Looking for impossible chess positions

1 and 2 are illegal. What was the d-pawn captured a piece on c6, which has to be a rook. The rook could only have come out when the pawn was on c4, which blocks the bishop in. With the pawn on h3 the other rook could have gone to c6.

#23 Position 2 is legal.

With the White pawns on b3 and c3, the Black a-pawn captures the a1 rook on b2, promotes to a bishop on b1, then moves to a6.

Next, White plays c4, clears the back rank (which requires moving the g-pawn) and moves the h1 rook to c6 via c1, c3, d3 and d6 where it is captured by the Black d-pawn.

@tmkroll - you're sensing the issues correctly. The "trapped" bishop on a6 is the key to the whole puzzle. But definitely not all 3 positions are illegal, otherwise why the puzzle? ;-) The missing h1-rook is not an immediate concern as there are more pressing formations.

A hint: Black is missing the dark bishop, and the a-pawn. Which of these could've caused me such grief with uncooking the puzzle?

And EDIT: @caveatcanis correct! As for the other 2?

#23 Position 3 is legal.

The black a-pawn captures the a1 rook on b2 and promotes on b1. The promoted piece is captured on h3, releasing the h1 rook, which is sacrificed on c6. The original Black bishop then travels to a6 before white plays c4.

I was so pissed-off with this, not having realised my diagrams started with had d4 instead of c4. lol. Nice one anyway. 23 is not legal, as rook can't exit if c4 is not played. Furtheremore, b3 must be played so bishop can exit so rook can exit afterwards. Also, the only thing the d pawn has possibly captured is a rook. Once d has captured the rook, it is moved on c6. Now the black light coloured bishop can exit. However, with all c4, b3 and c6 having been locked by pawns, the bishop can no longer move to the square shown.

The difficulty lies in the fact that a black B can't reach a6 if there are pawns on b3, c4 and c6. White is only missing two Rooks, and black needs a capture on c6. Black is missing the a7 pawn and Bf8.

The Ba6 is either the original B or a promoted pawn. If it was from c8, the only chance is for white to have played Pb2xBc3 to let the R out without stopping the Bc8 from reaching a6. But that requires another capture to fix the pawn structure, which is only possible if Pa7 can be sacrificed in a useful way. To do this, it would have to capture the other white Rook.

Pa7 might capture an R on b2 and promote. White could let it out and to a6 by playing c3! The only rub is that white's other Rook must be sac'ed on c6.

In either case, I need white to be able to sac both Rooks usefully for the position to be legal.

So let's approach each twin armed with that knowledge.

a) Rh1 never got out, so the position is not legal.

b) White could have played Bh3 and Kg2 to let Rh1 out. The scenario with Pa7 promoting works. Legal.

c) Black could have promoted a7 and sac'ed it on h3. Then Rh1->c6 and d7xc6 to let Bc8 out. Legal.

! Correct and wow, well done both caveatcanis and BigDoggProblem. I originally constructed a different puzzle with a very similar theme, but realised later it was cooked. Took a fair while to figure out how to elegantly separate the solutions by shifting the pawn.

Oh, and @ BigDoggProblem: One tiny flaw in your answer - Pb2xBc3 happens to be impossible. Since for the bBf8 to get to c3, Pe6 must have been played, which traps the original bBc8 at home! caveatcanis's construction is the only method for #2.

And as a bonus, this is a pair easily solved if you've solved the previous puzzle. 1) As per diagram, 2) Black king on g5 instead!

I like this one I saw in a math book of all places.

Not impoossible, though it may seem it at first.

White to mate in 4

Yes, that is why it was only a 'chance' and never got upgraded.

@remilion

Position 1 is impossible because the only way for the black bishop to get out is if white's rook was captured on c6; The only way to get the white rook to c6 is by playing c4... but if c4 has been played then the bishop can't make it to a6.

Still working on position #2

Yeah, I should have known better than to post with the line "unless I'm missing something."

