Computer Challenge 01 – analysis

by Karsten Müller
5/2/2023 – Last week we presented a Computer Challenge. It's a position in which even the best chess engines will not find the solution — even if you run them for a very long time. Our readers were asked to come up with the logic and strategy of the solution themselves, and then confirm everything with the help of the computer. The result: success? Here is the full truth on the historical endgame puzzle.
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When proofreading Mark Dvorestky's Endgame Manual for the 5th edition in 2019 I had noticed that Mark's winning proof was not convincing. As neither I nor my engines could find a win or drawing proof I decided to leave it out as there were other examples on such fortresses in the book.

The first correct and convincing published win was given by Peter Prohaszka in two excellent articles in American Chess Magazine 21/2021, p.67-73 (this is the second part with the solution). The following by Wolfram Schön based on the comments of the ChessBase readers is a version of it:

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This is a solution mainly based on comments made by EmilV, Werewolf and oxygenes. The basic idea is first to bring the white king to f1 to control the black bishop. Then White prepares a knight sacrifice on e5. Black can avoid zugzwang for some time, but not forever.
45...Qb6! 46.Qb1! EmilV's move. I consider it the most systematic, as it fixes the pawn structure even more. Other moves might shorten the winning process, but give Black some chances for counterplay. h4! Forced. 46...Kb8?! 47.Nxb6 cxb6 48.h4! gxh4 49.Qh1!+- 47.Qf1 So the black bishop is limited to the g3/h2 squares. Kb8 48.Kb2 Kc8 49.Kc2 Kb8 50.Kd2 Kc8 51.Ke2 Kb8 52.Qb1 Kc8 53.Kf1 The white king has replaced the queen as a guard. Kb8 54.Qb3 White prepares to release his knight from a4 to transfer it to d3 eventually. Kb7 55.Nb2 Qa7! A position of reciprocal zugzwang. 55...Kb8?! 56.a4! A key move to EmilV's winning plan. No black counterplay with a5-a4 anymore. Qa7 57.Qc3 Kb7 58.Nd3 Qa8 59.Kg1! EmilV's zugzwang. Black can't defend a5 and prevent Nd3xe5 at the same time. Kb6 60.Nxe5 Qh8?! 61.Nd7++- 56.Qc2! Waiting move - zugzwang. Not 56.a4? Kb6 57.Qc3 Qa8 58.Nd3 Qh8= Black has organised his defence to protect a5, c5 and e5. 56...Bh2 The only way for Black to hold the positions of Qa7 and Kb7. 56...Kb8?! 57.a4! A version of the 55...Kb8-line. Kb7 58.Qc3 Qa8 59.Nd3 Bh2 60.Kf2 Bg3+ 61.Kg1! EmilV's zugzwang again. 57.Na4! White threatens b5-b6 again. Qb6 58.g4! A key improvement for White on the kingside. The hiding square h2 will be taken from the black bishop. Bg3 58...hxg3?! 59.Nxb6 cxb6 60.h4!+- 59.Qb3 Bh2 60.Qa2 A triangle manoeuvre of the white queen. Bg3 61.Qc2! Zugzwang. Kb8 61...Qa7 62.b6! cxb6 63.Qb3 Qa6 64.Nc3+- 62.Nb2 Qa7 63.a4! Now White can realise EmilV's winning idea, Black is one move too far from the set-up with Kb6 and Qa8. Kb7 64.Nd3 Qb6 On 64...Qa8 White can play as in the mainline or: 65.Qc3 Bh2 66.Nxc5++- 65.Qc3 Bh2 66.Kf2 A king triangle. Bg3+ 67.Kg2 The final zugzwang. Qa7 67...Kc8 68.Nc1 Kb7 69.Nb3+- 68.Nxe5! dxe5 69.Qxe5 Qb6 70.Qxg5+- White wins easily with his passed pawns. The black bishop in the opponent's camp is not very helpful for the defence.
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Petrosian,A-Hazai,L-1970

Here's the full game in which the position occurred, and the solution worked out by Michael Taktikos. In the following you can click (or tap) on any move or any diagram to get a special replayer, where you can start an engine and explore the moves even further.

