Missing white peices:
Good work everyone, especially TheGrobe. Like often happens, I wasn't thinking of 0-0-0 so much, mainly because of the bishop's position. There was one ponit where it had crossed my mind, but it fell out because I thought of something else that looked more promising, and I didn't remember it afterwards. Furthermore, my reasoning that both bishops must've been promoted was of course wrong, I just didn't see it for some strange strange reason. Also, I knew that such positions are always possible, just improbable (so long as there haven't been any mistakes!), so I knew that my reasoning was wrong, I just didn't know where.
Missing white peices:
The queen-side Knight must have been captured by the Black pawn that promoted to a Bishop. The queen-side (black-squared) bishop must have been captured by a Black Knight. The Rook, the Queen, the e-Pawn and the king-side knight all seem plausible at the moment.I don't agree about the queen-side Knight. It could have moved out of the way, then the queen could have moved to b2 to be captured. Not that it really matters...
I'll be working on solving this mystery now...
Yes -- I realized the mistake and deleted the post, evidently not before you saw it.
Aha!
The black knight can't be a promoted pawn, because it would have taken two captures to arrive at it's square, meaning there should be seven missing White peices, but there are only six.
This means that it must have arrived there from e3, which also means that White's O-O-O must have hapened earlier than just the last move. Since there are no other legal previous moves for White based on the pieces on the board, there must be a missing White peice -- the piece on g4.
Now what is it?
There's a small error with this, too. The knight could have come from h2, either before the bishop made it there or whilst the bishop was on g1. This means that the last move COULD have been 0-0-0, which means that the missing piece COULD be black....
It can't have moved out of the way -- something was captured on b1, and it wasn't the queen-side Rook (accounted for) or Bishop (wrong colour). If the queen-side knight moved out of the way, the king-side knight must have moved in to be captured on b1, and the knights are, after all, pretty much interchangeable.
I'm leaning towards it being a White pawn on g4, but I can't prove it -- there were four captures that occurred on the queen-side for Black to obtain his promotion, and although the e-Pawn could have captured to the Queen-side to be involved, I'm thinking that there must be some reason that this is not the case.
Yes, SOMETHING must have been captured on b1, but you missed my suggestion; it could have been the white queen.
Yeah -- I saw that after I hit submit and immediately deleted the post due to the error. Not fast enough, though, evidently.
Aha!
The black knight can't be a promoted pawn, because it would have taken two captures to arrive at it's square, meaning there should be seven missing White peices, but there are only six.
This means that it must have arrived there from e3, which also means that White's O-O-O must have hapened earlier than just the last move. Since there are no other legal previous moves for White based on the pieces on the board, there must be a missing White peice -- the piece on g4.
Now what is it?
There's a small error with this, too. The knight could have come from h2, either before the bishop made it there or whilst the bishop was on g1. This means that the last move COULD have been 0-0-0, which means that the missing piece COULD be black....
Well it can't have been before the Bishop got there, because again it would have had to have been before the g2-g3 pawn push which has the same problem of not allowing the White bishop to get to h1.
The Black Bishop entering first and then sitting on g1 is an interesting possibility though. Hmmmm...
I'm still convinced the peice is White, though -- there must be a reason that the Knight can't have come from h2.
Aha!
The black knight can't be a promoted pawn, because it would have taken two captures to arrive at it's square, meaning there should be seven missing White peices, but there are only six.
This means that it must have arrived there from e3, which also means that White's O-O-O must have hapened earlier than just the last move. Since there are no other legal previous moves for White based on the pieces on the board, there must be a missing White peice -- the piece on g4.
Now what is it?
There's a small error with this, too. The knight could have come from h2, either before the bishop made it there or whilst the bishop was on g1. This means that the last move COULD have been 0-0-0, which means that the missing piece COULD be black....
Well it can't have been before the Bishop got there, because again it would have had to have been before the g2-g3 pawn push which has the same problem of not allowing the White bishop to get to h1.
The Black Bishop entering first and then sitting on g1 is an interesting possibility though. Hmmmm...
It could have been there first if the white bishop had already moved out across e2, then was later played to the a8-h1 diag and then to h1, with g3 coming at some point after the bishop comes out and before its final move.
Yes, I suppose that's so.... This is an infuriating problem.
