hi David,

...

I send you my analyzis of the solution.

this is my analisis of the position:

in the given position few black pieces can move.
if Nh3 moves white mates in 1 move: Nf2#
if Bg2 moves, white mates in 2 -  1. R:f1  Kg2, 2. Ne3#
so black can only move with his king side pawns, and with Ba7.
note that after black captures g3 with his pawn (and later, when he moves
other pawns to g3) he can move Nf4 next move, and no mate is possible.
thus immidiately after black has a pawn on g3, white must move:
  1. N:g3+ Kg1, 2. Ne2+ Kh1   to remove black's pawn from g3.

this means, that if white kills black's Ba7 and queen side pawns, black
must move with either Nh3 or Bg2, which result in mate in 2. but this is
not so easy for white to kill Ba7. the obvious way will be to move Kb7,
but then black move: B:c6+ and kg2 to release the position. then, black
has enough to win the game. also, if white's king moves to almost every
other white square on the board, black can check him with either Bg2 or
Nh3, and release the position.there are 2 exceptions: c8 (current
position) and a4.
now, white can only kill black's Ba7 on b8 and not on a7. but this can't
happen if all white's moves are king moves on black squares, because then
it takes white an even number of moves to return with his king to c8, and
during that time black moves Ba7-b8-a7-b8-a7, so that when white moves
Kd8-c8 black moves Bb8-a7. also, white can't move either of his knights
because this will let black move Nh3 and release the position, so white
can only move his king.
so, as explained, white can kill Ba7 only if he can move an odd number of
moves with his king, and return to c8. this can be done in 19 moves:
Kd8-e7-f8:g7-f6-e5-d4-c3-b4-a4-a3-b4-c3-d4-e5-f6-e7-d8-c8.
and after black's pawn g7 is gone, white can do it in 17 moves:
Kd8-e7-f6-e5-d4-c3-b4-a4-a3-b4-c3-d4-e5-f6-e7-d8-c8.
after each such sequence, black's bishop is on a7, and can't move to b8,
so black spare a pawn move.
black has 8 pawn moves to spare: h7-h6, h6-h5, h5-h4, h4:g3, g4-g3, g6-g5,
g5-g4, and g4-g3 again. as noted above, after the moves: h4:g3 and g4-g3
white must make the moves N:g3+  Kg1, Ne2+ kh1 so that black won't release
the position.
so the sequence for the mate is:
white's king 19 moves trip (including killing g7, meanwhile black moves
with his Ba7)
19. ... h7-h6
20.-36. white king's 17 moves trip
36. ... h6-h5
37.-53. white king's 17 moves trip
53. ... h5-h4
54.-70. white king's 17 moves trip
70. ... h4:g3
71. N:g3+ Kg1, 72. Ne2+ Kh1
73.-89. white king's 17 moves trip
89. ... g4-g3
90. N:g3+ Kg1, 91. Ne2+ Kh1
92.-108. white king's 17 moves trip
108. ... g6-g5
109.-125. white king's 17 moves trip
125. ... g5-g4
126.-142. white king's 17 moves trip
142. ... g4-g3
143. N:g3+ Kg1, 144. Ne2+ Kh1
145.-161. white king's 17 moves trip
162. ... Ba7-b8
163. K:b8  Bg7 - somewhere,
164. R:f1+  Kg2, and finally 165. Nf3#

black can save his g7 pawn by moving g6-g5, and g7-g6, but
this means that black has pawns on g6 and g7, and white can make shorter
odd moves trips to h7. he can do it only after black moves h7-h6 ohterwise
black moves Nf4+ or Nf2+. now, white has an eleven moves trip:
Kd8-e7-f8-g7-h8-h7-g7-f8-e7-d8-c8.

so, the count is:

1.-19. white king trip, meanwhile black moves g6-g5 and g7-g6
19.  ... h7-h6
20.-30. 11 moves trip, ... h6-h5
31.-41. 11 moves trip, ... h5-h4
42.-52. 11 moves trip, ... h4:g3
53. N:g3+  Kg1, 54. Ne2+  Kh1
55.-71. 17 moves trip, ... g4-g3
72. N:g3+  Kg1, 73. Ne2+  Kh1
74.-90. 17 moves trip, ... g5-g4
91.-107. 17 moves trip, ... g4-g3
108. N:g3+  Kg1, 109. Ne2+  Kh1
110.-126. 17 moves trip, ... g6-g5
127.-143. 17 moves trip, ... g5-g4
144.-160. 17 moves trip, ... g4-g3
161. N:g3+  Kg1, 162. Ne2+  Kh1
163.-179. 17 moves trip
179.  ... Bb8, 180. K:b8  Bg2-somewhere, 181 R:f1+  Kg2,  182. Ne3#

so, these are the whole 182 moves. another fabel's masterpiece.

Dror.