Arshak B Petrosian½–½Laszlo Hazai
6th international junior tournament
Schilde BEL25.07.1970[Taktikos,Michael]
1.c4 g6 2.Nc3 Bg7 3.d4 Nf6 4.e4 d6 5.f3 e5 6.d5 Na6 7.Be3 Nh5 8.Qd2 Qh4+ 9.Bf2 Qe7 10.0-0-0 0-0 11.Be3 f5 12.Bd3 f4 13.Bf2 Bf6 14.Nge2 Bh4 15.Bg1 Bd7 16.Kb1 b6 17.Nc1 Nc5 18.Bc2 a5 19.Bxc5 bxc5 20.Ba4 Bxa4 21.Nxa4 Rfb8 22.Qc2 Kh8 23.Rd3 Rb4 24.Ra3 Qe8 25.Nd3 g5 26.h3 Nf6 27.Rc1 Kg7 28.b3 Nd7 29.Qd2 Rb7 30.Ndb2 Nb6 31.Ka1 h5 32.Nd1 Nxa4 33.Rxa4 Rb4 34.Nc3 Qc8 35.Nb5 Bg3 36.Qe2 Kf6 37.Rb1 Ke7 38.a3 Rxa4 39.bxa4 Kd8 40.Nc3 Rb8 41.Rb5 Rxb5 42.axb5 Qa8 43.Na4 Qa7 44.Ka2 Kc8 45.Qb2 Qb6
46.Nxb6+ Mueller,Karsten: "The endgame is very deep as given by Peter Prohaszka in two excellent articles in American Chess Magazine 21/2021, p.67-73 (this is the second part with the solution)." The plan to smash the black fortress in a nutshell: 1) The knight "wants" (via b2) to square d3, from where it can occasionally sacrifice itself on e5 or c5, or quickly change direction from d3 and act via c1 and b3 on a5.2) The queen "wants" to support him from square c3, from where she can act on a5 and e5 3) The king wants to move to the kingside, last not least making room for the correct positioning of the former pieces. Which of the engines can find the win? Their evaluation is deceptive, since almost all show a win evaluation without recognizing the fortress, and with 1. Nxb6? are immediately caught up in the draw vortex. Among the engines that recognize the fortress and start with 1.Kb3!, Lc0 v0.30.0-rc1 with the NN BT2-768x15smolgen-12h-do-01-3650000 is the one who consistently follows the described plan and evaluates the planned move significantly better than other moves. From this one is the main variant in the pgn (on a GPU 3060ti, roughly 15 sec/move) The second best is Crystal 5 KWK, and the third place goes to Stockfish XI-r1 FD (fortress detection). The latter two also start right away with 1.Kb3, but in their 4 HV they show roughly the same (winning) evaluation for almost all moves and do not follow the plan in a straight line, but only after lengthy maneuvering and some repetitions of the position. The side variants mentioned in the pgn come from these engines (on a CPU Ryzen 9 3900x, set to 22 threads, roughly 15 sec/ move) 46.Kb3 Qa7 47.Kc3 47.Kc2 Qb6 47...Bf2 48.b6 Qa6 49.Nc3 a4 49...Qxb6 50.Qxb6 cxb6 51.Nb5 Kd7 52.Kb3 Be1 53.Na7 Kc7 54.Nc6 Bd2 55.Ka4 Bc1 56.Ne7 Bb2 57.Nf5+- 49...Qxc4 50.Qb5 Qxb5 51.Nxb5 cxb6 52.Nxd6++- 49...cxb6 50.Qb5 Qxb5 51.Nxb5 Kd7 52.Kb3 Be1 53.Na7 Kc7 54.Nc6 Bf2 55.Ka4 Be3 56.Kb5 Bf2 57.a4+- 50.Qb5 Qxb6 51.Qxa4 Kb7 52.Qd7+- 48.Kd2 Kd8 48...Kb8 49.Ke2 h4 50.Qb3 Kc8 51.Nb2 Qa7 52.Kf1 Qb6 52...Kb7 53.Qc2 Bh2 54.Na4 Qb6 55.g4 Bg3 56.Nb2 Qa7 57.Qb3 Bh2 58.Qa4 Kb8 59.Qc2 Bg3 60.a4 Kb7 61.Qc3 Qb6 62.Nd3 Bh2 63.Kf2 Bg3+ 64.Kg2 Qa7 65.Nxe5 dxe5 66.Qxe5 Qb8 67.Qxg5+- 53.a4 Qb8 54.Qc3 Qa8 55.Nd3 Kb7 56.Kg1 Kb6 57.Nxe5 dxe5 58.Qxe5+- Qd8 59.Qg7 Be1 60.Kf1 Bb4 61.Ke2+- 48...h4 49.Ke2 Bh2 50.Kf1 Bg3 51.Qb3 Kb7 52.Qc2 Kc8 53.Nb2 Qa7 54.a4 Qb8 55.Nd3 Kb7 56.Qc3 Qa8 57.Kg1 Kb6 58.Nxe5 dxe5 59.Qxe5 Qd8 60.Qg7 Be1 61.Kf1 Bb4 62.Ke2 Ba3 63.Kd3 Bc1 64.e5 Bb2 65.Ke4 Kb7 66.Kf5 Qc8+ 67.Kxg5 Qb8 68.d6+- 49.Ke2 Ke7 49...Qa7 50.b6+- 49...h4 50.Qb3 Kc8 51.Nb2 Qa7 52.a4 Kb7 53.Qc3 Bh2 54.Kf1 Bg3 55.Nd3 Qa8 56.Kg1 Kb6 57.Nxe5 dxe5 58.Qxe5+- 50.Kd3 Kd8 50...Qb7 51.b6 cxb6 52.Nxc5 Qa7 53.Ne6 Qa6 54.Ng7+- 51.Qb1 h4 52.Qc2 Kc8 53.Ke2 Kb7 54.Kf1 Bh2 55.g4 Bg3 56.Qb3 Bh2 57.Qc3 Bg3 58.Qc2 Kb8 59.Nb2 Qa7 60.a4 Qb6 61.Nd3 Ka7 62.Qc3 Kb7 63.Qd2 Qa7 64.Nxe5 dxe5 65.d6 cxd6 66.Qd5+ Kc7 67.Qc6+ Kb8 68.Qe8+ Kb7 69.Qe7+ Ka8 70.Qd8+ Qb8 71.Qxg5+- 47...Qb6 48.Kd2 h4 49.Qc2 Kb7 50.Ke2 Bh2 51.Qb3 Kb8 52.Kf1 Kc8 53.Nb2 Qa7 54.g4 Kb7 55.Qa4 Kb8 56.Qc2 Bg3 57.a4 Ka8 58.Nd3 Qb6 59.Qc3 Kb7 60.Kg1 Qa7 61.Nxe5 dxe5 62.Qxe5 Qa8 63.Qg7 Be1 64.Kf1 Bd2 65.Ke2 Bc1 66.e5 Bb2 67.Kd3 Qd8 68.Ke4 Kb8 69.Qf7 Qh8 70.Qe7+- 1-0 (25) Lc0 + Crystal-Lc0 + Crystal Riddle Analysis 2023 [Taktikos,Michael] 46...cxb6 47.h4 gxh4 48.Qd2 h3 49.gxh3 h4 50.Kb3 Kb7 51.Ka4 Ka7 52.Qg2 Kb7 53.Qb2 Ka7 54.Qc2 Kb7 55.Qc3 Ka7 ½–½

