Hard Puzzle for July

[Spliffjiffer]

Jesus Christ, this puzzle is absolutely amazing Hobacle !!!
congratulations that you were able to create such a masterpiece, just breathtaking !
a tip: without your tip i wouldnt have solved this...i still dont understand it honestly

so far only the solution with a Crytal 3.1 output after a short backtrack of my mainline:

[pgn]
[Event "?"]
[Site "?"]
[Date "2021.07.28"]
[Round "?"]
[White "Hobacle"]
[Black "Juli"]
[Result "0-1"]
[SetUp "1"]
[FEN "8/2p1r1P1/P1Pp1K2/b1pP4/1qBp2p1/3P1pPp/3N1P1B/k5BN b - -"]

1. ... Qxd2 2. Kxe7 Qg5+ 3. Kf7 Qf5+ 4. Ke7 Qg6 5. Kf8 Qf6+ 6. Kg8 Bb6 7.
Kh7 Qf7 8. Kh8 Qh5+ 9. Kg8 Ba7 10. Kf8 Qf5+ 11. Ke7 Qg6 12. Kf8 Qf6+ 13.
Kg8 Bb8 14. Kh7 Qf7 15. Kh8 Qh5+ 16. Kg8 Kb2 17. Kf8 Qf5+ 18. Ke7 Qg6 19.
Kf8 Qf6+ 20. Kg8 Kb1 21. Kh7 Qf7 22. Kh8 Qh5+ 23. Kg8 Kc1 24. Kf8 Qf5+ 25.
Ke7 Qg6 26. Kf8 Qf6+ 27. Kg8 Kd1 28. Kh7 Qf7 29. Kh8 Qh5+ 30. Kg8 Ke1 31.
Kf8 Qf5+ 32. Ke7 Qg6 33. Kf8 Qf6+ 34. Kg8 Kf1 35. Kh7 Qf7 36. Kh8 Qh5+ 37.
Kg8 Kg2 38. Kf8 Qf5+ 39. Ke7 Qg6 40. Kf8 Qf6+ 41. Kg8 Kxh1 42. Kh7 Qf7 43.
Kh8 Qh5+ 44. Kg8 Kg2 45. Kf8 Qf5+ 46. Ke7 Qg6 47. Kf8 Qf6+ 48. Kg8 Kf1 49.
Kh7 Qf7 50. Kh8 Qh5+ 51. Kg8 Ke1 52. Kf8 Qf5+ 53. Ke7 Qg6 54. Kf8 Qf6+ 55.
Kg8 Kd2 56. Kh7 Qf7 57. Kh8 Qh5+ 58. Kg8 Kc3 59. Kf8 Qf5+ 60. Ke7 Qg6 61.
Kf8 Qf6+ 62. Kg8 Kb4 63. Kh7 Qf7 64. Kh8 Qh5+ 65. Kg8 Ka5 66. Kf8 Qf5+ 67.
Ke7 Qg6 68. Kf8 Qf6+ 69. Kg8 Kb6 70. Kh7 Qf7 71. Kh8 Qh5+ 72. Kg8 Ka7 73.
Kf8 Qf5+ 74. Ke7 Qg6 75. Kf8 Qf6+ 76. Kg8 Ka8 77. Kh7 Qf7 78. Kh8 Qh5+ 79.
Kg8 Ba7 80. Kf8 Qf5+ 81. Ke7 Qg6 82. Kf8 Qf6+ 83. Kg8 Bb6 84. Kh7 Qf7 85.
Kh8 Qh5+ 86. Kg8 Ba5 87. Kf8 Qh6 88. Kf7 Qh7 89. Kf8 Bd2 90. g8=N
Bg5 91. Bb5 Ka7 92. Bc4 Kb6 93. Bb5 Kxb5 94. a7 Qf5+ 95. Kg7 Qc8 96. Nf6 Bxf6+ 97. Kxf6 c4 98. dxc4+ Kxc4 99. Kg5 d3
100. Kf4 d2 101. a8=Q Qxa8 102. Kxg4 d1=Q 103. Kf5 Qxd5+ 104. Kf4 Qe8 105.
Kg4 Qg6+ 106. Kxh3 Qgh5# 0-1

[/pgn]

Congratulations! You solved it!
Here is my solution:
[pgn][Event ""]
[Site ""]
[Date "2021.07.28"]
[Round "?"]
[White ""]
[Black ""]
[Result "0-1"]
[FEN "8/2p1r1P1/P1Pp1K2/b1pP4/1qBp2p1/3P1pPp/3N1P1B/k5BN b - - 0 1"]
[SetUp "1"]

