Well we can be certain that it's not a promoted pawn....
Deductive Puzzle #10
Two missing white peices (one Knight, one pawn -- or white Bishop if the pawn promoted to the one on e2) and Black has captured on b6 and h6.
Black is only missing two peices that could have been used to change the pawn's rank (Black's e-file pawn could not have), so either the pawn captured over to the g file, or promoted through the e-file capturing on f2 -- if it did so, it could only have promoted to a Knigh or Bishop as otherwise the King would have had to have moved.
Black's light squared bishop was captured on its starting square, necessarily by a Knight from either a7 or b6 (d6 would have caused the Black King to move).
I think if it can be proven that the pawn didn't promote on f8, but was rather captured on g6 then it's clear that it's the King side rook that remains on the board, as the Queen-side rook would have been required as fodder to get White's e-pawn to the g file prior to the h-file opening up for the King side rook.
If, on the other hand, White's pawn did promote on f8, it could only have done so by capturing Black's King-side rook on that square (as it's clear that Black's dark squared Bishop is not a promoted pawn), so the remaining Rook would be the Queen side rook.
Becuase the Bishop was captured in place, and the second rook couldn't have gotten out there's no way the pawn captured it's way over the the g file.
This means that out of necessity the pawn promoted on f8 (likley to a Knight) by capturing Black's king-side rook on that square and that the remaining rook is Black's Queen side rook.
The question before us is which White pieces were captured on b6 and g6 and when. Looking at White's pieces, there are only two that are missing. One White Knight (the other was offered as odds) and one White "pawn." I will investigate later if the pawn could have promoted. For now, we note the number of missing Black pieces. One Rook, one "pawn," and one light-squared Bishop, for a total of three. As noted by TheGrobe, the Bishop was captured by a Knight on its home square. Thus, only two pieces could have been captured by the pawn. Clearly, then, the pawn could not have made it to b6. Thus, it was the pawn that was captured on g6, either as a pawn or as a promoted piece. If the pawn was not promoted, then it made two captures to get to g6. These would have to be the e-pawn, which would have had to capture to the f-file, and a Rook. However, since the e-pawn would have to have captured a piece to make it to the f-file, this leaves no piece left to be captured on b6. Thus, the White pawn must have promoted.
In order to promote, White's pawn must have captured onto f8 and become either a Bishop or a Knight (otherwise, the Black King would have had to have moved). This is easy enough to achieve by having Black's e-pawn captured by a piece sufficiently far away from the Black Royalty to avoid needing to move them. This would allow the White pawn an unobstructed path down the e-file. Since there exists no more Black pawns, and since White is not missing enough pieces for the pawn to have captured off the e-file as noted before, this means all of Black's remaining pieces are the original ones. This means that the King-side Rook must have been captured by the White pawn. Thus, it is the Queen-side Rook that remains on the board.
Sorry for the delay in getting up the next puzzle, but I was away for the weekend. Anyways, here it is!
White has given Black odds of a knight, and none of the royalty has yet moved. Does the Black castle in play belong to the King or the Queen?