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2 is NOT.
Not sure about 1.
Both are illegal because in both the king is in check (black).
My inuition tells me both are legal. But my retrograde problem skills stink.
We start by noting the differences between the two positions: one has a pawn on g3, whereas the other has a pawn on h3. This implies that, in the first position, an extra capture on a dark square has occurred.
We also note that black is currently in a double check. The only way this check could have occurred is via an en passant capture axb6. Before this capture, black was only missing a pawn, a rook, and a dark-squared bishop. So, the question before us is 'can we determine that all of these pieces were removed from the board somewhere other than g3?'
Considering that the b7 pawn had not moved until black's last move and that the d7 pawn has yet to move, this means that the black bishop on h7 is the result of a promotion. Also, since the g2 pawn has not moved, this promotion occurred somewhere other than h1 (the resulting light-squared bishop couldn't have gotten out). Two possibilities are that the promotion occurred on f1 with the h-pawn or that it occured on b1/d1 with the c-pawn. In each of these cases, at least 4 captures must be made by black pawns to account for the pawn structure (either 3 for the h-pawn to get to the e-file plus one for the c-pawn, or 2 for the h-pawn to get to the f-file and 2 for the c-pawn to get to the e-file). White seems to be missing plenty of pieces to account for these captures, so is there any other way to restrict our options going forward?
I know note that, in order to create the bishop check (which prompted black's response of b5 leading to the en passant), we must find some way for white to uncover the bishop's line of sight. The only way I can see to do that is via another en passant double check, only this time by black.
1. ... Black move resulting in check by bishop on h7.
(To account for this move, all we need is Nf5-d6 to block the check from the rook, which implies the white rook on e6 captured a piece. Since it wasn't the pawn (it promoted to a bishop) or the original light-squared bishop (it was captured on its home square) or the dark-squared bishop (e6 isn't a dark square), it must be the rook.)
2. e4 dxe3 ep
3. Kd3-c3+ b5
4. axb6+ ep
What this means is that the black pawn on e3 came from the c-file, and the pawn which promoted to the light-squared bishop came from h7. However, with the e-pawn still in place, the promotion could not have occurred on f1. This means the h-pawn must have made at least 6 captures to promote on d1/b1. Adding the 2 captures needed for the c-pawn gives a total of 8 captures, 6 of which must be on light squares.
White is missing: 4 pawns, 2 knights, the queen, and the dark-squared bishop. This is exactly the number of captures needed by black. One of the pawns (the e-pawn) was captured on a dark square. The dark-squared bishop was also captured on a dark-square. For the b-pawn to account for one of the 8 captures, it must have promoted (possibly to replace one of the rooks and is currently on the board).
So, summing up what I currently know:
Black is missing:
original light-squared bishop (captured on home square by a knight)
b-pawn (was just captured via en passant)
h-pawn (was promoted to light-squared bishop currently on board)
rook (was captured on e6 by rook to create a check which was then blocked by the knight currently on d6)
dark-squared bishop (no known capture location at this time)
white is missing:
b-pawn (was promoted either to rook on board or to a piece which was captured on a light square)
c-pawn (was captured on a light square or was promoted to a rook on board or promoted to a piece captured on a light square)
e-pawn (was captured via en passant on e3)
f-pawn (similar to c-pawn)
Two knights (captured on light squares)
Queen (captured on light square)
dark-squared bishop (captured on dark square by c-pawn)
Now, for the white b-pawn to promote, it must have made at least one capture (since the black b-pawn was in the way at the time). The only piece this could be is black's dark-squared bishop. This means that there is no longer any possible piece that white could capture to account for the pawn on g3.
Therefore, the first position is illegal.
The last few moves must have been:
1. e2e4 d4xe3+ 2. Kd3c3+ b7b5 3. a5xb6+
Black had just given discovered check by moving his knight from e4 or f5 to d6. White's previous move had been Rxe6+, moving the Rook from f6 or g6 to capture a black rook.
This tells us that the Black bishop on h7 must have been promoted from the h-pawn on b1 or d1.
That requires 6 white-squared captures. The h pawn captured the queen, two knights, c and f pawns and one other piece.The other piece is provided by promoting White's b pawn on c8. (Either directly or as a replacement for one of White's original rooks.)
Position 2 is illegal because White needs a second promotion to replace the trapped rook on h1, but none is available,
Ok, so combining my analysis with that of caveatcanis, neither position is legal.
Nice analysis guys. That's not quite what I wanted. I am working on correcting it.
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