1... Qxd2 $3 (1... Re8 $2 2. Ne4 $11 (2. Kf7 $2 2... Qxd2 3. Kxe8 Qg5)) (1...
Rxg7 $2 2. Kxg7 Qxd2 3. Kg6 $11) 2. Kxe7 (2. g8=Q Re5 3. Qg6 Bb6 4. Kf7 Qg5 5.
Qxg5 Rxg5 6. Ke7 (6. Kf6 Re5 $19) 6... Kb1 7. Kd7 Rg8 8. Ke7 Kc1 9. Kf7 Rg5 10.
Ke7 Kd1 11. Kd7 Rg8 12. Ke7 Ke1 13. Kf7 Rg5 14. Ke7 Kf1 15. Kd7 Rg8 16. Ke7 Kg2
17. Kf7 Rg5 18. Ke7 Kxh1 19. Kd7 Rg8 20. Ke7 Kg2 21. Kf7 Rg5 22. Ke7 Re5+ 23.
Kd7 Re1 24. Kc8 Rxg1 $19) 2... Qg5+ 3. Kf7 (3. Kd7 Qxg7+ 4. Kc8 Bb6 5. Kb7 Kb2
6. a7 Bxa7 7. Kxa7 Kc3 8. Kb7 Qe7 9. Kc8 Kb4 10. Kb7 Ka5 11. Ka7 Qe1 12. Kb7
Qe2 13. Ba6 (13. Kc8 Kb6 14. Kd8 Qe5 15. Kd7 Qf6 16. Ke8 Qg7 17. Kd8 Qf7 18.
Ba2 Qf6+ 19. Kd7 Qf5+ 20. Ke7 Qxd3 21. Kf6 Qc2 22. Kg5 d3 23. Kxg4 d2 24. Bb3
Qxb3 25. Kg5 d1=Q 26. g4 Qd2+ 27. Bf4 Qbxd5+ 28. Kh4 Qxf4 29. Ng3 Qdd4 30. Kxh3
Qxg4+ 31. Kh2 Qh4#) 13... Qb2+ $19) 3... Qf5+ 4. Ke7 Qg6 (4... Qc8 $2 5. a7 Bd2
6. Kf7 Qf5+ 7. Ke7 $11) 5. Kf8 Qf6+ (5... Bd2 $2 6. a7 Qf5+ 7. Ke7 Qc8 8. Kf7
$11) 6. Kg8 Bb6 (6... Bd2 $4 7. a7 $18) 7. Kh7 (7. a7 Bxa7 8. Ba6 Bb6 9. Kh7
Qf7 10. Kh8 (10. Bc8 c4 $19) 10... Qh5+ 11. Kg8 Qxd5+ $19) (7. Bb3 Kb2 8. Bd1
c4 $19) 7... Qf7 8. Kh8 (8. Kh6 Kb2 9. Kg5 Qxg7+ 10. Kf5 Kc3 $19) (8. Bb3 Kb2
9. Bd1 c4 $19) 8... Qh5+ 9. Kg8 Kb2 10. Kf8 Qf5+ 11. Ke7 Qg6 12. Kf8 Qf6+ 13.
Kg8 Kc3 14. Kh7 Qf7 15. Kh8 Qh5+ 16. Kg8 Ba7 17. Kf8 Qf5+ 18. Ke7 Qg6 19. Kf8
Qf6+ 20. Kg8 Bb8 21. Kh7 Qf7 22. Kh8 Qh5+ 23. Kg8 Kd2 $3 (23... Kb4 $2 24. Kf8
Qf5+ 25. Ke7 Qg6 26. Kf8 Qf6+ 27. Kg8 Ka5 28. Kh7 (28. Bb3 $2 28... Kxa6 29.
Bd1 c4 $19) 28... Qf7 29. Kh8 Qh5+ 30. Kg8 Kb6 31. Kf8 (31. Bb3 $2 31... Kxa6
32. Bd1 c4 $19) 31... Qf5+ 32. Ke7 Qg6 33. Kf8 Qf6+ 34. Kg8 Ka7 35. Kh7 Qf7 36.
Kh8 Qh5+ 37. Kg8 Ka8 38. Kf8 Qf5+ 39. Ke7 Qg6 40. Kf8 Qf6+ 41. Kg8 Ba7 42. Kh7
Qf7 43. Kh8 Qh5+ 44. Kg8 Kb8 (44... Bb6 45. Kf8 Qf5+ 46. Ke7 Qg6 47. Kf8 Qf6+
48. Kg8 Ba5 49. Kh7 Qf7 50. Kh8 Qh5+ 51. Kg8 Bd2 52. Kf8 $11) 45. Kf8 Qf5+ 46.
Ke7 Qg6 47. Kf8 Qf6+ 48. Kg8 Kc8 49. Kh7 Qf7 50. Kh8 {(50.Kh6 can make DTZ
longer but not necessary)} 50... Qh5+ 51. Kg8 Kd8 52. Kf8 $11 {50}) 24. Kf8 Qf5+
25. Ke7 Qg6 26. Kf8 Qf6+ 27. Kg8 Ke1 28. Kh7 Qf7 29. Kh8 Qh5+ 30. Kg8 Kf1 31.
Kf8 Qf5+ 32. Ke7 Qg6 33. Kf8 Qf6+ 34. Kg8 Kg2 35. Kh7 Qf7 36. Kh8 Qh5+ 37. Kg8
Qh6 $3 (37... Kxh1 $2 38. Kf8 Qf5+ 39. Ke7 Qg6 40. Kf8 Qf6+ 41. Kg8 Kg2 (41...
Qh6 42. Kf7 Qc1 $2 43. g8=Q Qxg1 44. Bxg1 Kxg1 45. Qxb8 Kxf2 46. a7 h2 47. Qb1
Kg2 48. a8=Q h1=Q 49. Qaa2+ f2 50. Qxf2+ $18) (41... Qe7 42. Kh8 Qe1 $2 43.
g8=Q Qxg1 44. Bxg1 Kxg1 45. Qxb8 Kxf2 46. a7 h2 47. a8=Q h1=Q+ 48. Kg8 Kg2 49.
Qxc7 f2 50. Qg7 f1=Q 51. c7 $18) 42. Kh7 Qf7 43. Kh8 Qh5+ 44. Kg8 Kf1 45. Kf8
Qf5+ 46. Ke7 Qg6 47. Kf8 Qf6+ 48. Kg8 Ke1 49. Kh7 Qf7 50. Kh8 Qh5+ 51. Kg8 Kd2
52. Kf8 Qf5+ 53. Ke8 Qg6+ (53... Qxd3 $2 54. Bxd3 Kxd3 55. g8=Q c4 56. Qxg4 c3
57. Qxf3+ Kd2 58. Qf4+ $18) 54. Kf8 Qf6+ 55. Kg8 Kc3 56. Kh7 Qf7 57. Kh8 Qh5+
58. Kg8 Kb4 59. Kf8 Qf5+ 60. Ke8 Qg6+ 61. Kf8 Qf6+ 62. Kg8 Ka5 63. Kh7 Qf7 64.
Kh8 Qh5+ 65. Kg8 Kb6 66. Kf8 Qf5+ 67. Ke7 Qg6 68. Kf8 Qf6+ 69. Kg8 Ka7 70. Kh7
Qf7 71. Kh8 Qh5+ 72. Kg8 Ka8 73. Kf8 Qf5+ 74. Ke7 Qg6 75. Kf8 Qf6+ 76. Kg8 Ba7
77. Kh7 Qf7 78. Kh8 Qh5+ 79. Kg8 Kb8 (79... Bb6 80. Kf8 Qf5+ 81. Ke7 Qg6 82.
Kf8 Qf6+ 83. Kg8 Ba5 84. Kh7 Qf7 85. Kh8 Qh5+ 86. Kg8 Bd2 (86... Be1 87. Kf8
Bxf2 $2 88. g8=Q $18) 87. Kf8 Bh6 $11) 80. Kf8 Qf5+ 81. Ke7 Qg6 82. Kf8 Qf6+
83. Kg8 Kc8 84. Kh7 Qf7 85. Kh6 $1 (85. Kh8 $2 85... Qh5+ 86. Kg8 Kd8 87. Bb5
c4 $19 {49.5}) 85... Kd8 86. Kh7 Qh5+ 87. Kg8 Ke8 $11 {50}) 38. Kf7 Qh7 39. Kf8
Qf5+ 40. Ke7 Qg6 41. Kf8 Qf6+ 42. Kg8 Kxh1 43. Kh7 Qf7 44. Kh8 Qh5+ 45. Kg8 Kg2
46. Kf8 Qf5+ 47. Ke7 Qg6 48. Kf8 Qf6+ 49. Kg8 Kf1 50. Kh7 Qf7 51. Kh8 Qh5+ 52.
Kg8 Ke1 53. Kf8 Qf5+ 54. Ke7 Qg6 55. Kf8 Qf6+ 56. Kg8 Kd2 57. Kh7 Qf7 58. Kh8
Qh5+ 59. Kg8 Kc3 60. Kf8 Qf5+ 61. Ke7 Qg6 62. Kf8 Qf6+ 63. Kg8 Kb4 64. Kh7 Qf7
65. Kh8 Qh5+ 66. Kg8 Ka5 67. Kf8 Qf5+ 68. Ke7 Qg6 69. Kf8 Qf6+ 70. Kg8 Kb6 71.
Kh7 Qf7 72. Kh8 Qh5+ 73. Kg8 Ka7 74. Kf8 Qf5+ 75. Ke7 Qg6 76. Kf8 Qf6+ 77. Kg8
Ka8 78. Kh7 Qf7 79. Kh8 Qh5+ 80. Kg8 Ba7 81. Kf8 Qf5+ 82. Ke7 Qg6 83. Kf8 Qf6+
84. Kg8 Bb6 $1 (84... Kb8 $2 85. Kh7 Qf7 86. Kh8 Qh5+ 87. Kg8 Kc8 88. Kf8 Qf5+
89. Ke7 Qg6 90. Kf8 Qf6+ 91. Kg8 Kd8 92. Kh7 Qf7 $11 {50}) 85. Kh7 Qf7 86. Kh8
Qh5+ 87. Kg8 Ba5 88. Kf8 Qf5+ 89. Ke7 Qg6 90. Kf8 Bd2 91. g8=N (91. Kg8 Bh6 92.
Kf8 Qxg7+ $19 {49.5!!!}) 91... Bg5 92. Bc4 Kb6 93. Bb5 Kxb5 94. a7 Qf5+ 95. Kg7 Qc8 96. Nf6 Bxf6+ 97. Kxf6 c4 98. dxc4+ Kxc4 99. Kg5 d3
100. Kf4 d2 101. a8=Q Qxa8 102. Kxg4 d1=Q 103. Kf5 Qxd5+ 104. Kf4 Qe8 105.
Kg4 Qg6+ 106. Kxh3 Qgh5# 0-1
[/pgn]

