Anti Cheating Puzzle :D

what if h6 Bh2 a3 Kf7?

I saw this idea of having passed f and h pawns, but Kf7 just shuts that idea down completely no?

I think I get it. After 1.h6 Kf7 you do the triangulation thing to deprive black of bishop waiting moves and then black's king needs to go back to the back rank and then white plays g6 hxg6 f6. Yes, I think you solved it!

Not so simple. What if Black plays 1. h6 a3! themselves? Now if your plan somehow involved triangulation, the King no longer has room for it and this is only IF triangulation via Ka2 and Ka1 is relevant in this puzzle.

Hate to rain on the parade, but 1. h6 Bh2 is still an alternative (as is Black advancing a Queenside pawn too) as the Black King isn't forced to budge. So what if White can create one passed pawn? The Black King can still hold it based on its defensive location. Even if White had two connected passed pawns, it isn't enough to win on its own.

No clue why chess.com is glitching on me with the pgn moves, so here is the annotated pgn I put if someone has more luck than me posting:

[Event "?"]
[Site "?"]
[Date "????.??.??"]
[Round "?"]
[White "?"]
[Black "?"]
[Result "*"]
[SetUp "1"]
[FEN "6k1/p1p4p/2p5/5PPP/p5p1/2p2pP1/P1P2P2/5KbN w - - 0 1"]

1. h6 {Just illustrating how the Black King can defend against White passed
pawns.} 1... Kf7 (1... a3 {Starting with h6 also allows ...a3 to ruin any King
triangulation ideas based on the a1 and a2 squares, but ignoring this fact and
just focusing on the White pawn majority, I still don't see the way...}) 2. f6
(2. g6+ Kg8 3. g7 (3. gxh7+ Kxh7 4. f6 Kxh6 5. f7 Kg7 6. f8=Q+ Kxf8 {Nothing.})
3... Kf7 4. f6 Kg8 {Also no visible progress despite White's 2 connected
passers.}) 2... Kg6 {How does White's passed pawn help them make progress $2} *

then we take the bishop and triangulation is not needed as our king will be free and black will run out of pawn moves...

edit: same idea applies to 1..Kf7; black cannot give away the bishop as it's very easy for white then to force the black king away from defending the queening squares (this is also a line that stockfish understands btw, as it immediately gives winning advantage to white indeed after 2.Kxg1)

2.KxB and White thanks Black profusely for saving him about 100 moves.

Chess dot com broke its PGN viewer.

Here is the solution. Can paste this into other PGN viewers to see it.

[Variant "From Position"]
[FEN "6k1/p1p4p/2p5/5PPP/p5p1/2p2pP1/P1P2P2/5KbN w - - 0 1"]

1. h6 Bh2 2. a3 Kf7 3. Ke1 Bg1 4. Kd1 Bh2 5. Kc1 Bg1 6. Kb1 Bh2 7. Ka2 Bg1 8. Ka1 Bh2 9. Kb1 Bg1 10. Kc1 Bh2 11. Kd1 Bg1 12. Ke1 Bh2 13. Kf1 c5 14. Ke1 Bg1 15. Kd1 Bh2 16. Kc1 Bg1 17. Kb1 Bh2 18. Ka2 Bg1 19. Ka1 Bh2 20. Kb1 Bg1 21. Kc1 Bh2 22. Kd1 Bg1 23. Ke1 Bh2 24. Kf1 c4 25. Ke1 Bg1 26. Kd1 Bh2 27. Kc1 Bg1 28. Kb1 Bh2 29. Ka2 Bg1 30. Ka1 Bh2 31. Kb1 Bg1 32. Kc1 Bh2 33. Kd1 Bg1 34. Ke1 Bh2 35. Kf1 c6 36. Ke1 Bg1 37. Kd1 Bh2 38. Kc1 Bg1 39. Kb1 Bh2 40. Ka2 Bg1 41. Ka1 Bh2 42. Kb1 Bg1 43. Kc1 Bh2 44. Kd1 Bg1 45. Ke1 Bh2 46. Kf1 c5 47. Ke1 Bg1 48. Kd1 Bh2 49. Kc1 Bg1 50. Kb1 Bh2 51. Ka2 Bg1 52. Ka1 Bh2 53. Kb1 Bg1 54. Kc1 Bh2 55. Kd1 Bg1 56. Ke1 Bh2 57. Kf1 a6 58. Ke1 Bg1 59. Kd1 Bh2 60. Kc1 Bg1 61. Kb1 Bh2 62. Ka2 Bg1 63. Ka1 Bh2 64. Kb1 Bg1 65. Kc1 Bh2 66. Kd1 Bg1 67. Ke1 Bh2 68. Kf1 a5 69. Ke1 Bg1 70. Kd1 Bh2 71. Kc1 Bg1 72. Kb1 Bh2 73. Ka2 Bg1 74. Ka1 Bh2 75. Kb1 Bg1 76. Kc1 Bh2 77. Kd1 Bg1 78. Ke1 Bh2 79. Kf1 Kf8 80. g6 hxg6 81. f6 Kg8 82. Ke1 Bg1 83. Kd1 Bh2 84. Kc1 Bg1 85. Kb1 Bh2 86. Ka2 Bg1 87. Ka1 Bh2 88. Kb1 Bg1 89. Kc1 Bh2 90. Kd1 Bg1 91. Ke1 Bh2 92. Kf1 g5 93. Ke1 Bg1 94. Kd1 Bh2 95. Kc1 Bg1 96. Kb1 Bh2 97. Ka2 Bg1 98. Ka1 Bh2 99. Kb1 Bg1 100. Kc1 Bh2 101. Kd1 Bg1 102. Ke1 Bh2 103. Kf1 Bg1 104. Kxg1 Kf7 105. h7

