Is this position legal?

0-0

Last move? (two solutions):

• -1.Ne2xQg1
• -1.Ne2xNg1

Alright, three solutions then :)

-1.d2-d3

d2-d3 had to be played earlier as otherwise the bishop would not have been able to come out ..

How about Kg5 to Kh4 or Kh5 to h4? Those moves look possible ..

The bishop on h2 can be the promoted a-pawn so the last move could have been d2-d3. Kg5-h4 or Kh5-h4 can't be the last move as there is no way to unlock the position after retracting either of those moves.

My mind is blocked with regards to position #1324. If it is possible that this position could exist, then are there much more moves possible in this position. For instance:

50. Qc5 Qb6 51. Qd5 Qc6 ... 59.Qg5 Qf6 60.Qh4 Qxh4 61.Kxh4.

I don't know how the black king exactly could get to that field, but I will give it a try. He must have walked there using the h-file. The white dsb should have arrived at h2 before the king was there, because the g-pawn must be played in time to let the black king pass the h3-field. That implies that the white h-pawn was at g4 before the black king was at h4. It has captured a black piece at g4, giving the black king the possibility to walk to h3.

Therefor should the white lsb come from f1 to h3. The moment the black king moved to h3 was it definitely not on that diagonal and to provide the black king the possibility to come to e1 and itself to h3, should they pass each other at e1 for the black king and e2 for the white lsb.

With the lsb via e2 to f1 and h3 is that problem solved. Now should the black king get to the second row in order to give room to the queen side rook to go to g2. Furthermore must the white knight on e1 be able to go to that square. That is possible from g2, leaving e1 an impossible square for black to stand on. The black king stands therefor a while on e2, the rook is played from a1 to g1, the knight from g2 to e1 and the black king from e2 to f1.(The white knight on g2 can come from e3, f4 and h4 respectively and arrive there whenever appropriate.)

However, before the black king is moving to f1, must there be a black piece at g1 to block the check from the rook on h1. That piece must be a knight or a queen as those pieces are the only one to get to g1. Why a bishop can not get to the field is obvious and for a rook to get there should the knight get out for a moment. That however, is not possible after the white queenside rook is at his place.

The white king should have got at h4 and or h5 via g5. That field is now blocked by the h-pawn. Therefor is the other last move h7-h6.

@LoekBergman: Good analysis on how the bK got in, but take it a step further. You counted that white made three captures total: exf, hxg and Nx~g1. These all occur outside black's pawn wall, so only the 2 bN and bQ could be captured. Therefore white did not make any other captures unless it is inside black's pawn wall, since the bB and bRs cannot get out. Therefore the last move cannot have been wKxQh4.

Also, black cannot have made the last move h7-h6 nor a7-a6. Consider the board position. If both black pawns were back on a7 and h7, the bK couldn't have got out. So black cannot have just played both pawn moves. Meanwhile the bK cannot have just come from e2 (impossible knight check). In fact, it has been at least 2 moves since black's king was free (e.g. last 2 moves were 60. Nc1-d3+ Ke1-f1 61. Nd3-e1 h7-h6 62. d2-d3 or 75. Nb3-c1+ Ke2-f1 76. Nc1-e2 h7-h6 77. Ne2xg1.) Black can only make one waiting move with the pawns, so it must be that white moved last.

In terms of retractions it becomes simpler to understand. White takes at least 2 retractions to free the bK and give black retromoves (-1. Ne2xg1 ~ -2. N~-e2 or -1. d2 ~ -2. Nd3), and black only has 1 tempo move; therefore, white moved last, either Ne2xN/Qg1 or d2-d3. In the event d2-d3 was the last move, white's captures of N/Q were exf, hxg and axb, while capturing b6xRa7 and a7xRb8=B inside black's pawn wall.

Thanks, got it. Should have been more precise in keeping the score in captures made.

I don't exactly understand the capturing of the a-pawn. You started saying that white made three captures outside the black pawn wall. Those are exf, hxg and Ne2xg1. I agree. How can the white a-pawn then have another capture? The a-pawn must have been captured by one of the four black pieces that came out of the black pawn wall.

Ah, I should have been more exact in phrasing that one. If the last move was white's d2-d3, then the capture Ne2xg1 is not necessary because the wK came from e1. Therefore in that case the 3 captures outside the wall were exf, hxg and axb (axb to allow the white a-pawn to promote on b8, as retracting d2-d3 implies that the wBh2 is promoted.)

And since shoopi posted some older works, I'd share one of mine too.
Legal? And with wPg3 -> h3?

It is legal. The pawn on h3 instead of g3 looks illegal to me, because there are only two pieces of black gone: the a-pawn and the DSB. You can never capture a DSB on a white square. Hence illegal.

The variation wPg3->h3 is legal as well. I won't say how, so others can try.

Well, the a-pawn could have promoted to a queen and got sacrificed at h3 :)

Yeah. the a pawn captured the queenside rook of white, promoted to a queen, and got sacrificed at h3.

LoekBergman got the diagrammed position right, Uncia_Uncia (solution by PM) and prashanth got the variation right. Both are legal; the status of the bBa6 changes in the twinning.

I apologise for posting this next one a third time, but it is simply my favourite original (slightly modified for aesthetics.) Can white castle?

I don't think white can... It's not possible to get the king there without making the queen not hold control of the 1st rank.. But if the queen moves somewhere else at first and then the black king enters, then the queen can't enter d1..

My conclusion : White cannot castle..

It's a bit more complicted than that.  It's impossible with the queen and bishop there, sure.  But what if the bishop on c1 was somewhere else and there was a Black knight there instead?

...hmm, but a knight could only come and go from d3, and that would put the White king in check, which is a problem.

... but what if that bishop on c1 is promoted via hxg6 (taking a bishop) and gxf8 (taking a rook)?  If that's a promoted bishop, Black could have played Nb3xBc1 while the pawn was still on b2.  Only then does White push b3 to allow the Black rook to promote and to allow his promoted bishop back in.

... but THAT runs into the problem where if White plays Bxc1, there's no unaccounted material left for White to capture fxe3. 

If instead there's a white knight on c1 blocking the queen, then if the Black king started on b2 and the Black rook started on b1, White could play Nc3+ Ka1 Ne4 to get the knight out, but now there's no way for the king to get back to b1, and you're left with the king and rook swapped.

So white cannot castle.  The White bishop must have gone to that square when the White queen, and therefore the White king, were off somewhere else.

All answers still wrong so far. Everyone's still missing something.
@DavidMertz1, good reasoning, but you overlooked something too. You almost had it by considering the different types of pieces on c1, but didn't look fully at one of the possibilities.

...also, it was solved by Uncia_Uncia (via PM a few weeks ago).