And I have a question, did you find 20...Kb1!!(equivalent is 37...Qh6!! in my line) on your own?

hello Hobacle,
i did find the triangulation completely by my own but everything after the king reached a8 everything was made by corechess.engine in the beginning and later i checked it with Crystal separately....we can say that i solved it until Ka8...and everything after that is pure computer-analysis...originally i took a deep thought when my king reached g2 (to take on h1...due to your tip of course)...i thought: what the hell else can i do than taking the knight...conclusion:nothing...so i thought about 5-10 minutes to optimize the move-count and and had the idea that it should be shorter if the black king would be on the h/g-complex because on the other side (e/f/g-complex) i need more time to get him to g8...so i took my moves back until king reached e1 and made the triangulation there(e2)...its the same of course (we can make the triangulation wherever we want to)....i needed about 1- 2 min to reach the position where the king is on a8 finally and switched on the engine...the rest u know already

[Hobacle]

hello Hobacle,
i did find the triangulation completely by my own but everything after the king reached a8 everything was made by corechess.engine in the beginning and later i checked it with Crystal separately....we can say that i solved it until Ka8...and everything after that is pure computer-analysis...originally i took a deep thought when my king reached g2 (to take on h1...due to your tip of course)...i thought: what the hell else can i do than taking the knight...conclusion:nothing...so i thought about 5-10 minutes to optimize the move-count and and had the idea that it should be shorter if the black king would be on the h/g-complex because on the other side (e/f/g-complex) i need more time to get him to g8...so i took my moves back until king reached e1 and made the triangulation there(e2)...its the same of course (we can make the triangulation wherever we want to)....i needed about 1- 2 min to reach the position where the king is on a8 finally and switched on the engine...the rest u know already

You're a genius!

Here is the complete solution:
[pgn][Event ""]
[Site ""]
[Date "2021.07.28"]
[Round "?"]
[White ""]
[Black ""]
[Result "0-1"]
[FEN "8/2p1r1P1/P1Pp1K2/b1pP4/1qBp2p1/3P1pPp/3N1P1B/k5BN b - - 0 1"]
[SetUp "1"]

