An idea for a different type of tournament

[Uri Blass]

analysis after 1.e4 with multi-pv without learning also showed that 1.e4 is a mate in at most 13 moves for white.


FEN: 4k3/pppppppp/8/8/4P3/8/PPPP1PPP/RNBQKBNR b KQ e3 0 1

ShashChess15.1-general-64:

65/21 5:04:02 238,042,751k 13,049k -M10 e7-e5 Qd1-h5 Ke8-e7 Qh5xe5+ Ke7-d8 Qe5xg7 c7-c6 d2-d4 Kd8-c7 Bc1-f4+ d7-d6 Qg7xf7+ Kc7-b6 Bf4xd6 a7-a6 Qf7-c7+ Kb6-a7 Bd6-c5+ Ka7-a8 Qc7-c8+
65/21 5:04:02 238,042,751k 13,049k -M10 c7-c5 Qd1-h5 b7-b6 Ng1-f3 a7-a6 Bf1xa6 b6-b5 Ba6xb5 g7-g5 Nf3-e5 Ke8-d8 Qh5xf7 Kd8-c7 Ne5xd7 Kc7-d6 Qf7-g8 e7-e6 Qg8-b8+ Kd6-e7 Qb8-f8+
65/21 5:04:02 238,042,751k 13,049k -M10 f7-f5 Qd1-h5+ Ke8-d8 Qh5-f7 d7-d6 Ng1-f3 f5xe4 Nb1-c3 c7-c6 Nf3-d4 Kd8-d7 Nc3-d5 c6xd5 Bf1-b5+ Kd7-c8 Nd4-e6 b7-b6 Bb5-c6 e4-e3 Qf7-e8+
65/21 5:04:02 238,042,751k 13,049k -M10 h7-h5 Qd1xh5 d7-d6 Bf1-c4 g7-g6 Qh5-b5+ c7-c6 Qb5xb7 Ke8-f8 d2-d4 Kf8-g7 Qb7xe7 d6-d5 Ng1-f3 d5xc4 Nf3-e5 Kg7-g8 Qe7xf7+ Kg8-h8 Ne5xg6+
65/21 5:04:02 238,042,751k 13,049k -M10 d7-d5 Qd1-g4 Ke8-d8 Ng1-f3 f7-f6 Qg4-e6 c7-c6 e4xd5 a7-a6 d5xc6 b7xc6 Qe6xc6 e7-e6 Bf1-c4 Kd8-e7 Bc4xe6 Ke7-f8 Qc6-d7 g7-g6 Qd7-f7+
65/23 5:04:02 238,042,751k 13,049k -M11 b7-b6 Qd1-h5 b6-b5 Qh5xh7 e7-e6 Qh7xg7 a7-a6 Ng1-f3 c7-c6 Nf3-e5 Ke8-d8 Qg7xf7 Kd8-c7 Qf7xd7+ Kc7-b6 Qd7xc6+ Kb6-a7 Ne5-d7 a6-a5 Qc6-b6+ Ka7-a8 Qb6-a6+
65/23 5:04:02 238,042,751k 13,049k -M11 g7-g5 Qd1-h5 h7-h6 Ng1-f3 d7-d6 Bf1-c4 Ke8-d7 Qh5xf7 a7-a6 d2-d3 c7-c6 Bc4-e6+ Kd7-c7 Qf7xe7+ Kc7-b6 Qe7-d8+ Kb6-a7 Qd8-c7 Ka7-a8 Be6-c8 a6-a5 Qc7xb7+
65/23 5:04:02 238,042,751k 13,049k -M11 f7-f6 Qd1-h5+ g7-g6 Qh5xh7 d7-d6 Bf1-c4 Ke8-d7 Qh7-g8 a7-a5 Nb1-c3 d6-d5 Qg8xd5+ Kd7-c8 Qd5-e6+ Kc8-b8 Nc3-b5 b7-b6 Qe6-c6 e7-e6 Qc6xc7+ Kb8-a8 Qc7-a7+
65/23 5:04:02 238,042,751k 13,049k -M11 d7-d6 Qd1-g4 Ke8-d8 Qg4xg7 Kd8-d7 Qg7xf7 a7-a6 Ng1-f3 b7-b5 Nf3-d4 c7-c6 Nd4-f5 Kd7-c7 Qf7xe7+ Kc7-b6 Qe7-d8+ Kb6-a7 Nf5xd6 a6-a5 Qd8-c7+ Ka7-a6 Qc7-b7+
