An original retro problem #23

I'm putting these in a new thread for easier tracking in the future.

Original:

Legal or illegal?

Variation:

Legal or illegal?

Solved!

For the variation, legal?? Nd1xNb2? (Knight came from d1 taking on a knight on b2.) I'm kinda new to these retro problems.

I'm not used to this kind of puzzles, but here is my try for the first one "Original":

White has lost its queen and the lsb.
Black has lost the dsb and 2 pawns from a and h file.

All captures done by white has to be done by pawns since there are three doubled pawns.
The pawn on e4 has to come from the d file. It's not possible that it came forn g2 since there is still a pawn on f3 (and f2). e4 cannot come from c2 because then also the d pawn woould have captured dxc3/dxc4. That would be too much captures.
It's not possible that e4 pawn had taken a pawn (that would have been too many captures by black's pawn), so there was the move d2xBe3.
So, f3 pawn must have done gxf3, taking a pawn.
So, black's h pawn captured twice to get to f3.
So, white b3 or b4 pawn must have taken the black's a pawn. But there are no further possible captures for black to get its a pawn to the b file.

--> It's not a legal position.

Some good analyse, however, this is wrong. gxf3 could also take a piece! (and don't forget, pawns can promote).

I think enough time has passed that I'll post my analysis. I think this position is illegal.

The first position is illegal.

To prove this, first some observations:

(1) Black is missing 3 men, which were captured by White (a x b, d x e, g2 x f3).

(2) White is missing 2 pieces (because all 8 pawns are present): wQ, wB.

(3) White is in check; this check must have been administered via bQd? x w?c2.

(4) There is one or two retrocage(s); the "Southwest" consists of most of the men in the rectangle whose corners are a1 and f4; the "Northwest" consists of the men in the rectangle whose corners are a6 and c8. The only pieces which can retract moves are the wQ, and the wPf3. The wPf3 can't "uncapture" until the wB is on f1; wPb2 - b3 is an "uncheck" which is illegal.

Now for some deductions:

(5) All three white pawns captured pieces (possibly promoted pawns). The missing Black pawns came from a7 and h7. The wPf3 and wPe4 couldn't have captured a bP, because at least 2 captures were required to get the bP to that file; this is too many. (The bQ had to capture on c2.) Now, if the wPb4 captured a bP on the b-file, then that bP would have had to capture a White man. But since the bP on h7 needs a capture to get off that file, Black has made at least 3 captures; again, this can't be. Thus, all three wPs captured Black pieces.

(6) Both Black pawns on a7 and h7 promoted. The bPh7 captured a White piece on the g-file and marched directly to g1; the bPa7 marched directly to a1.

(7) This balances out the White pieces; thus, the Black queen can't uncapture any pieces, when moves are being retracted.

The threat of a retro-stalemate is imminent; the wPh5 is the only one which can retract moves, and there are only three left. Thus, Black has to move its queen and open up a retrocage.

At this point, the southwest retrocage must remain static; the northwest retrocage must be opened first, as soon as possible. Thus, the previous moves were:

Black: wQd1 x w?c2 +

White: wPh4 - h5

Black: wQg1 - d1

White: wPh3 - h4

Black: wQg5 - g1

White: wPh2 - h3

Black: wQa5 - g5

Retro-stalemate. Thus, the position is illegal.

I am surprised that Shoopi didn't include the variant where the wP was on h6; in this case, it appears that the position is legal. (At least, if I've carried the analysis far enough ... I never know how far is far enough for some retrograde analysis problems!)

In that case, the bQ would have had to capture a wQ on c2; otherwise, no wB could get to f1. Then, after a sequence of moves similar to the above, 

Black: Nc5 - a6

White: Ra6 - a7 

Black: Nd7 - c5

White: Rb6 - a6

and White can now retract the wR to h1. Then one of the Black pieces can retract back to g1, unpromote, and then uncapture a White bishop at g2. (The wBb8 can shuttle between a7 and b8, to prevent a retro-stalemate.) White can retract the bishop back to f1, uncapture the Black piece at f3, and open the southwest retrocage. Hopefully, without further complications.

Your analyse is mostly correct, you're just not considering every possibility.

What if the black queen is in fact promoted?

Good point. (In an earlier try, I did account for this; I assumed that the bQ was promoted and ran into a wall somewhere. I also wrote wQ when I meant bQ.) Let's see:

Black: bQd1 x wQc2 +

White: wPh4 - h5

Black: bQg1 - d1

White: wPh3 - h4

Black: bPg2 - g1(bQ)

White: wPh2 - h3

Black: bPh3 x wBg2

Just in time! (As intended, I bet.) Now the wB is the one that has to block the a-file, so that the bN can move. Note that the bPe6 can't retract its move until the bB is back on f8. Thus, the wB has to get to a4 via e8, h5, g4, and h3.