1... Qxd2 $3 (1... Re8 $2 2. Ne4 $11 (2. Kf7 $2 2... Qxd2 3. Kxe8 Qg5 {back to the main line})) (1...
Rxg7 $2 2. Kxg7 Qxd2 3. Kg6 $11) 2. Kxe7 (2. g8=Q Re5 3. Qg6 Bb6 4. Kf7 Qg5 5.
Qxg5 Rxg5 6. Ke7 (6. Kf6 Re5 $19) 6... Kb1 7. Kd7 Rg8 8. Ke7 Kc1 9. Kf7 Rg5 10.
Ke7 Kd1 11. Kd7 Rg8 12. Ke7 Ke1 13. Kf7 Rg5 14. Ke7 Kf1 15. Kd7 Rg8 16. Ke7 Kg2
17. Kf7 Rg5 18. Ke7 Kxh1 19. Kd7 Rg8 20. Ke7 Kg2 21. Kf7 Rg5 22. Ke7 Re5+ 23.
Kd7 Re1 24. Kc8 Rxg1 $19) 2... Qg5+ 3. Kf7 (3. Kd7 Qxg7+ 4. Kc8 Bb6 5. Kb7 Kb2
6. a7 Bxa7 7. Kxa7 Kc3 8. Kb7 Qe7 9. Kc8 Kb4 10. Kb7 Ka5 11. Ka7 Qe1 12. Kb7
Qe2 13. Ba6 (13. Kc8 Kb6 14. Kd8 Qe5 15. Kd7 Qf6 16. Ke8 Qg7 17. Kd8 Qf7 18.
Ba2 Qf6+ 19. Kd7 Qf5+ 20. Ke7 Qxd3 21. Kf6 Qc2 22. Kg5 d3 23. Kxg4 d2 24. Bb3
Qxb3 25. Kg5 d1=Q 26. g4 Qd2+ 27. Bf4 Qbxd5+ 28. Kh4 Qxf4 29. Ng3 Qdd4 30. Kxh3
Qxg4+ 31. Kh2 Qh4#) 13... Qb2+ $19) 3... Qf5+ 4. Ke7 Qg6 (4... Qc8 $2 5. a7 Bd2
6. Kf7 Qf5+ 7. Ke7 $11) 5. Kf8 Qf6+ (5... Bd2 $2 6. a7 Qf5+ 7. Ke7 Qc8 8. Kf7
$11) 6. Kg8 Bb6 (6... Bd2 $4 7. a7 $18) 7. Kh7 (7. a7 Bxa7 8. Ba6 Bb6 9. Kh7
Qf7 10. Kh8 (10. Bc8 c4 $19) 10... Qh5+ 11. Kg8 Qxd5+ $19) (7. Bb3 Kb2 8. Bd1
c4 $19) 7... Qf7 8. Kh8 (8. Kh6 Kb2 9. Kg5 Qxg7+ 10. Kf5 Kc3 $19) (8. Bb3 Kb2
9. Bd1 c4 $19) 8... Qh5+ 9. Kg8 Kb2 10. Kf8 Qf5+ 11. Ke7 Qg6 12. Kf8 Qf6+ 13.
Kg8 Kc3 14. Kh7 Qf7 15. Kh8 Qh5+ 16. Kg8 Ba7 17. Kf8 Qf5+ 18. Ke7 Qg6 19. Kf8
Qf6+ 20. Kg8 Bb8 21. Kh7 Qf7 22. Kh8 Qh5+ 23. Kg8 Kd2 $3 (23... Kb4 $2 24. Kf8
Qf5+ 25. Ke7 Qg6 26. Kf8 Qf6+ 27. Kg8 Ka5 28. Kh7 (28. Bb3 $2 28... Kxa6 29.
Bd1 c4 $19) 28... Qf7 29. Kh8 Qh5+ 30. Kg8 Kb6 31. Kf8 (31. Bb3 $2 31... Kxa6
32. Bd1 c4 $19) 31... Qf5+ 32. Ke7 Qg6 33. Kf8 Qf6+ 34. Kg8 Ka7 35. Kh7 Qf7 36.
Kh8 Qh5+ 37. Kg8 Ka8 38. Kf8 Qf5+ 39. Ke7 Qg6 40. Kf8 Qf6+ 41. Kg8 Ba7 42. Kh7
Qf7 43. Kh8 Qh5+ 44. Kg8 Kb8 (44... Bb6 45. Kf8 Qf5+ 46. Ke7 Qg6 47. Kf8 Qf6+
48. Kg8 Ba5 49. Kh7 Qf7 50. Kh8 Qh5+ 51. Kg8 Bd2 52. Kf8 $11) 45. Kf8 Qf5+ 46.
Ke7 Qg6 47. Kf8 Qf6+ 48. Kg8 Kc8 49. Kh7 Qf7 50. Kh8 {50.Kh6 can make DTZ
longer but not necessary} 50... Qh5+ 51. Kg8 Kd8 52. Kf8 $11 {50}) 24. Kf8 Qf5+
25. Ke7 Qg6 26. Kf8 Qf6+ 27. Kg8 Ke1 28. Kh7 Qf7 29. Kh8 Qh5+ 30. Kg8 Ke2 $3
(30... Kf1 31. Kf8 Qf5+ 32. Ke7 Qg6 33. Kf8 Qf6+ 34. Kg8 Kg2 35. Kh7 Qf7 36.
Kh8 Qh5+ 37. Kg8 Kxh1 $2 {(37...Qh6 still wins but one move slower than the
main line)} 38. Kf8 Qf5+ 39. Ke7 Qg6 40. Kf8 Qf6+ 41. Kg8 Kg2 (41... Qh6 42.
Kf7 Qc1 $2 43. g8=Q Qxg1 44. Bxg1 Kxg1 45. Qxb8 Kxf2 46. a7 h2 47. Qb1 Kg2 48.
a8=Q h1=Q 49. Qaa2+ f2 50. Qxf2+ $18) (41... Qe7 42. Kh8 Qe1 $2 43. g8=Q Qxg1
44. Bxg1 Kxg1 45. Qxb8 Kxf2 46. a7 h2 47. a8=Q h1=Q+ 48. Kg8 Kg2 49. Qxc7 f2
50. Qg7 f1=Q 51. c7 $18) 42. Kh7 Qf7 43. Kh8 Qh5+ 44. Kg8 Kf1 45. Kf8 Qf5+ 46.
Ke7 Qg6 47. Kf8 Qf6+ 48. Kg8 Ke1 49. Kh7 Qf7 50. Kh8 Qh5+ 51. Kg8 Kd2 52. Kf8
Qf5+ 53. Ke8 Qg6+ (53... Qxd3 $2 54. Bxd3 Kxd3 55. g8=Q c4 56. Qxg4 c3 57.
Qxf3+ Kd2 58. Qf4+ $18) 54. Kf8 Qf6+ 55. Kg8 Kc3 56. Kh7 Qf7 57. Kh8 Qh5+ 58.
Kg8 Kb4 59. Kf8 Qf5+ 60. Ke8 Qg6+ 61. Kf8 Qf6+ 62. Kg8 Ka5 63. Kh7 Qf7 64. Kh8
Qh5+ 65. Kg8 Kb6 66. Kf8 Qf5+ 67. Ke7 Qg6 68. Kf8 Qf6+ 69. Kg8 Ka7 70. Kh7 Qf7
71. Kh8 Qh5+ 72. Kg8 Ka8 73. Kf8 Qf5+ 74. Ke7 Qg6 75. Kf8 Qf6+ 76. Kg8 Ba7 77.
Kh7 Qf7 78. Kh8 Qh5+ 79. Kg8 Kb8 (79... Bb6 80. Kf8 Qf5+ 81. Ke7 Qg6 82. Kf8
Qf6+ 83. Kg8 Ba5 84. Kh7 Qf7 85. Kh8 Qh5+ 86. Kg8 Bd2 (86... Be1 87. Kf8 Bxf2
$2 88. g8=Q $18) 87. Kf8 Bh6 $11) 80. Kf8 Qf5+ 81. Ke7 Qg6 82. Kf8 Qf6+ 83. Kg8
Kc8 84. Kh7 Qf7 85. Kh6 $1 (85. Kh8 $2 85... Qh5+ 86. Kg8 Kd8 87. Bb5 c4 $19
{49.5}) 85... Kd8 86. Kh7 Qh5+ 87. Kg8 Ke8 $11 {50}) 31. Kf8 Qf5+ 32. Ke7 Qg6
33. Kf8 Qf6+ 34. Kg8 Kf1 35. Kh7 Qf7 36. Kh8 Qh5+ 37. Kg8 Kg2 38. Kf8 Qf5+ 39.
Ke7 Qg6 40. Kf8 Qf6+ 41. Kg8 Kxh1 42. Kh7 Qf7 43. Kh8 Qh5+ 44. Kg8 Kg2 45. Kf8
Qf5+ 46. Ke7 Qg6 47. Kf8 Qf6+ 48. Kg8 Kf1 49. Kh7 Qf7 50. Kh8 Qh5+ 51. Kg8 Ke1
52. Kf8 Qf5+ 53. Ke7 Qg6 54. Kf8 Qf6+ 55. Kg8 Kd2 56. Kh7 Qf7 57. Kh8 Qh5+ 58.
Kg8 Kc3 59. Kf8 Qf5+ 60. Ke7 Qg6 61. Kf8 Qf6+ 62. Kg8 Kb4 63. Kh7 Qf7 64. Kh8
Qh5+ 65. Kg8 Ka5 66. Kf8 Qf5+ 67. Ke7 Qg6 68. Kf8 Qf6+ 69. Kg8 Kb6 70. Kh7 Qf7
71. Kh8 Qh5+ 72. Kg8 Ka7 73. Kf8 Qf5+ 74. Ke7 Qg6 75. Kf8 Qf6+ 76. Kg8 Ka8 77.
Kh7 Qf7 78. Kh8 Qh5+ 79. Kg8 Ba7 80. Kf8 Qf5+ 81. Ke7 Qg6 82. Kf8 Qf6+ 83. Kg8
Bb6 $1 (83... Kb8 $2 84. Kh7 Qf7 85. Kh8 Qh5+ 86. Kg8 Kc8 87. Kf8 Qf5+ 88. Ke7
Qg6 89. Kf8 Qf6+ 90. Kg8 Kd8 91. Kh7 Qf7 $11 {50}) 84. Kh7 Qf7 85. Kh8 Qh5+ 86.
Kg8 Ba5 87. Kf8 Qh6 88. Kf7 Qh7 89. Kf8 Bd2 90. g8=N (90. Kf7 Bh6 91. Kf6 Qxg7+
$19 {49.5!!!}) 90... Bg5 91. Bb5 Ka7 92. Bc4 Kb6 93. Bb5 Kxb5 94. a7 Qf5+ 95.
Kg7 Qc8 96. Nf6 Bxf6+ 97. Kxf6 c4 98. dxc4+ Kxc4 99. Kg5 d3 100. Kf4 d2 101.
a8=Q Qxa8 102. Kxg4 d1=Q 103. Kf5 Qxd5+ 104. Kf4 Qe8 105. Kg4 Qg6+ 106. Kxh3
Qgh5# 0-1