65/23 5:04:02 238,042,751k 13,049k -M11 b7-b5 Qd1-h5 c7-c6 Qh5xh7 Ke8-d8 Qh7xg7 f7-f5 d2-d4 f5xe4 Qg7-f8+ Kd8-c7 Bc1-f4+ e7-e5 Bf4xe5+ d7-d6 Be5xd6+ Kc7-b6 Qf8-b8+ Kb6-a6 Bd6-c5 e4-e3 Qb8xa7+
65/23 5:04:02 238,042,751k 13,049k -M11 Ke8-d8 Qd1-h5 c7-c6 Qh5xf7 g7-g5 Qf7-f8+ Kd8-c7 Qf8xe7 Kc7-c8 Ng1-f3 Kc8-c7 Nf3-e5 a7-a5 Qe7xd7+ Kc7-b6 Qd7-d8+ Kb6-a7 Ne5-d7 b7-b5 Qd8-c7+ Ka7-a8 Qc7-b8+
65/23 5:04:02 238,042,751k 13,049k -M11 Ke8-f8 Qd1-h5 d7-d6 Qh5-b5 c7-c6 Qb5xb7 f7-f6 Qb7xc6 d6-d5 Qc6xd5 g7-g6 Qd5-d8+ Kf8-f7 Bf1-c4+ Kf7-g7 Qd8xe7+ Kg7-h6 d2-d3+ Kh6-h5 Qe7xh7+ Kh5-g4 h2-h3+
65/23 5:04:02 238,042,751k 13,049k -M11 a7-a5 Qd1-h5 c7-c6 Qh5xh7 d7-d6 Qh7-g8+ Ke8-d7 d2-d4 g7-g5 Qg8-b8 Kd7-e6 Qb8-e8 d6-d5 Ng1-f3 f7-f6 Bf1-c4 d5xc4 Nf3-e5 f6xe5 Bc1xg5 e5xd4 Qe8xe7+
65/25 5:04:02 238,042,751k 13,049k -M12 e7-e6 Qd1-h5 h7-h6 Qh5-e5 d7-d6 Qe5xg7 Ke8-e7 d2-d4 a7-a6 Qg7-g8 b7-b5 Qg8-c8 c7-c5 Bc1xh6 c5xd4 Bh6-g5+ f7-f6 Qc8-c7+ Ke7-e8 Bg5xf6 Ke8-f8 Qc7-e7+ Kf8-g8 Qe7-g7+
65/25 5:04:02 238,042,751k 13,049k -M12 c7-c6 Qd1-g4 g7-g6 Qg4-f4 d7-d6 Bf1-c4 f7-f5 Bc4-e6 Ke8-d8 Qf4-h6 Kd8-c7 Qh6-f8 a7-a5 Qf8xe7+ Kc7-b8 d2-d3 f5xe4 Bc1-e3 c6-c5 Be6-d5 Kb8-c8 Be3-g5 e4xd3 Qe7xb7+
65/25 5:04:02 238,042,751k 13,049k -M12 g7-g6 Qd1-f3 c7-c6 Qf3-f4 d7-d6 Bf1-c4 f7-f5 Bc4-e6 Ke8-d8 Qf4-h6 Kd8-c7 Qh6-f8 a7-a5 Qf8xe7+ Kc7-b8 d2-d3 f5xe4 Bc1-e3 c6-c5 Be6-d5 Kb8-c8 Be3-g5 e4xd3 Qe7xb7+
66/25 5:04:02 238,042,751k 13,049k -M12 a7-a6 Qd1-g4 g7-g6 Qg4-f4 d7-d6 Bf1-c4 e7-e6 Qf4-f6 h7-h6 d2-d4 Ke8-d7 Qf6xf7+ Kd7-c6 Bc4xe6 Kc6-b6 a2-a4 Kb6-c6 Be6-d5+ Kc6-b6 a4-a5+ Kb6-a7 Qf7xc7 h6-h5 Qc7xb7+
66/27 5:04:02 238,042,751k 13,049k -M12 h7-h6 Qd1-h5 a7-a6 d2-d4 Ke8-f8 Bc1xh6 Kf8-e8 Bh6xg7 d7-d6 Qh5-h8+ Ke8-d7 d4-d5 f7-f6 Qh8-b8 a6-a5 Bg7-f8 f6-f5 Bf1-b5+ c7-c6 Qb8xb7+ Kd7-d8 Bb5xc6 f5xe4 Bf8xe7+

[jp]

analysis after 1.e4 with multi-pv without learning also showed that 1.e4 is a mate in at most 13 moves for white.