The plan to smash the black fortress in a nutshell:

  1. The knight "wants" (via b2) to square d3, from where it can occasionally sacrifice itself on e5 or c5, or quickly change direction from d3 and act via c1 and b3 on a5.
  2. The queen "wants" to support him from square c3, from where she can act on a5 and e5
  3. The king wants to move to the kingside, last not least making room for the correct positioning of the former pieces.

Which of the engines can find the win? Their evaluation is deceptive, since almost all show a win evaluation without recognizing the fortress, and with 1. Nxb6? are immediately caught up in the draw vortex. Among the engines that recognize the fortress and start with 1.Kb3!, Lc0 v0.30.0-rc1 with the NN BT2-768x15smolgen-12h-do-01-3650000 is the one who consistently follows the described plan and evaluates the planned move significantly better than other moves. From this one is the main variant in the pgn (on a GPU 3060ti, roughly 15 sec/move)

The second best is Crystal 5 KWK, and the third place goes to Stockfish XI-r1 FD (fortress detection). The latter two also start right away with 1.Kb3, but in their 4 HV they show roughly the same (winning) evaluation for almost all moves and do not follow the plan in a straight line, but only after lengthy maneuvering and some repetitions of the position. The side variants mentioned in the pgn come from these engines (on a CPU Ryzen 9 3900x, set to 22 threads, roughly 15 sec/ move).

Karsten Müller is considered to be one of the greatest endgame experts in the world. His books on the endgame - among them "Fundamentals of Chess Endings", co-authored with Frank Lamprecht, that helped to improve Magnus Carlsen's endgame knowledge - and his endgame columns for the ChessCafe website and the ChessBase Magazine helped to establish and to confirm this reputation. Karsten's Fritztrainer DVDs on the endgame are bestsellers. The mathematician with a PhD lives in Hamburg, and for more than 25 years he has been scoring points for the Hamburger Schachklub (HSK) in the Bundesliga.

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