f2-f3 is not what took me two weeks to find, but still it is a very hard one. Here is what I have so far. A real hint would be to correct me where I am wrong:
- the black h-pawn promoted h2xNg1B and then moved as bishop to h2. This took 6 moves and was finished after white's king moved to h1 via a long walk (d3 and in front of the pawns)
- the black a-rook is now on d5 via a5, and before c5 was played and after the a-pawn moved two times
- the black h-rook sacrificed in two moves Rg8 - Rg3 and was captured hxg3
- the knight on d8 came from b8 in 2 moves (via c6)
- the other knight made 3 moves Nh6 (or Nf6?) - Ng4 - Ne5
- the bishop on c8 made two moves Bh6 - Be3 and was captured dxBe3
- the black queen made three moves (probably a5-c3-b2 or c7-e5-b2 or even b6-d4-b2)
- the king made 5 moves, could be via f8-g7-g6-f5-e4 or even via d8-c7-d6(or c6) - d5(or e5)-e4
- the black g-pawn captured the white bishop gxBf6
- the above defines black's 29 moves, but not the order in which they were made
- white's knight on g1 comes from b1
- not sure which rook is on a3 and on a1. White has made 5 'spare' moves, i.e. moves that could be avoided if there were no ennemy obstacles (other than the ones to capture on g3 and e3). That could have been extra rook moves to circumvent either the c4-pawn or the black a5-pawn or both. Possibly the white queen made one (or more) extra moves, or also the knight from b1.
- majour issue: after the capture on f6, and one (or more) moves of the black kingsknight, the white king cannot reach the g-file anymore, unless hxg3 has not happened yet. That makes the sequence of the moves extremely difficult.
I read your analysis of the Black moves and could find nothing wrong with it (which is not to say that there isn't).
For White, we can account for either 4 or 5 pawn moves (including 2 captures), and a minimum of 3 rook moves, 3 knight moves, 2 bishop moves, and 1 queen move.
Total = 13 or 14 moves
This leaves either 16 or 17 White moves to play with.
And the White king must get to h1 via h3.
I agree. There is that problem I mentioned about the white king accessing the g-file. On the other hand, since the white king has to make a long journey, the d-file should be opened early in the game. Possible only after Bf8-Bh6-Be3, which in turn is possible (only?) after b3-Bb2-Bf6. My best guess is that these are white's three first moves. Since I only see one blavck move on the queenside which can be made just like that (a7-a5), black must have made at least one knight move in the first two moves (Nh6 - probably, or Nf6).
But that puts a hold on the rook maneuver (Rg8 - Rg3) because of the initial issue. So, what other moves (in sufficient quantity) can black make early in the game?
Since none of us has yet found the solution perhaps we should start with the reasonable assumption that White first three moves are indeed 1.b3 2.Bb2 3.Bf6 and then start discarding the possible answers of Black.
E.g Black could not have moved the h pawn.
Black could not have moved the a pawn to a4, etc.
OK. In the first two moves, black must play at least one knight move (as you said a5 is possible, but not a4). But since the bishop is needed at h6 to sacrifice itself on e3, also as soon as possible, I suggest to try the following: 1.b3 Nh6 2.Bb2 Ng4 3.Bf6 gxf6.
The knight could go via f6 to g4 as well, but it had to be Nh6 if the other of the first two moves was a5 and not Ng4.
The problem now is that white has not many moves until the d-file is open. The only 'final' move would be c4, which has to be considered. But that means that the rook (whichever one) may have to make a detour to get to a3. And in any case, the next move would be a tempo consumption move, because the fastest way to open the d-file would be: 1.b3 Nh6 2.Bb2 Ng4 3.Bf6 gxf6 4 ... Bh6 5. ... Be3 6.dxe3.
So, what are the 'best' 4th and 5th move for white? Is it with c4 or something else. One idea I tried was Qc1 - Qb2 and after the d-file opened, Nd2-Nf3 and bringing the a-rook to d1 aiming at a3. But so far not with success.
Originally posted by BigDoggProblem After this, how will the white King access the g-file?
The only possibility for that is that the white king goes to the g-file before the h-pawn has taken the rook on g3. Probelm with that is that the king would still have to make room for the rook to leave h1 and move away from the h-file. It is because of that that I started to look at the possibility of the a1-rook going to a3 via the d-file. Because black has not enough moves on the queenside before his a-pawn moves to a4.
But all this sounds very doubtful in my own ears 😕