Illegal Position Contest!

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

you can get the top part but I think it is impossible to get the king to a1 without moving the paws on b2 or d2

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.

Here is a lengthy try in which the cage is unlocked and the Kings freed. It fails because there is no way to unpromote a black N on g1. Black would have had to play ...hxg2 to get the pawn through, but I cannot retract wPh4 back to h2 with the wB on g1/h2.

The reason that occurred to me for the failure of this try is that Black must play hxg2 while leaving all of White's units on the board.

The white piece blocking at e2 doesn't have to be captured. Although it can't be wRg6 due to the pawns/pieces in the way, it could be wNg4 via un-pinning: the white knight can retract out of g4, and black could have a piece ready to retract to g4 (blocking the bQh3 check). The white piece can also be captured by black's a-pawn on its route to promotion.

Additionally, the black a-pawn doesn't have to be captured at c7 - an original (i.e. not promoted) knight/queen could be captured instead. It could be on the board now, replacing the original piece captured on c7. Also, bNg2 is not fixed in place: it could be un-pinned by a white knight, blocking wQh1 from checking the bK.

Yeah, minor detail. Will work on a better try.

Your first try is still pretty good! I didn't even notice that black's h-pawn couldn't promote due to wBh2. The only issue with this is that there are other tries (with variations) which fall just short as well but I'm sure you'll be able to find them.

I need a hint. In your valid illegality proof, how many possibilities are there that have to be ruled out?