Deductive Puzzle #15

Sorry for the sporadic posting schedule I'm doing these on, just been a hectic week unfortunately.

Ok, so a White pawn has promoted. Know that a)Black's Queen was captured on her own row, White Queen's rook captured on its own file, no king has moved, and black can castle. Where is the promoted White pawn? 

Black - missing bishop, queen

white - missing piece that the pawn promoted to, knight, rook

White's h-pawn promoted meaning black g and h pawns need 1 capture each. doubled black a-pawns account for other missing white piece. Promoted to knight/white squared bishop. Doubled white a-pawns and piece captured by promoted pawn to get to g-file account for missing black pieces.

Edit: looks like most plausible order of events would be something along this line:

Black: h7xg6

White: h7x(Black queen)g8=N

White: Ng8-h6-g4

Black: g7xh6

I assume the promoted piece is still in game because we have to find it. I don't know how to prove this, but it looks likely. Nor do I know how the other pieces fit in yet.

Let’s start with investigating how White’s pawn promoted:

 

• It can’t have captured it’s way over to the hole on the b-0file as there aren’t enough missing black pieces to support this.
• Likewise, it can’t have gone around the g pawn via the f file for the same reason
• It can’t have captured from g6 to h7 because this would require two black captures off the 8th rank and we know the Queen was captured there.
• It can’t have passed by the Black pawns via the h-file while the g and h pawns were doubled on the g file because this would require two captures of White pieces on g6 and h6 and we know that one was captured on a6, and the Rook was captured on the a file so there are not enough missing white pieces that are candidates to support this.
• The only other option is that the white pawn captured from h6 to g7 after black had played g6.  Because we know the Queen was captured on the 8th rank, the only piece that could have been captured here is the dark squared Bishop, and because Black can castle, we can conclude that the pawn promoted either on f8 (by capturing the Queen) or g8 without a capture.

Hmm, something's wrong though, because the Black bishop was clearly captured on a3....

Ahh, I counted the Rook and the capture on a6 as two captures -- it must have been one, so the g and h pawn-swap scenario is in fact the correct one.

White's pawn promoted on g8 by capturing the Queen as RyanMK suggested after being allowed through on the h-file.

And it must have promoted to a knight, as a bishop would be unable to get out. The promoted piece is on f3.

So we know that if White promoted to a Bishop or a Rook on g8, it was captured before it escaped (on g8 or f8 for the rook, or alternatively g8 or h7 for the Bishop, but this can't be the case because of the three missing White peices we know that White's Rook was captured on a6, and the other two on g6 and h6.

This means that White's pawn promoted to either a Queen or a Knight and that it either escaped to replace its counterpart (which was captured on g6 or h6) or that it itself was captured on h6.

I'm pretty sure it's the queen. As RyanMK stated, hxg6, pawn to h7 seems to be the only way to get the pawn promoted, as the black bishop can't be sacrificed since it needs to be captured at a3.

The problem is how to get the Queen to g8. The queen can escape through the b7 hole and f6-g7-g8. This implies that the pawn structure must be fixed before promotion. This leaves us with the piece either being the white Queen or the white knight. Now the piece must be the queen since the only escape square for the Knight is f6, which would be check, ruining castling or the knight still being on the board. So white's promoted piece is the Queen.

A Knight or a Queen -- it could also have been subsequently captured on h6 as opposed to finding its way back to d1 or f3.  We do know, however, that White's Bishops and rook are the originals.

Since we are supposed to find the piece, I assume it's on the board. It would be unable to promote to a queen without giving check making black unable to castle. Therefore it must have promoted to the knight that's now on f3. And the knight could escape through h6 as well as f6. h6 would not give check.

Why can't the Queen have come in via h7-g8 while the pawns were doubled on the g-file the same way the pawn eventially did?  Both seem to be possible.

Here is an example game:

I hate to say it, but I really dislike the sample game approach -- it turns it from deductive reasoning to demonstrative reasoning.  It doesn't tell is why it must be so, just that it must be so.

So why can't the Queen have come in via the h-file?

As far as why the Queen couldn't have come in while the pawns where doubled, this is because the queen couldn't have escaped. Based on the number of white pieces, there are 3 missing white pieces. Also there are three necessary black captures. So for black to have got his queen in while the pawns were doubled, the capture on a6 would have already happened. This implies that b-file was opened (since the white king can't move and the rook must be captured on the a-file). If the b-file was opened, the black bishop must have escaped which it can't do with the pawn still on g7. 

Sequence of events

1. White piece captured on g6.

2. White h-pawn arrives on h7.

3. White piece captured on h6.

4. Black Bishop captured on a3.

5. White Queen-Rook captured on a6.

6. Black Queen snakes it's way around to g8.

7. White pawn captures Black's Queen and promotes.

8. White's newly promoted piece escapes.

Some notes:

1. Both captures on g6 and h6 were made before the White Queen, Queen-Rook, and Queen Bishop could escape.

2. Pawn structure at time of promotion indicates White did not promote to a Knight, otherwise it would have checked the Black King when it escaped via f6.

3. Pawn structure also indicates it could not have promoted to a Rook because it would be unable to escape.

4. Pawn structure also indicates that it could not have been promoted to a light-squared Bishop.  Again, there is no escape route.

5. White dark-squared Bishop could not have escaped prior to the capture on h6, thus it was not captured there.  Thus, the one on the board is an original piece.

This leaves only the Queen left as an option.  This is viable if a Black Knight or White Bishop was on f8 at the time of promotion.  Because both Queen and Knight can move on either color, either one could have been captured on g6 and either could have been captured on h6.

Ah-ha -- because in order to do so, the Rook capture first had to have occurred on a6, which means that the Bishop capture had to occur before that on a3 as the White Queen side Rook couldn't have escaped otherwise (this is where the info about neither King having moved is key)  but in order for the Bishop to have escaped, the pawns had to have been first undoubled, so the Queen couldn't have gotten out while the pawns where still doubled on the g-file and must have entered via the a1-h8 diagonal.

And we have our retrograde solution.

 Thanks, that must be it! good job!

Evidently you type faster than me.  Nice work guys.

Same here.  I started typing mine between TheGrobe's comments at 6:20 and 6:22.

Just one final note:  The knowledge that the pawn structure was fixed before the promotion means that it was in fact the original Queen that was captured on h6 or g6 and not the promoted pawn.  The h-file scenario had left open the possibility that it could have been either (with h6 being the capture square if it were the promoted pawn that was captured).