Is three knights versus knight really winning?

Question

Take a look at the following position, reached in the #40974 Tactics Problem on chess.com

[fen "4N1K1/5P2/6k1/8/6n1/8/N7/8 w - - 0 1"]

The engine gives +6.22 for white in this position. I already understand that any move other than f8=N+ draws. (For example, promoting to a queen allows perpetual Nh6+ and Nf7+ unless white gives up the queen which would also draw)

My question is: Is white really winning after promoting the pawn to a knight with check?

How does one proceed to checkmate in this position, with three knights against a knight? And worse, in less than 50 moves? (otherwise it's a draw, of course)

I know that three knights versus lone king is possible (although quite hard!), but it seems to me that three knights versus knight is extremely unlikely to be winning upon perfect play...

Answer 1

It is winning for white. You can enter all positions of six pieces into a tablebase, like this online one.

It's mate in at most 39 moves, for example:

[FEN "4N1K1/5P2/6k1/8/6n1/8/N7/8 w - - 0 1"]

1.f8=N+ Kf5 2.Kg7 Ke4 3.Nb4 Kd4 4.Kf7 Kc5 5.Na6 Kd4 6.Nac7 Ne5 7.Kf6 Nc4 8.Ke6 Na5 9.Nd6 Nc4 10.Nf5 Kd3 11.Nd5 Na5 12.Nf6 Nc4 13.N8d7 Kd2 14.Nd5 Ke2 15.Kf6 Kd2 16.Nc5 Kc2 17.Nd4 Kd2 18.Kg5 Na5 19.Kf5 Nc4 20.Ke4 Kd1 21.Kd3 Ne5 22.Kc3 Ke1 23.Ne4 Ng6 24.Kc2 Ne5 25.Nf4 Nc4 26.Kc3 Ne5 27.Nf5 Kf1 28.Kc2 Nc4 29.Nfg3 Kg1 30.Kd3 Na5 31.Ke2 Nc6 32.Nf5 Ne5 33.Nd2 Kh1 34.Kf2 Ng4 35.Kg3 Nh6 36.Nxh6 Kg1 37.Ng4 Kh1 38.Nf2 Kg1 39.Ne2#.

There were many other choices along the way for both sides that would lead to the same amount of moves.

Edit: I think this is probably a quite modern study, where the composer used tablebases in the process of composing the study, but I might be wrong.

Answer 2

The tablebase says that it is mate in 38 moves after f8N+. That is for perfect play of both sides. If black does not play perfectly it might be shorter (same for white).

Since it is a chess problem/study, it does not really matter how easy it is to win, though I would not expect it to be too difficult (against a human player).

Answer 3

As long ago as 1895 KNNNvKN was treated as winning:

[Title "Moritz Lewitt; Deutsches Wochenschach 14 Jul 1895"]
[fen "3K2NN/8/8/8/8/8/1P6/n2k4 w - - 0 1"]
[StartFlipped "0"]

1.b4 Nc2 2. b5 Na3 3. b6 Nc4 4. b7 Na5 5. b8=N!

FEN: 3K2NN/8/8/8/8/8/1P6/n2k4

This is in PDB with no further moves given, even though the bK is out of play and mate is 34 moves off. Clearly it was thought that showing White safely promoting was enough to prove White wins.

Solution below:

1.b4 Nc2 2. b5 Na3 3. b6 Nc4 4. b7. Now 4 ... Na5 threatens 5 ... Nxb7 and 5 ... Nc6+, so White must push the pawn, and must promote to knight to counter the threatened fork.

According to the Syzygy tablebase White (with 3 knights) wins in 72.4% of positions, and 27.6% are drawn. The winning process is indeed hard in the worst cases: according to Kirill Kryukov, White wins in the following position, though Black can make the win last 90 moves:

[Title "KNNNvKN longest win. White to move."]
[fen "7N/N7/1k6/8/8/8/2K1n3/1N6 w - - 0 1"]
[StartFlipped "0"]

1.Nc8+! Kc7 2.Ne7! Kd7 3.Neg6! Ke6 4.Kd3! Ng3 5.Nd2! Kf6 6.Nf3! Nf5 7.Nf4! Ng3 8.Nd4 Nf5 9.Nde6 Nh4 10.Nd8! Nf3 11.Nd5! Ke5 12.Nc3 Kf6 13.Ne4+ Kf5 14.Ke3 Ng5 15.Ng3+! Kf6 16.Nh5! Kf5 17.Nf4 Kf6 18.Kf2 Kg7 19.Nfg6! Ne4+ {Nh7} 20.Kf3 Nc5 21.Kg3 {Ke3}

FEN: 7N/N7/1k6/8/8/8/2K1n3/1N6