[/pgn]

[peter]

Jesus Christ, this puzzle is absolutely amazing Hobacle !!!
congratulations that you were able to create such a masterpiece, just breathtaking !
a tip: without your tip i wouldnt have solved this...i still dont understand it honestly

so far only the solution with a Crytal 3.1 output after a short backtrack of my mainline:

[pgn]
[Event "?"]
[Site "?"]
[Date "2021.07.28"]
[Round "?"]
[White "Hobacle"]
[Black "Juli"]
[Result "0-1"]
[SetUp "1"]
[FEN "8/2p1r1P1/P1Pp1K2/b1pP4/1qBp2p1/3P1pPp/3N1P1B/k5BN b - -"]

1. ... Qxd2 2. Kxe7 Qg5+ 3. Kf7 Qf5+ 4. Ke7 Qg6 5. Kf8 Qf6+ 6. Kg8 Bb6 7.
Kh7 Qf7 8. Kh8 Qh5+ 9. Kg8 Ba7 10. Kf8 Qf5+ 11. Ke7 Qg6 12. Kf8 Qf6+ 13.
Kg8 Bb8 14. Kh7 Qf7 15. Kh8 Qh5+ 16. Kg8 Kb2 17. Kf8 Qf5+ 18. Ke7 Qg6 19.
Kf8 Qf6+ 20. Kg8 Kb1 21. Kh7 Qf7 22. Kh8 Qh5+ 23. Kg8 Kc1 24. Kf8 Qf5+ 25.
Ke7 Qg6 26. Kf8 Qf6+ 27. Kg8 Kd1 28. Kh7 Qf7 29. Kh8 Qh5+ 30. Kg8 Ke1 31.
Kf8 Qf5+ 32. Ke7 Qg6 33. Kf8 Qf6+ 34. Kg8 Kf1 35. Kh7 Qf7 36. Kh8 Qh5+ 37.
Kg8 Kg2 38. Kf8 Qf5+ 39. Ke7 Qg6 40. Kf8 Qf6+ 41. Kg8 Kxh1 42. Kh7 Qf7 43.
Kh8 Qh5+ 44. Kg8 Kg2 45. Kf8 Qf5+ 46. Ke7 Qg6 47. Kf8 Qf6+ 48. Kg8 Kf1 49.
Kh7 Qf7 50. Kh8 Qh5+ 51. Kg8 Ke1 52. Kf8 Qf5+ 53. Ke7 Qg6 54. Kf8 Qf6+ 55.
Kg8 Kd2 56. Kh7 Qf7 57. Kh8 Qh5+ 58. Kg8 Kc3 59. Kf8 Qf5+ 60. Ke7 Qg6 61.
Kf8 Qf6+ 62. Kg8 Kb4 63. Kh7 Qf7 64. Kh8 Qh5+ 65. Kg8 Ka5 66. Kf8 Qf5+ 67.
Ke7 Qg6 68. Kf8 Qf6+ 69. Kg8 Kb6 70. Kh7 Qf7 71. Kh8 Qh5+ 72. Kg8 Ka7 73.
Kf8 Qf5+ 74. Ke7 Qg6 75. Kf8 Qf6+ 76. Kg8 Ka8 77. Kh7 Qf7 78. Kh8 Qh5+ 79.
Kg8 Ba7 80. Kf8 Qf5+ 81. Ke7 Qg6 82. Kf8 Qf6+ 83. Kg8 Bb6 84. Kh7 Qf7 85.
Kh8 Qh5+ 86. Kg8 Ba5 87. Kf8 Qh6 88. Kf7 Qh7 89. Kf8 Bd2 90. g8=N
Bg5 91. Bb5 Ka7 92. Bc4 Kb6 93. Bb5 Kxb5 94. a7 Qf5+ 95. Kg7 Qc8 96. Nf6 Bxf6+ 97. Kxf6 c4 98. dxc4+ Kxc4 99. Kg5 d3
100. Kf4 d2 101. a8=Q Qxa8 102. Kxg4 d1=Q 103. Kf5 Qxd5+ 104. Kf4 Qe8 105.
Kg4 Qg6+ 106. Kxh3 Qgh5# 0-1
[/pgn]

