Illegal Position Contest!

I think any starting position with an odd number of knights and white to move is impossible because neither the knights or rooks can lose a move.

It's White's turn to move next.

No, it's Black's turn to move next.

Oh I totally overlooked the knight going to the a8 square and being captured by the rook lol nvm.

A black queen, rook, bishop, or knight.

I agree with n9 - it does not matter what piece was captured on f8. Here is a retraction with bQ as that piece:

Can white mate in 1?

White can mate in 1 only if 1.0-0# is legal.

White's missing Q, R and Black's missing Ph7 and N.

Ph7 must have promoted and sac'ed on e3 or d3, or replaced the piece that did. The bN sac'ed on the 'other' square. Ph7 captured something at g2 to reach promotion. The other white unit was captured on f6.

To unlock/retract the southwest corner, we must get the 2nd wR back on the board and back to b1 or a1, and the Bc1 home. Only then can we retract b2-b3 to let the two light-square B's out. Once they are out, the bB can return home to c8, then retract ...b7-b6, then retract Ba7 home [and also Q, K and QR], then retract ...gxf6 to bring the 2nd white unit back.

This means options are limited for unpromoting on g1. One possible idea is bring a bR or bQ to g1:

We had to uncapture the Q on g2 because the wR could not get out. However, now we cannot get the wR back home. It needs to get home before retracting Bb1 to c8, but Bb1 needs to retract to c8 before wR can be put back on the board [at f6] - impossible.

A wN can't reach f1 unless we retract either the pawn from e3 [which leads to the impossible situation above] or g3. g2-g3 can't retract until Bf1 is home, meaning the N can't get there anyway.

bNa1 can't get there [or even to g1 directly] because c2 is off limits [checks wK, which would have had to move, nullifying castling] and b2-b3 cannot be retracted soon enough.

Bb1 and Bc2 cannot get to f1 because of the difficulties retracting b2-b3 as explained above.

Thus we are out of options for retractions that preserve white's castling right. He cannot mate in 1.

A puzzle from my friend that has stumped me:

White to move:

Neither.

In the top one, black has nine units and White's pawns made seven captures. However, the White h pawn never left its file and neither did black's h pawn. There is no way to include the black h7 pawn in the capturing.

In the second one, black has all 16 units, so no White pawn left its file. The pawn on c7 means the two black pawns behind it must both leave that file. Given this, I count a minimum of seven captures and white has 10 units showing which is too many captures.

Completely Illegal

Neither king can be first to reach its final square. If the white king arrives first, the black king has no path. If the black king arrives first, the white king can only arrive if the g7 pawn hasn't captured yet, but then the f8 bishop will never get to d8 since the white b-pawn will already be on c7 to let the black b-pawn move so the c8 bishop can reach e2, else the black king couldn't have arrived.

Good start, but this proof doesn't quite work. What if there was a white piece on e2 (before the kings were in place)? The white piece could then move to create a discovered check, and the black LSB could block on e2. When retracting moves, this would be "un-pinning" the black LSB. (Similarly, bNg2 and wNg4 could be un-pinned too.)

e.g. with some pieces removed

I hadn't thought of that. But if both kings are in place before the f8 bishop is released, the white e-pawn will still be at e3 to let that bishop come around, so the black knights at f1 and g2 will be in place. Then Black needs a capture at f6 and another capture by the a-pawn to promote to a black piece to be captured at c7, and has no captures left to remove the white piece that was blocking at e2.