FEN: 4k3/pppppppp/8/8/4P3/8/PPPP1PPP/RNBQKBNR b KQ e3 0 1

So currently 1.e4 is better than 1.e3?

[Uri Blass]

I guess both are mate in at most 13 moves but I did not try to prove mate in 13 with e3

The only line that we need to analyze is 1.e3 a6 and if there is mate in 12 after it then 1.e3 is mate in 13.

[Uri Blass]

After 1.e3 a6 2.Qg4 the only hope is g6


FEN: 4k3/1ppppppp/p7/8/6Q1/4P3/PPPP1PPP/RNB1KBNR b KQ - 1 2

ShashChess15.1-general-64:

52/23 05:31 4,218,636k 12,714k -M11 h7-h6 Ng1-f3 h6-h5 Qg4xg7 d7-d6 Qg7-g8+ Ke8-d7 Qg8xf7 c7-c6 Nf3-d4 c6-c5 Qf7-e6+ Kd7-c7 Qe6xe7+ Kc7-b6 Nd4-e6 Kb6-c6 Bf1xa6 b7xa6 Nb1-c3 Kc6-b6 Qe7-c7+
52/23 05:31 4,218,636k 12,714k -M11 h7-h5 Qg4xg7 d7-d6 Qg7-g8+ Ke8-d7 Qg8xf7 h5-h4 Ng1-f3 b7-b5 Nf3-d4 c7-c5 Nd4-f5 d6-d5 Nf5xe7 d5-d4 Ne7-d5+ Kd7-d6 e3-e4 a6-a5 Qf7-c7+ Kd6-e6 Qc7-e7+
52/23 05:31 4,218,636k 12,714k -M11 a6-a5 Ng1-f3 h7-h5 Qg4xg7 d7-d6 Qg7-g8+ Ke8-d7 Qg8xf7 h5-h4 Nf3-d4 c7-c5 Bf1-b5+ Kd7-c8 Qf7-e8+ Kc8-c7 Nd4-c6 b7xc6 Qe8xc6+ Kc7-b8 Bb5-a6 d6-d5 Qc6-b7+
52/23 05:31 4,218,636k 12,714k -M11 c7-c6 Qg4xg7 d7-d6 Ng1-f3 h7-h6 Qg7-g8+ Ke8-d7 Qg8xf7 b7-b5 a2-a4 b5-b4 Bf1xa6 d6-d5 Nf3-d4 c6-c5 Nd4-f5 Kd7-c6 Qf7xe7 Kc6-b6 Qe7-b7+ Kb6-a5 Qb7-b5+
52/25 05:31 4,218,636k 12,714k -M12 g7-g6 Qg4-b4 b7-b5 Qb4-c3 e7-e6 Qc3xc7 g6-g5 b2-b3 f7-f6 Bc1-a3 b5-b4 Ba3xb4 g5-g4 Qc7-c8+ Ke8-f7 Qc8xd7+ Kf7-g6 Bf1-d3+ f6-f5 Qd7xe6+ Kg6-g7 Qe6xf5 h7-h5 Qf5-f8+



This hope does not work because 3.Nf3 is mate in at most 11 moves so
1.e3 is also solution for mate in 13 moves if you believe the mate scores of the engine.

FEN: 4k3/1ppppp1p/p5p1/8/6Q1/4P3/PPPP1PPP/RNB1KBNR w KQ - 0 3

ShashChess15.1-general-64:

44/26 00:57 743,728k 12,850k +M13 Qg4-c4 c7-c6 Qc4-d4 d7-d6 Qd4-b6 a6-a5 Qb6xb7 Ke8-f8 b2-b3 a5-a4 Bc1-b2 f7-f6 Bf1-c4 d6-d5 Bb2-a3 c6-c5 Qb7xd5 Kf8-g7 Qd5-g8+ Kg7-h6 h2-h4 a4xb3 Qg8-f8+ Kh6-h5 Bc4-e2+
44/26 00:57 743,728k 12,850k +M13 Qg4-e4 c7-c6 Qe4-d4 d7-d6 Qd4-b6 a6-a5 Qb6xb7 Ke8-f8 b2-b3 a5-a4 Bc1-b2 f7-f6 Bf1-c4 d6-d5 Bb2-a3 c6-c5 Qb7xd5 Kf8-g7 Qd5-g8+ Kg7-h6 h2-h4 a4xb3 Qg8-f8+ Kh6-h5 Bc4-e2+
45/24 00:57 743,728k 12,850k +M12 Qg4-d4 d7-d6 Qd4-a7 Ke8-f8 Qa7xb7 d6-d5 Qb7xc7 d5-d4 e3xd4 a6-a5 Ng1-f3 h7-h5 Nf3-e5 f7-f6 Qc7-d8+ Kf8-g7 Qd8xe7+ Kg7-h6 Qe7xf6 Kh6-h7 Qf6xg6+ Kh7-h8 Ne5-f7+
45/24 00:57 743,728k 12,850k +M12 Qg4-b4 b7-b5 Qb4-c3 e7-e6 Qc3xc7 g6-g5 b2-b3 f7-f6 Bc1-a3 b5-b4 Ba3xb4 g5-g4 Qc7-c8+ Ke8-f7 Qc8xd7+ Kf7-g6 Bf1-d3+ Kg6-g5 Qd7xh7 e6-e5 Bd3-g6 e5-e4 Qh7-h5+
45/22 00:57 743,728k 12,850k +M11 Ng1-f3 h7-h5 Qg4-c4 Ke8-d8 Qc4xf7 d7-d6 Nf3-g5 c7-c6 Qf7-e6 a6-a5 Ng5-f7+ Kd8-c7 Qe6xe7+ Kc7-b6 Qe7-d8+ Kb6-a7 Nf7xd6 h5-h4 Qd8-c7 Ka7-a8 Qc7xb7+

[Uri Blass]

Note that by statistics it seems that 1.e4 and 1.d4 are the best moves when you need to mate no pawn or single pawn or 2 pawns.

Here is the statistics so far(did not do it for more than 2 pawns)

[d]4k3/8/8/8/8/8/PPPPPPPP/RNBQKBNR w KQ - 0 1 bm 1.e4 dm 6
[d]4k3/p7/8/8/8/8/PPPPPPPP/RNBQKBNR w KQ - 0 1 bm 1.e4 1.d4 1.e3 1.b3 1.Nc3 dm 7
[d]4k3/1p6/8/8/8/8/PPPPPPPP/RNBQKBNR w KQ - 0 1 bm 1.e4 1.d4 1.Nc3 dm 7
[d]4k3/2p5/8/8/8/8/PPPPPPPP/RNBQKBNR w KQ - 0 1 bm 1.e4 1.d4 dm 7
[d]4k3/3p4/8/8/8/8/PPPPPPPP/RNBQKBNR w KQ - 0 1 bm 1.e4 1.d4 dm 7
[d]4k3/4p3/8/8/8/8/PPPPPPPP/RNBQKBNR w KQ - 0 1 bm 1.e4 1.d4 dm 7
[d]4k3/5p2/8/8/8/8/PPPPPPPP/RNBQKBNR w KQ - 0 1 bm 1.e4 1.d4 dm 7
[d]4k3/6p1/8/8/8/8/PPPPPPPP/RNBQKBNR w KQ - 0 1 bm 1.e4 1.d4 1.e3 1.Nc3 dm 7
[d]4k3/7p/8/8/8/8/PPPPPPPP/RNBQKBNR w KQ - 0 1 bm 1.e4 1.d4 1.e3 1.Nc3 1.b3 1.b4 dm 7