^ wtf is this. is black supposed to attempt to deny the mate?

Anyway, diagram for 23) 2 coming soon

I just read though and realized that someone else has already solved these.

2: if i am correct, the whole idea of the g3 pawn, is to let black castle. However, a rook still can't exit without the black light bishop being blocked. In this case, the white bishop doesn't allow any rook to exit.

EDIT: i was wrong lol. You are a genius.

It not possible; knight cannot lose a tempo.  Got ahead play around with it and see. No matter how much dancing around you do with the knight, when they come back to the starting position it will always be white to move.

#22: Oh crap, nice. I didn't think of a pawn capture as the last step; I keep thinking of Black king move as the last of Black's move.

#34, position 1: (By the way I haven't read through the analysis of people for #23, so this is kind of from scratch. Sorry if anything is repeated.)

Impossible.

Black is missing a dark bishop and the a-pawn; he also made the capture dxc6. White is missing both rooks and made the captures xg3 and xg4+. Also xg4+ is the last move by White, and it's Black to move.

The dark bishop of Black must be captured on g3. Thus we need the sequence of moves (with possible moves in between): Black e6, Black Bg3, White xg3.

Since White's captures are on g3 and g4, the light bishop of Black leaves its home. So we need black dxc6 to be earlier than black e6, otherwise the bishop is trapped. This, in turn, requires White's rook to come to c6.

Note that White xg3 occurs after White Rc6, so the rook cannot come from h1. Hence the rook must be from a1, getting through c3. This means White c4 is played before White Rc6.

Recap: White c4, White Rc6, Black dxc6, Black e6, Black Bg3, White xg3.

White's Ra1 must go through c3. Either we need White b3, or we need White's bishop to go first, which in turn requires White b3. So White b3 comes before White Rc6. But these, combined, makes White b3, White b4, and Black dxc6 to come before Black moves Bc8, which blocks off a6 from bishops. So Bc8 doesn't go to a6; it is captured on g4, and the bishop on a6 is Pa7 underpromoted, and comes before dxc6.

Finally, Pa7 needs to have axb2 in order to be able to promote. But it can only capture a rook. The rook in h1 is sealed, and the rook in a1 is required to go to c6; there's nothing to capture on b2, so the position is impossible.

I think I missed something here, because it means in position 2 (black Kh5 -> Kg5) the position is still impossible and it's boring. :P

@sonya_c - Not one of Martin Gardiner's books perhaps? "Lord Dunsany's Chess Problem" as it was listed in the book I have at least. 1. Nc6! wins for white; 1. Nc6 Nf3 2. Nb4 Ne5 3. Qxe5 and 4. Nd3#.

@jdcannon - Very good job! And the proof games one might say are the most convincing way to prove legality (although the most tedious to communicate.) And you're right that knights can't lose tempi, although if any single chessmen were off the board it becomes impossible to tell whose move it is - e.g. starting position minus the g8 knight, it could be either player's turn.

@computo200 - Black castling is an optical illusion. It may not even have been played; the king and rook could walk around each other. For puzzles like this, it is not the appearances (looks like black castled, looks like white played c4 and b3 quickly) but the certainties (Pd7xc6, Pc4, Pb3 creating a wall, Pg2xh3 needing a capture to release the rook vs. Pg3-Bh3 etc.) that really matter.

@chaoticiak - #22 My reasoning was somewhat wrong, it could've also been Nxsomething or Kxsomething for white's last move too. Of course if you specified black to move there it's illegal.

#34 - immaculate reasoning to prove #34-1 is illegal (which it is.) However it relies almost entirely on one statement - "Also xg4+ is the last move by White..." - which may not be the case in position 2?

@Remellion if any single chessmen were off the board it becomes impossible to tell whose move it is - e.g. starting position minus the g8 knight, it could be either player's turn.

Not quite true. You could lose either or both queens (captured by knights, of course), and still not be able to gain/lose a move.