Yet somewhat shorter, maybe with a few moves short-cutted, ...

Edit, edit, edit: (cannot delete anymore) the .pgn I had here, was a wrong one chosen from past trials mixed with newer ones, forget about the shortcuts till I'll be back...

[amanjpro]

Somehow this puzzle uncovered a bug in my aspiration windw+prob-cut implementation... thank you
After solving the issue, I'll try the puzzle with Zahak

[peter]

Jesus Christ, this puzzle is absolutely amazing Hobacle !!!
congratulations that you were able to create such a masterpiece, just breathtaking !
a tip: without your tip i wouldnt have solved this...i still dont understand it honestly

so far only the solution with a Crytal 3.1 output after a short backtrack of my mainline:

[pgn]
[Event "?"]
[Site "?"]
[Date "2021.07.28"]
[Round "?"]
[White "Hobacle"]
[Black "Juli"]
[Result "0-1"]
[SetUp "1"]
[FEN "8/2p1r1P1/P1Pp1K2/b1pP4/1qBp2p1/3P1pPp/3N1P1B/k5BN b - -"]

1. ... Qxd2 2. Kxe7 Qg5+ 3. Kf7 Qf5+ 4. Ke7 Qg6 5. Kf8 Qf6+ 6. Kg8 Bb6 7.
Kh7 Qf7 8. Kh8 Qh5+ 9. Kg8 Ba7 10. Kf8 Qf5+ 11. Ke7 Qg6 12. Kf8 Qf6+ 13.
Kg8 Bb8 14. Kh7 Qf7 15. Kh8 Qh5+ 16. Kg8 Kb2 17. Kf8 Qf5+ 18. Ke7 Qg6 19.
Kf8 Qf6+ 20. Kg8 Kb1 21. Kh7 Qf7 22. Kh8 Qh5+ 23. Kg8 Kc1 24. Kf8 Qf5+ 25.
Ke7 Qg6 26. Kf8 Qf6+ 27. Kg8 Kd1 28. Kh7 Qf7 29. Kh8 Qh5+ 30. Kg8 Ke1 31.
Kf8 Qf5+ 32. Ke7 Qg6 33. Kf8 Qf6+ 34. Kg8 Kf1 35. Kh7 Qf7 36. Kh8 Qh5+ 37.
Kg8 Kg2 38. Kf8 Qf5+ 39. Ke7 Qg6 40. Kf8 Qf6+ 41. Kg8 Kxh1 42. Kh7 Qf7 43.
Kh8 Qh5+ 44. Kg8 Kg2 45. Kf8 Qf5+ 46. Ke7 Qg6 47. Kf8 Qf6+ 48. Kg8 Kf1 49.
Kh7 Qf7 50. Kh8 Qh5+ 51. Kg8 Ke1 52. Kf8 Qf5+ 53. Ke7 Qg6 54. Kf8 Qf6+ 55.
Kg8 Kd2 56. Kh7 Qf7 57. Kh8 Qh5+ 58. Kg8 Kc3 59. Kf8 Qf5+ 60. Ke7 Qg6 61.
Kf8 Qf6+ 62. Kg8 Kb4 63. Kh7 Qf7 64. Kh8 Qh5+ 65. Kg8 Ka5 66. Kf8 Qf5+ 67.
Ke7 Qg6 68. Kf8 Qf6+ 69. Kg8 Kb6 70. Kh7 Qf7 71. Kh8 Qh5+ 72. Kg8 Ka7 73.
Kf8 Qf5+ 74. Ke7 Qg6 75. Kf8 Qf6+ 76. Kg8 Ka8 77. Kh7 Qf7 78. Kh8 Qh5+ 79.
Kg8 Ba7 80. Kf8 Qf5+ 81. Ke7 Qg6 82. Kf8 Qf6+ 83. Kg8 Bb6 84. Kh7 Qf7 85.
Kh8 Qh5+ 86. Kg8 Ba5 87. Kf8 Qh6 88. Kf7 Qh7 89. Kf8 Bd2 90. g8=N
Bg5 91. Bb5 Ka7 92. Bc4 Kb6 93. Bb5 Kxb5 94. a7 Qf5+ 95. Kg7 Qc8 96. Nf6 Bxf6+ 97. Kxf6 c4 98. dxc4+ Kxc4 99. Kg5 d3
100. Kf4 d2 101. a8=Q Qxa8 102. Kxg4 d1=Q 103. Kf5 Qxd5+ 104. Kf4 Qe8 105.
Kg4 Qg6+ 106. Kxh3 Qgh5# 0-1
[/pgn]

SlowChess has a very fine mate- search. So far I'm back to 73. move of your .pgn, sparing 6 from there onwards:

[d]1b3K2/k1p3P1/P1Pp4/2pP3q/2Bp2p1/3P1pPp/5P1B/6B1 b - - 0 1

Analysis by SlowChess Blitz 2.6 sse:

73...Qf5+ 74.Ke7 Qg6 75.Kf8 Qf6+ 76.Kg8 Ka8 77.Kh7 Qf7 78.Kh8 Qh5+ 79.Kg8 Ba7 80.Kf8 Qf5+ 81.Ke7 Qg6 82.Kf8 Qf6+ 83.Kg8 Kb8 84.Bb3 Kc8 85.Kh7 Qf7 86.Kh6 Kd8 87.Bd1 Ke7 88.Kg5 Qxg7+ 89.Kf5 c4 90.Bxf3 gxf3 91.g4 cxd3 92.Bf4 Kf7 93.Ke4 Qxg4 94.Kxd3 Qxg1 95.Kc2 Qa1 96.Kd3 Qb1+ 97.Kc4 Qa2+ 98.Kb4 Bc5+ 99.Kb5 Qb3+ 100.Ka5 Qb4#
-+ (-#28) Depth: 55/85 00:06:47 2913MN, tb=2411064

That transposes to your line up to 83.Kg8, there ...Bb6 makes the final detour relatively to ...Kb8. Hobacle there has a line leading to 50 moves without capture nor pawn- move. Maybe 84.Bb3 is blunder, yet we have it a mating line of an engine counting the 50 moves. That reminds me of something we had at the shorter way too, doesn't it?