table:
1)e4 9
2)d4 8
3)Nc3 4
4)e3 3
5)b3 2
6)b4 1

competition continue against 2 pawns.
[d]4k3/pp6/8/8/8/8/PPPPPPPP/RNBQKBNR w KQ - 0 1 bm 1.d4 dm 7
[d]4k3/p1p5/8/8/8/8/PPPPPPPP/RNBQKBNR w KQ - 0 1 bm 1.e4 1.d4 dm 7
[d]4k3/p2p4/8/8/8/8/PPPPPPPP/RNBQKBNR w KQ - 0 1 bm 1.e4 1.d4 dm 7
[d]4k3/p3p3/8/8/8/8/PPPPPPPP/RNBQKBNR w KQ - 0 1 bm 1.e4 dm 7
[d]4k3/p4p2/8/8/8/8/PPPPPPPP/RNBQKBNR w KQ - 0 1 bm 1.e4 1.d4 dm 7
[d]4k3/p5p1/8/8/8/8/PPPPPPPP/RNBQKBNR w KQ - 0 1 bm 1.e4 1.d4 dm 7
[d]4k3/p6p/8/8/8/8/PPPPPPPP/RNBQKBNR w KQ - 0 1 bm 1.e4 1.d4 1.Nc3 1.e3 1.b3 dm 7
[d]4k3/1pp5/8/8/8/8/PPPPPPPP/RNBQKBNR w KQ - 0 1 bm 1.e4 1.d4 1.Nc3 1.e3 1.d3 1.g3 1.g4 1.a4 1.Na3 dm 8
[d]4k3/1p1p4/8/8/8/8/PPPPPPPP/RNBQKBNR w KQ - 0 1 bm 1.e4 1.d4 1.Nc3 1.e3 1.b4 1.a4 1.Na3 dm 8
[d]4k3/1p2p3/8/8/8/8/PPPPPPPP/RNBQKBNR w KQ - 0 1 bm 1.e4 1.d4 1.Nc3 1.e3 1.b3 1.b4 1.d3 1.a4 1.Na3 1.h4 dm 8
[d]4k3/1p3p2/8/8/8/8/PPPPPPPP/RNBQKBNR w KQ - 0 1 bm 1.e4 1.d4 1.Nc3 1.e3 1.d3 1.b4 1.Na3 dm 8
[d]4k3/1p4p1/8/8/8/8/PPPPPPPP/RNBQKBNR w KQ - 0 1 bm 1.e4 1.d4 1.Nc3 1.e3 1.b3 1.b4 1.d3 1.a4 1.h4 1.g3 1.g4 1.c3 1.Na3 dm 8
[d]4k3/1p5p/8/8/8/8/PPPPPPPP/RNBQKBNR w KQ - 0 1 bm 1.e4 1.d4 dm 7
[d]4k3/2pp4/8/8/8/8/PPPPPPPP/RNBQKBNR w KQ - 0 1 bm 1.e4 1.d4 1.Nc3 1.e3 1.g3 dm 8
[d]4k3/2p1p3/8/8/8/8/PPPPPPPP/RNBQKBNR w KQ - 0 1 bm 1.e4 1.d4 1.Nc3 1.e3 1.b4 1.b3 1.Na3 1.a4 Nh3 dm 8
[d]4k3/2p2p2/8/8/8/8/PPPPPPPP/RNBQKBNR w KQ - 0 1 bm 1.e4 1.d4 1.Nc3 1.e3 dm 8
[d]4k3/2p3p1/8/8/8/8/PPPPPPPP/RNBQKBNR w KQ - 0 1 bm 1.e4 1.d4 1.Nc3 1.e3 1.b4 1.b3 1.g3 1.g4 1.h4 1.c4 1.c3 1.a4 1.Na3 1.Nf3 1.Nh3 dm 8
[d]4k3/2p4p/8/8/8/8/PPPPPPPP/RNBQKBNR w KQ - 0 1 bm 1.e4 1.d4 1.Nc3 1.e3 1.b4 1.b3 1.Na3 1.a4 1.d3 1.g3 1.g4 1.h4 1.c3 1.c4 1.Nh3 1.Nf3 dm 8(16 options)
[d]4k3/3pp3/8/8/8/8/PPPPPPPP/RNBQKBNR w KQ - 0 1 bm 1.e4 1.d4 1.Nc3 1.e3 1.b3 1.Na3 dm 8
[d]4k3/3p1p2/8/8/8/8/PPPPPPPP/RNBQKBNR w KQ - 0 1 bm 1.e4 1.d4 1.Nc3 1.e3 1.b3 dm 8
[d]4k3/3p2p1/8/8/8/8/PPPPPPPP/RNBQKBNR w KQ - 0 1 bm 1.e4 1.d4 1.Nc3 1.e3 1.b3 1.b4 1.Nh3 dm 8
[d]4k3/3p3p/8/8/8/8/PPPPPPPP/RNBQKBNR w KQ - 0 1 bm 1.e4 1.d4 1.Nc3 1.e3 1.b3 1.b4 1.Na3 1.a4 1.d3 1.g3 1.Nh3 1.h4 1.Nf3 1.c4 dm 8
[d]4k3/4pp2/8/8/8/8/PPPPPPPP/RNBQKBNR w KQ - 0 1 bm 1.e4 1.d4 1.Nc3 1.e3 1.b3 1.b4 dm 8
[d]4k3/4p1p1/8/8/8/8/PPPPPPPP/RNBQKBNR w KQ - 0 1 bm 1.e4 dm 7
[d]4k3/4p2p/8/8/8/8/PPPPPPPP/RNBQKBNR w KQ - 0 1 bm 1.e4 1.d4 dm 7
[d]4k3/5pp1/8/8/8/8/PPPPPPPP/RNBQKBNR w KQ - 0 1 bm 1.e4 1.d4 1.e3 1.b3 1.b4 dm 8
[d]4k3/5p1p/8/8/8/8/PPPPPPPP/RNBQKBNR w KQ - 0 1 bm 1.e4 dm 7
[d]4k3/6pp/8/8/8/8/PPPPPPPP/RNBQKBNR w KQ - 0 1 bm 1.e4 dm 7