White: wBf1 - g2

Black: bPh4 - h3

White: wBh3 - f1

Black: bPh5 - h4

White: wBg4 - h3

Black: bPh6 - h5

White: wBh5 - g4

Black: bPh7 - h6

White: wBe8 - h5

And Black is almost in retro-stalemate. The bPd6 can't retract its move until the bB returns to c8. Thus, if the position is legal, the next (or previous) move has to be

Black: bPf7 - f6

White: wBa4 - e8

Black: bNa6 - c5

Once again, just in the nick of time.

Now, the wR can get back to h1 easily enough; however, the route for the wB to get back to f1 has evidently been cut off, so the position might be illegal in this case as well.

Of course, if there's a resource I can't see here, this problem could go on for another stage or two. (What was captured on f3?) It would be nice to see how you came up with this problem in the first place!

((Later, after I got up to get some aspirin for a headache:))

I just realized that I forgot about unchecking. The last three moves listed above could be replaced with:

Black: bNa6 - c5

White: wBa4 - e8 +

and now, after retracting the wR to h1, the wB can be retracted to f1 the way it "came in". This ends Phase 1; the next phase starts with the question: What piece was captured on f3?

It appears that at this point, the position can be released, and it doesn't matter what piece was captured at f3. (Of course, it can't be a bB, because a1 isn't a white-colored square.)

Yes, I agree.

If you would like, you could try and solve the variation as well. It has a similar structure, perhaps slightly harder though.

This one was indeed more complicated. I've also had a busy week with teaching, so I wasn't able to post this earlier.

The position appears to be illegal (barring any missing analysis, of course). Once again, there are a few observations to make first:

(1) White is only missing his black-squared bishop.

(2) Black is missing six pieces.

(3) There is a huge retrocage, which consists of every piece except for a WN and the two BPs on the sixth rank.

(4) Black's e-, f-, and g-pawns are behind White's pawns.

The fact that White's eight pawns are all on the board means we can determine how many captures the pawns made. This number must be odd, so is 1, 3, or 5. One capture was made by WPa?xb? (from the a-file to the b-file). It seems impossible for the WPs now on e4, f3, and g3 to have made two captures and let in the BPs on e2, f2, and g2, so the WPs have made 5 captures.

The only Black pawns which might have captured are the ones on c6 and g2; otherwise, there would be double pawns on some file, or one of the doubled pawns would have been captured, leaving a "gap". The BPc6 did not capture, because c6 is a white square. The BPg2 might have captured the bishop, but it did not make the capture BPh3xg2.

Thus, Black is two moves away from retro-stalemate. White has to uncapture a Black piece and allow more moves, and the retrocage needs to be opened/released.

Uncapturing a Black piece which can then move is easy to do. White retracts WNa4-b2 then WN??xB?a4 (leaving a piece on the a-file); or WNa4-b2 followed by WNb6-a4 and WN??xB?b6. Thus, the WN captured a piece on the a-file or the b-file.

The other task is opening up the retrocage. This stumped me until I remembered how "unchecking" works:

White: WBd1-c2

Black: BRc2-a2 +

White: WPa2xB?b3 +

This is the easy part 8-). The next step would logically be to determine which pieces were uncaptured. There doesn't seem to be any information that suggests that a particular piece was captured, so I made a few "tries", which didn't really help.

However, then I realized that I could prove the position was illegal without this information. The idea is to show that a BP made a capture on the queenside, and a BP made a capture on the kingside.

The kingside is easy: Consider the h-pawn, which is not there. Thus, it was either captured or it promoted. If it promoted, then it had to have made at least one capture to get around the WPh2. If it did not promote, it was captured on e4, f3, or g3; but in each case, the BP had to make at least one capture.

Now for the queenside: The WN did not capture both the a- and b-pawns, so let X be the pawn it didn't capture. If it promoted, it had to get past the WP on the same file, which requires a capture. If it was captured, it was captured on b3 after the WP moved to b4. Once again, at least one capture was required (one if, X is the a-pawn; two, if X is the b-pawn).

Thus, Black had to have captured at least two White pieces, which contradicts (1).

First, I'm impressed by the detailed reasoning. Very good analysis in there. However, there are still some oversights. Trying not to give away too much, the proof of illegality is unsound - it implicitly assumes something about the unlocking.