Ah, the location of the promotion of the white pawn doesn't matter...
It's the promotion of the black pawn that's impossible. Black must have promoted his A pawn to a light squared bishop (since his original LSB couldn't have moved because the B7 and D7 pawns are still at home). Now, this promotion could have occurred at 3 different squares: B1, D1, and F1 (H1 is impossible because of white's unmoved pawns on the right side). Further, it could not have promoted FROM D2 since it would have been attacking the white king from that square (since the king presumably hasn't moved). In this eventuality, white would have had to move his king or capture the pawn, preventing its promotion. So the black pawn had to promote from either C2 or E2, and in doing so must have captured a piece on the promotion square.
To promote at any of these squares, the A pawn would have to capture at least 3 of white's pieces. White is missing both bishops, a pawn and (presumably) his King side rook. The rook couldn't have been captured by black's A pawn because it can't have moved past the white king. So the A pawn must have captured a pawn and both bishops. White's LSB must be have been captured on the promotion square as it's the only one of the three pieces which could occupy any of the promotion squares. So, which pawn did the black A pawn capture? It can't have been white's A or B pawns (as shown above, they are on A3 and B2) or the F, G or H pawns.
If it was white's C pawn, then white's D pawn is on B4, and must have captured black pieces on C3 and D4, both dark squares. Black is missing a knight, a queen and both of his original bishops. However, both bishops had to be captured at home (C8 and F8) since they can't have moved past the pawns on the 7th rank, and they can't have been captured by a white pawn for the same reason. So the black queen and knight were captured by the white D pawn, the white c pawn was captured by the black A pawn, and the black bishops were captured by non-pawns and the white e pawn must have been promoted to a white rook... where? Either via A7, F7 or H7, all of which would have required the E pawn to capture more material. But then black would have less pieces.
Ok, so the white C pawn must be at B4, and it must have been the white D pawn that got captured by the black A pawn. But how? The A pawn can get to the B file by capturing white's DSB, but it has to capture white's D pawn on the C file, so the D pawn must capture something on the C file as well. It can't be either of the bishops. So it was either a knight or the queen. Meanwhile, the white C pawn captured the other black piece (knight or queen) on either B3 or B4. And that leaves the white E pawn to promote.. but again how? It has to move off it's file by capturing another black piece, but there aren't any left.
EDIT: Oh, black's A pawn can't have captured white's E pawn for similar reasons
So, white can't castle because it's impossible that both sides have promoted a pawn.
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[Last edited by stigant at 12-22-2008 08:12 PM]
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Excellent answer!