table so far for mating potential:
1)e4 36
2)d4 32
3)Nc3 20
4)e3 20
5)b3 14
6)b4 13
7)Na3 10
8)a4 8
9)d3 6
10)g3 6
11)Nh3 5
12)g4 4
13)h4 4
14)c3 3
15)c4 3
16)Nf3 3

[Uri Blass]

I did a mistake and gave a point to b4 instead of h4 correct table for mating against no pawn single pawn or 2 pawns is

1)e4 36
2)d4 32
3)Nc3 20
4)e3 20
5)b3 14
6)b4 12
7)Na3 10
8)a4 8
9)d3 6
10)g3 6
11)Nh3 5
12)h4 5
13)g4 4
14)c3 3
15)c4 3
16)Nf3 3

[jp]

[d]4k3/pppppppp/8/8/8/8/PPPPPPPP/RNB1KBNR w KQkq

No mate score found for this position at low to moderate depths.

I looked at the position after removing three or four Black flank Pawns... and the fastest mate SF was finding used five moves pushing and promoting the a- or h-Pawn to the Queen White does not have.

[Uri Blass]

If I include mating against 3 pawns the table for
faster mates is
1)e4 89
2)d4 79
3)e3 61
4)Nc3 49
5)b3 35
6)b4 26
7)Na3 20
8)a4 18
9)h4 16
10)g3 14
11)Nh3 11
12-13)c3 Nf3 10
14)d3 9
15)c4 7
16)g4 6
17-19)f4 h3 a3 1
20)f3 0

The only mate with in 6 is against no pawns.
mates in 7 are against a single pawn
and also against 2 pawns in the following cases
a7+b7(only d4),a7+c7(d4,e4),a7+d7(e4,d4),a7+e7(e4),a7+f7(e4,d4),a7+g7(e4,d4),a7+h7(e4,d4,Nc3,e3,b3),b7+h7(e4,d4),e7+g7(e4),e7+h7(e4,d4)
f7+h7(e4),g7+h7(e4)

I found only 2 cases when 3 pawns lose by mate in 7
a7+b7+h7(d4) and a7+g7+h7(e4)
probably no case of mate in 7 against 4 pawns.

so this is the table of mates in 7:
1)e4 20(a7,b7,c7,d7,e7,f7,g7,h7,a7+c7,a7+d7,a7+e7,a7+f7,a7+g7,a7+h7,b7+h7,e7+g7,e7+h7,f7+h7,g7+h7,a7+g7+h7)
2)d4 16(a7,b7,c7,d7,e7,f7,g7,h7,a7+b7,a7+c7,a7+d7,a7+f7,a7+g7,a7+h7,b7+h7,e7+h7,a7+b7+h7)
3)Nc3 5(a7,b7,g7,h7,a7+h7)
4)e3 4(a7,g7,h7,a7+h7)
5)b3 3(a7,h7,a7+h7)
6)b4 1(h7)

for 3 pawns we usually have mate in 8 and I mentioned 2 exceptions when we have mate in 7.
exceptions when we have mate in 9 with a lot of solutions are:
1)c7,d7,e7
2)c7,d7,f7
3)c7,e7,f7
4)d7,f7,g7

[jp]

[d]4k3/8/8/8/8/8/PPPPPPPP/RNB1KBNR w KQkq

SF10 & K13.2 find mates in 9, all of them involving promoting the a- or h-pawn to a Queen.

For example:
1. a4 Kd7 2. a5 Kd6 3. a6 Ke6 4. a7 Ke7 5. a8=Q Kd7 6. Ra6+ Kc7 7. Rc6+ Kd7 8. Qb7+ Ke8 9. Rc8#

[jp]

If we give Black a- and h-pawns to make it harder for White to mate by promoting a pawn (as above, for 4k3/8/8/8/8/8/PPPPPPPP/RNB1KBNR w KQkq) ...

[d]4k3/p6p/8/8/8/8/PPPPPPPP/RNB1KBNR w KQkq

... then SF11 fails to announce mate at depth 30, PV=2.