Only 5 white pieces and 6th pawn/promoted piece missing = total 6 white pieces missing,
5 captures made by B currently at a2
N could not have been a promoted piece (white's K was at e1 last move) and at f1 must have captured something there or at e3. total 6 captures accounted for so far.
B currently at h2 cannot have been promoted as there was no other white piece captured.
6 white pieces/pawns have been captured = 6 missing.
white's last move must have been 0-0-0 (cannot be K move, otherwise it moved from check of an immobile B)
Did white promote something? If he did, it was captured.
therefore the piece at g4 is black, NOT B or Q (then white's last move 0-0-0 impossible). It cannot be a black R as already 2 rooks are accounted for and no other promotion occured (and no way of promoted R to go out). So it
must be a black N or pawn. It cannot be a white piece, as all has been accounted for.
So black can castle if his K, Rs have not moved (not possible to ascertain) and white did not promote a pawn (highly improbable).
N could not have been a promoted piece (white's K was at e1 last move) and at f1 must have captured something there or at e3. total 6 captures accounted for so far.
1st of all, how do you know that white's K was at e1 last move. Furthermore, why did the knight have to capture something on f1 or e3?
Only 5 white pieces and 6th pawn/promoted piece missing = total 6 white pieces missing,
5 captures made by B currently at a2
N could not have been a promoted piece (white's K was at e1 last move) and at f1 must have captured something there or at e3. total 6 captures accounted for so far.
B currently at h2 cannot have been promoted as there was no other white piece captured.
6 white pieces/pawns have been captured = 6 missing.
white's last move must have been 0-0-0 (cannot be K move, otherwise it moved from check of an immobile B)
Did white promote something? If he did, it was captured.
therefore the piece at g4 is black, NOT B or Q (then white's last move 0-0-0 impossible). It cannot be a black R as already 2 rooks are accounted for and no other promotion occured (and no way of promoted R to go out). So it
must be a black N or pawn. It cannot be a white piece, as all has been accounted for.
So black can castle if his K, Rs have not moved (not possible to ascertain) and white did not promote a pawn (highly improbable).
I don't know that this reasoning holds -- how do we know the White King was at e1 last move? If there was another white piece out there Whit's last move could have been made by that piece. Also, how do we know that there was a capture by the Knight, either on e3 or f1?
I don't know that we can prove that there was a sixth white piece captured.
Yeah -- what WanderingWinder said.
The only way the black white-square bishop could be in that position is if it is on h7?
so g4 would be... the white queen? and of course black couldn't castle.
No, there is another explanation for the Bishop on a2 that has been explored in depth in some of the previous posts.
N could not have been a promoted piece (white's K was at e1 last move) and at f1 must have captured something there or at e3. total 6 captures accounted for so far.
1st of all, how do you know that white's K was at e1 last move. Furthermore, why did the knight have to capture something on f1 or e3?
white's last move was either 0-0-0 or by the missing piece at g4 - K move from check of immobile B not possible. Now if there is a missing white piece at g4, then neither of the N, B(h2), Q has captured anything - which means black B went to h2 first, then white played g3, then the white B somehow managed to enter to h1, and the black Q followed the white bishop all the time exposed to its attack to g2 without capturing anything - does not make sense, does it? So there is no missing white piece at g4, so white's last move must have been 0-0-0, so white's king must have been at e1 last move. If that was so, how can black's supposed e-pawn promotion have taken place - exf1=Q#
If the N have not captured anything, then is there any valid logic it went to e3 in front of 2 pawns (it had not given check there), then to f1?
Hard to answer, since Black has a white-squared bishop which could not reach the position it is in through any sequence of legitimate moves . . .
I disagree. Blacks e-pawn could have captured its way over there, and captured a piece on b2 while promoting to a white-squared bishop. So there!
Can't solve the puzzle though.
You're right - I thought of the same thing on the way to the movie. Good catch - 
Aha!
The black knight can't be a promoted pawn, because it would have taken two captures to arrive at it's square, meaning there should be seven missing White peices, but there are only six.
This means that it must have arrived there from e3, which also means that White's O-O-O must have hapened earlier than just the last move. Since there are no other legal previous moves for White based on the pieces on the board, there must be a missing White peice -- the piece on g4.
Now what is it?