@KeSetoKaiba We have already discussed h6 Bh2 it is winning for white. You suggested 1.h6 a3 but that fails to white immediately capturing the bishop 1.h6 a3 2.Kxg1 and black is deprived of bishop waiting moves (which was the whole point of a3 and triangulation, a3 triangulation was necessary to capture the bishop in the 1.h6 Bh2 line). 

If white can immediately capture the bishop then white doesn't need triangulation. On the other hand, if 1.h6 Bh2 there is no way for white to capture the bishop except for the a3 triangulation. The reason why white wants to capture the bishop is because white wants to deprive black of bishop waiting moves and play h6 and then g6, if hxg6 then f6 and when black's king eventually moves from g8 that's when you push & promote either the h pawn or the f pawn. Then Kf7 doesn't help as black will eventually run out of pawn moves and he will have to move his king from f7. 

1.h6 Bh2 2.a3 is the solution to enable triangulation at a2 in the future so we can "catch" the bishop during a waiting move. Notice that after 1.h6, the construct on the top right is winning for white if we have the waiting moves while black doesn't have any. So the only way to capture the black bishop after Bh2 is to enable triangulation in the future. 

Edit: oh, I notice the discussion above has already found the solution after posting. 

Here's the final cycle after all the triangulations and waiting moves (I tried to post the moves but chess.com somehow doesn't show any moves after posting):

We gave solution long ago, but thanks for contributing!

You only gave 24 moves of the solution. This is the full solution.

So, you're welcome.

Since there are many variations to address, here is a step by step guide how to win.

Step 1: Play h6 on the first move.

Step 2: If your opponent doesn't move the bishop then you should take it. If black plays Bh2 then play a3 and win the bishop by triangulating on b1-a2-a1. 

Step 3: If black king isn't on f7 play g6. If black king is on f7 make as much waiting moves as you need before black runs out of waiting moves and moves his king from f7. In that moment play g6.

Step 4: After hxg6 play f6 if black king is not on e7 nor on e5. If black king is on e7 or e5 then push the h pawn to promotion and you won (in the next steps I assume black king was not on e7 nor e5 in this position).

Step 5:  Now you have pawns on f7 and h7. If black king is not on g8 then you can immediately push and promote either the h pawn or the f pawn.  If black king is on g8 make waiting moves until black king moves to f8 or h8. Then you should push and promote either the h pawn or the f pawn.

Step 6: Easy win from this point onwards.

Should add to step #2 that you have to triangulate over and over again - once for every push of a black pawn, until the black pawns have exhausted their moves.

Why would black play a pointless 3. - c4? Play hxg6 and now what?

Also,  I don't understand why not simply 1. Kxg1.

I know my first posts were long and detailed, but 1. Kxg1 leaves the White King unable to infiltrate and make progress.

then f6 just like my variation continues... a bit of thinking is required ofc.

There is no way for the white king to pass through the pawns anyway.

I am 100% convinced that the diagram is wrong and the black bishop should be on h2.