There's still some doubt if the long one for sure works >50 moves rule, as long as we don't know the exakt DTM. DTM wouldn't matter if 50 moves rule wouldn't, yet counting the moves makes the whole study some kind of mate in x-problem regards

[Spliffjiffer]

Jesus Christ, this puzzle is absolutely amazing Hobacle !!!
congratulations that you were able to create such a masterpiece, just breathtaking !
a tip: without your tip i wouldnt have solved this...i still dont understand it honestly

so far only the solution with a Crytal 3.1 output after a short backtrack of my mainline:

[pgn]
[Event "?"]
[Site "?"]
[Date "2021.07.28"]
[Round "?"]
[White "Hobacle"]
[Black "Juli"]
[Result "0-1"]
[SetUp "1"]
[FEN "8/2p1r1P1/P1Pp1K2/b1pP4/1qBp2p1/3P1pPp/3N1P1B/k5BN b - -"]

1. ... Qxd2 2. Kxe7 Qg5+ 3. Kf7 Qf5+ 4. Ke7 Qg6 5. Kf8 Qf6+ 6. Kg8 Bb6 7.
Kh7 Qf7 8. Kh8 Qh5+ 9. Kg8 Ba7 10. Kf8 Qf5+ 11. Ke7 Qg6 12. Kf8 Qf6+ 13.
Kg8 Bb8 14. Kh7 Qf7 15. Kh8 Qh5+ 16. Kg8 Kb2 17. Kf8 Qf5+ 18. Ke7 Qg6 19.
Kf8 Qf6+ 20. Kg8 Kb1 21. Kh7 Qf7 22. Kh8 Qh5+ 23. Kg8 Kc1 24. Kf8 Qf5+ 25.
Ke7 Qg6 26. Kf8 Qf6+ 27. Kg8 Kd1 28. Kh7 Qf7 29. Kh8 Qh5+ 30. Kg8 Ke1 31.
Kf8 Qf5+ 32. Ke7 Qg6 33. Kf8 Qf6+ 34. Kg8 Kf1 35. Kh7 Qf7 36. Kh8 Qh5+ 37.
Kg8 Kg2 38. Kf8 Qf5+ 39. Ke7 Qg6 40. Kf8 Qf6+ 41. Kg8 Kxh1 42. Kh7 Qf7 43.
Kh8 Qh5+ 44. Kg8 Kg2 45. Kf8 Qf5+ 46. Ke7 Qg6 47. Kf8 Qf6+ 48. Kg8 Kf1 49.
Kh7 Qf7 50. Kh8 Qh5+ 51. Kg8 Ke1 52. Kf8 Qf5+ 53. Ke7 Qg6 54. Kf8 Qf6+ 55.
Kg8 Kd2 56. Kh7 Qf7 57. Kh8 Qh5+ 58. Kg8 Kc3 59. Kf8 Qf5+ 60. Ke7 Qg6 61.
Kf8 Qf6+ 62. Kg8 Kb4 63. Kh7 Qf7 64. Kh8 Qh5+ 65. Kg8 Ka5 66. Kf8 Qf5+ 67.
Ke7 Qg6 68. Kf8 Qf6+ 69. Kg8 Kb6 70. Kh7 Qf7 71. Kh8 Qh5+ 72. Kg8 Ka7 73.
Kf8 Qf5+ 74. Ke7 Qg6 75. Kf8 Qf6+ 76. Kg8 Ka8 77. Kh7 Qf7 78. Kh8 Qh5+ 79.
Kg8 Ba7 80. Kf8 Qf5+ 81. Ke7 Qg6 82. Kf8 Qf6+ 83. Kg8 Bb6 84. Kh7 Qf7 85.
Kh8 Qh5+ 86. Kg8 Ba5 87. Kf8 Qh6 88. Kf7 Qh7 89. Kf8 Bd2 90. g8=N
Bg5 91. Bb5 Ka7 92. Bc4 Kb6 93. Bb5 Kxb5 94. a7 Qf5+ 95. Kg7 Qc8 96. Nf6 Bxf6+ 97. Kxf6 c4 98. dxc4+ Kxc4 99. Kg5 d3
100. Kf4 d2 101. a8=Q Qxa8 102. Kxg4 d1=Q 103. Kf5 Qxd5+ 104. Kf4 Qe8 105.
Kg4 Qg6+ 106. Kxh3 Qgh5# 0-1
[/pgn]

SlowChess has a very fine mate- search. So far I'm back to 73. move of your .pgn, sparing 6 from there onwards:
[d]1b3K2/k1p3P1/P1Pp4/2pP3q/2Bp2p1/3P1pPp/5P1B/6B1 b - - 0 1

Analysis by SlowChess Blitz 2.6 sse:

73...Qf5+ 74.Ke7 Qg6 75.Kf8 Qf6+ 76.Kg8 Ka8 77.Kh7 Qf7 78.Kh8 Qh5+ 79.Kg8 Ba7 80.Kf8 Qf5+ 81.Ke7 Qg6 82.Kf8 Qf6+ 83.Kg8 Kb8 84.Bb3 Kc8 85.Kh7 Qf7 86.Kh6 Kd8 87.Bd1 Ke7 88.Kg5 Qxg7+ 89.Kf5 c4 90.Bxf3 gxf3 91.g4 cxd3 92.Bf4 Kf7 93.Ke4 Qxg4 94.Kxd3 Qxg1 95.Kc2 Qa1 96.Kd3 Qb1+ 97.Kc4 Qa2+ 98.Kb4 Bc5+ 99.Kb5 Qb3+ 100.Ka5 Qb4#
-+ (-#28) Depth: 55/85 00:06:47 2913MN, tb=2411064

yes, i dont believe really that i found the shortest mate maybe ...after i spotted this increadable idea by Hobacle i spent maybe 15 min...trying to verify that the whole thing is actually real ...i thought (also) about to configure Chest after this (it didnt make much sense to me after our old mainline because the end of the line "opened up everything" and that is not a good sign normally when in the end of a mainline there are suddenly 3+ kingsqures and a bunch of pieces finally that can move)---in the new mainline it seems to me that everything is much more consistent but i heavily fear that there are to much kingsqures towards the end of the line still---ill give it a go at some point but my expectations are not really promising so to say...probably your approach with "normal" engines is just better !?...lets see

[peter]

...probably your approach with "normal" engines is just better !?...lets see

Well, SlowChess acts somehow in between "normal engine" and specialised mate searching one, therefore prefers shortest mating line to longer one leading to 50 moves draw now and then obviously, of course that doesn't help then here as for (only one) reliabled output.

