 should be drawn (1.08, depth 55). Or 3.f3 gxf3 4.Bxf3 Ke6 5.Bxh5 Nc6 6.Bg4+ (6.Be2 Nxa5 7.Bxa6 is drawn in the tablebase) 6...Kf6 7.Bf3 Nxa5 8.h5, and here 8...Nc4+ or 8...Nb3+ both work, bringing the knight over to the centre/kingside with tempo. Yet another 3rd move is 3.Kc1, hoping to attack the knight with the White king. But after 3...Kd6 4.Kb2 (4.Bg6 Ke7 5.Bxh5 Nd3+ 6.Kd2 Ne5 should hold) 4...Nc6, the game is drawn.