Illegal Position Contest!

Here's another way I just thought of to get to the legal position.

You haven’t mentioned about what happens if a white piece is added to g8. Then the last move must be the uncapture of a black unit on g8 (to give black a last move), but a black unit on g8 is illegal as you have stated. But again, well done!

Illegal because the rook could not have gotten there in the last move.

Illegal. To get the white king to h1, the f2 pawn moved forward and the h1 rook was captured. That leaves White with only five pieces to be captured, but the a7 pawn needs six captures on dark squares to promote on g1. The a2 pawn can only promote on a light square. The f2 pawn could make two captures to promote, except that Black is only missing one man. If the f7 pawn moved out of the way, it would need a second capture to return to the f-file. With the six captures by the a7 pawn and the capture of the a1 rook, that adds up to nine captures, but White is only missing eight men.

Note that this illegality proof works even if the black king is somewhere else, so that White has a last move of a piece to g1 to be captured.

Not quite, I’m afraid. Just because the a pawn promotes on a light square does not mean it must be captured on a light square (unless it promotes to bishop, but it can’t escape anyway). The king is on g5 for a specific reason, as I have a proof game for if he were anywhere else. Also the f pawn could capture fxe3 to get captured by the black pawn, or promote via fxg7 behind the black g pawn (but these are illegal for another reason!)

You're right. I must have been up too long when I worked out my proof.

This is the complete illegality proof for #6520:

The last move was fxg1=R#. So the pawn must have come from a7 and made 6 captures. White is missing 8 units, but we will find out that 3 of these units cannot get captured by this black pawn.

White needs a last move before fxg1=R#. Only the d3 pawn or piece on g1 moved last. Suppose the white unit on g1 moved last. A queen or bishop could not move last; a knight would come from f3 or h3 but that would check the black king. A rook is illegal for the white king has no way to get there. But that means the king rook could not get captured by the black pawn (as mentioned by n9531|1). Anyway, a queen, bishop or knight cannot move last, so white must retract d2-d3. This means that the bishop on c1 cannot be captured.

The f2 pawn must get captured by the black pawn, as that is one of the 6 remaining units that can get captured by it. If it were captured on f2, then there would always be a white or black pawn on f2, preventing the white king’s path to h1. If it captured fxe3, then it must capture the only missing black unit, but that is a light squared bishop which could never get captured on a dark square. The f pawn must promote, and it can capture max 1 black piece. Also black’s pawns cannot capture anymore (apart from the a pawn) as wBc1 and wRh1 could not get captured. So bPf5 came from f7, and the only way the white f pawn can promote with 1 capture is fxg7 behind the black f pawn, but that too is on a dark square and the light squared bishop cannot get captured there. Hence illegal.

A position by N. Petrovic, but it was without bPs on a7, b7 and g7.

Illegal. The position below, where a white queen has just moved to g4, shows the only legal way to reach the final position without the added pawns. With the added pawns, every square on the sixth rank is guarded by an unmoved black pawn, and the white king can't pass the fifth rank.

I would like other people to post challenging illegal positions, but I can still keep posting if no one else wants to do so.

Has anyone come up with a position which is not reachable, but ONLY due to the 50 move rule?

I guess the main issue with creating or finding such a position is pawn moves reset the move counter, and such a position needs to leave absolutely no possibility that there has been a recent capture or pawn move, or that if there has, that there was a 50 move gap at some point during the game.

This makes bishops a slightly headache-inducing piece, as they have to remain in place (and hold other pieces in place as a result) unless a capture or pawn move was made. Fortunately, we do know that every move involving the bishop or the pieces it had blocked must have happened after that capture or pawn move. Unfortunately, any position that lets a rook fully escape is not possible to determine whether the pawn advanced one square or two.

I found this position that T. Volet created. Here’s the link: https://www.janko.at/Retros/Glossary/FiftyMoves.htm

Or here as a board

First part: game here ended by 50 move rule

Second part (as you can see the position is legal, but would have ended by the 50 move rule)

Interesting post.

Your final position would be illegal under the 75 move rule under FIDE competition rules, but not under USCF rules where the application of the rule depends on the mood of the TD.

The 50 move rule would make a position unreachable only under some software GUIs that work according their own rules (arguably not chess at all) and make the 50 move rule mandatory. Under FIDE competition rules or USCF rules the 50 move rule wouldn't render any positions illegal because it applies only if it's claimed and under FIDE basic rules since 2017 it also couldn't because it's not there. 

Your proof game doesn't prove that the position can't be reached within the 75 move rule, of course.

I’ve improved the proof game but after 32. hxg4 there are 80 non-capture, non-pawn moves that follow. I don’t think it is possible to do 75 moves

https://www.janko.at/Retros/Glossary/FiftyMoves.htm

The link describes why the rook(s) must follow the long path, but is not very descriptive about why b7-b6 unlocks the position, and more. Here’s my analysis:

All the pawns have captured the missing pieces. bPb6 must come from b7, otherwise there will always be a black pawn on a7 or b6, preventing a pathway for bBa7. So retracting c6-c5 is illegal because it forces bPb6 to come from a7. But retracting b7-b6 immediately is illegal for the bBc8 cannot get captured by the white pawns, which made 4 captures. So we must uncapture this bishop somewhere. (The link shows where)

The black king can only get there from the whole b3. From b3, the black king must come from a4, b4 or c4. But c4 requires wPc4 to retract, and it must be from b3. This makes the black king unable to get there. Now a4 and b4 will be shown in the next paragraph.

We must at some point retract b7-b6 to unlock the position so the wNa4 and wBa3 can escape, as well as the black king. Retracting a move with bPc5 requires bPb6 to move. If we retract c2-c3, then the black king must escape the first 3 ranks, otherwise he cannot escape. If the black king came from b4, then either the white bishop is occupying b4 or is on a3 or a5, so wPc3 and the wB give illegal double check. If the king came from a4, then the wNa4 must retract, and since that retracted before c2-c3, the wN must escape via b6. This is the case for the black king retracting from b4 and a4 even if white does not retract c2-c3.

So before the retraction of b7-b6, the wNa4 is frozen. This also means that bNa8 is frozen. We cannot retract d2xe3 before b7-b6 as the wB cannot get to c1, and of course it could not be promoted. Therefore wNd1 is also frozen in position. These frozen knights create a winding path for the rook if it were to travel, which it must because it acts like a shield for the wK against bRe7. More about that explained by the link

Your reasoning seems good to me, but MARattigan can refute it by showing a proof game that reaches the position within the 75-move rule.