But thats the main problem to me with such long- liners based on cyclic zugzwang anyhow, those normally don't care for exact DTM but neither for 50 moves rule.
If you start counting moves you make the problem a mate in x one, such must have exactly known DTM, because that's the thematic main point. And then you have to decide here which one solution's indeed the correct one, the one with shortest DTM, which would be the point of mate in x- problems, or the one that avoids 50 moves by longer lines with DTM on their own.

Sorry, I'm just somewhat overcharged with that piece again

[Hobacle]

Sorry, I'm just somewhat overcharged with that piece again

I think you'd better spend more time in understanding the main line, not on the DTM.

[peter]

I think you'd better spend more time in understanding the main line, not on the DTM.

The main line, which one ever that might be (cyclic zugzwang always means the order of the triangles is exchangable, so there isn't any one and only main line except the author's one arbitrarily chosen by him) isn't hard to understand. Black has to make always one step with the King, when Black's King is forced in the triangle- maneuver to block his g-pawn from moving. The direction of White's King is even more clear, it has to be brought to the g- pawn, the exchange of places of Black's only Bishop and the King at a- line both stopping White's a- pawn till Black's King can support the Queen is the only really somewhat tricky point, but because there are no other figures except Black's King and Bishop to make any progress at all, humans should find that sooner or later as a plan too. Regard: a plan isn't a playing- line, the opponent's moves are missing.

What makes the piece really damned tricky, is to find out how many moves are to be made till mate if 50 moves- rule is to be regarded too, so I understand well, that you don't want to resign that. If you don't mind 50 moves rule you wouldn't even have to mind DTM, Nh1 wouldn't be necessary, it would be just one more long lined cyclic zugzwang puzzle.

But what you have to give then, if move counting gets the main point as this is so in mate in x- problems, you then have to give as a full solution- line the exact DTM without crossing 50 moves.

And this one point you and nobody else will manage to solve without engines' support, will you?

That's the difference between "understanding" and "planning" and playing chess, (about choss, I'm not quite sure, if I got you right, you're playing it with rules of you're very own only) humans are great in understanding and planning, as for playing lines to 50 moves draw or mate shortly before draw, they like to take some help of engines, don't they?

[Hobacle]

I think you'd better spend more time in understanding the main line, not on the DTM.

The main line, which one ever that might be (cyclic zugzwang always means the order of the triangles is exchangable, so there isn't any one and only main line except the author's one arbitrarily chosen by him) isn't hard to understand. Black has to make always one step with the King, when Black's King is forced in the triangle- maneuver to block his g-pawn from moving. The direction of White's King is even more clear, it has to be brought to the g- pawn, the exchange of places of Black's only Bishop and the King at a- line both stopping White's a- pawn till Black's King can support the Queen is the only really somewhat tricky point, but because there are no other figures except Black's King and Bishop to make any progress at all, humans should find that sooner or later as a plan too. Regard: a plan isn't a playing- line, the opponent's moves are missing.

What makes the piece really damned tricky, is to find out how many moves are to be made till mate if 50 moves- rule is to be regarded too, so I understand well, that you don't want to resign that. If you don't mind 50 moves rule you wouldn't even have to mind DTM, Nh1 wouldn't be necessary, it would be just one more long lined cyclic zugzwang puzzle.

But what you have to give then, if move counting gets the main point as this is so in mate in x- problems, you then have to give as a full solution- line the exact DTM without crossing 50 moves.

And this one point you and nobody else will manage to solve without engines' support, will you?

That's the difference between "understanding" and "planning" and playing chess, (about choss, I'm not quite sure, if I got you right, you're playing it with rules of you're very own only) humans are great in understanding and planning, as for playing lines to 50 moves draw or mate shortly before draw, they like to take some help of engines, don't they?

Engine didn't help me from 2...Qg5+ to 90.g8N at all.
Have you considered a question, how can Ottó Bláthy find the exact DTMs of his puzzles 140 years ago?
And this puzzle is #106 undoubtedly.

I can tell you why Black King has to get the White Knight at h1 first. When BK is at a1 and BB is at a5, the way for Black to move his King to a8 is:
Ba5-b6-a7-b8, Ka1-b2-c3-b4-a5-b6-a7-a8, 10 moves.
And when BK is at h1 and BB is at b8, the way is:
Kh1-g2-f1-e1-d2-c3-b4-a5-b6-a7-b8, also 10 moves.
There're two normal cycling, when BQ is at h5 and WK is at g8:
Kf8 Qf5+ Ke7 Qg6 Kf8 Qf6+ Kg8
This is called the left cycling.
When BQ is at f6 and WK is at g8:
Kh7 Qf7 Kh8 Qh5+ Kg8
This is called the right cycling. It's shorter than the left cycling.
You can see that in the position after 1...Qxd2 2.Kxe7, 2...Qg5+ 3.Kf7 costs one extra move(0.5 ply) to get into the normal cycling, but when BK is at h1 and BB is at b8, the normal cycling can be gotten into inmediately.
So BK can reach a8 0.5 ply faster when BK is at h1 and BB is at b8.
The most brilliant move in this puzzle is 30...Ke2!!(equivalent to 20...Kb1!! in Spliffjiffer's line).
If BK just goes to h1 then goes to a8, the first cycling after Kxh1 is a left cycling(also after 2...Qg5+). Finally the left cycling will be one more than the right cycling, as we know, that's slow.
So BK must wait for something, yes, if it goes to h1 one cycling later, then the first cycling after Kxh1 is a right cycling, finally the right cycling will be one more than the left cycling. This is quicker.