Can White Castle?

Question

This is the first chess puzzle I composed in the retrograde genre. I originally posted this in a chess dedicated forum. Hope you like it!

In the following position, is it possible that White could still castle?

• To prove it's possible, all you have to do is provide a legal game.
• If you believe it's impossible, you need to provide your reasoning.

Answer 1

Now that we have three increasingly complex proofs (two deleted, one of them mine) that it's impossible, it's pretty clear that it must be

possible after all!

Here's why:

1. b3 Nf6
2. Bb2 Ng4
3. Bf6 gxf6
4. Na3 Ne3
5. Nc4 Nxf1!
6. Ne5 fxe5
7. h4 Rg8
8. h5 Rg6
9. hxg6 Bh6
10. g7 Be3
11. g8=N Bc5
12. Nh6 Ba3
13. Nf5 Ng3
14. Nd4 exd4
15. Qb1 Bc1!
16. Qb2 Nh5
17. Qc3 dxc3
18. Rb1 Nf6
19. Rb2 cxb2
20. Nf3 b1=R
21. Nd4 Ra1
22. Nb5 Ng8
23. Na3 Bb2+
24. Nb1 Bg7
25. e3!! Bf8
26. O-O

In case you haven't already done so, you should totally check out @greenturtle3141's thorough answer (that unfortunately tripped up mere inches before the finish line) to see why the highlighted moves are absolutely essential.
Note that the notation has been edited to work with most PGN viewers, for instance https://chesstempo.com/pgn-viewer.html

This is, without doubt, the most refreshing chess problem I've ever tried to solve. Thanks, OP!

Answer 2

(Edit: so this is wrong..)

Brilliant puzzle! The answer is:

No! White can't castle.

[Spoiler alert! Scroll down at your own risk]

We proceed by contradiction. Assume that indeed, White can castle. We have the following undeniable facts:

• Neither the White King nor the White h-Rook have moved.
• Since neither of them have moved, the only way the Black Rook could have gotten to a1 is via a promotion.
• The only Black pawn that ever moved was the Black g-pawn. This is the piece that promoted.

We now ask: On which square did this pawn promote? It's not so obvious!

• The pawn could not have promoted on a1, because this would require 6 captures. Exactly 6 white pieces are missing. That means the pawn captured all of them. But this is impossible because the pawn can only capture on dark squares, and one of the captured pieces would have been a light-squared Bishop.
• The pawn theoretically could have promoted on b1 via 5 captures.
• The pawn could not have promoted on c1, for the same reason as that for a1.
• The path could not have promoted on d1 or f1, because it has to promote to a Rook, and this would check the king, and either the rook would be taken or the King would have to move.
• Obviously, it could not have promoted on e1.
• It could not have promoted on g1, because then it cannot get to a1 without the King moving.
• Obviously, it could not have promoted on h1.

We conclude that the pawn promoted on b1 via 5 captures. Each of these captures occurred on a dark square, so the five pieces captured were: 1. The a-Rook 2. The dark-squared Bishop 3. The Queen 4. A Knight 5. The h-pawn

The first four captures are easy enough. The problem is the h-pawn. If this pawn stays on its column, it could never be captured by the Black g-pawn in the south-west direction. So it changed columns. To change columns, it must capture a piece. That means that the White h-pawn captured the Black Rook. Even so, this isn't enough to get captured by the Black g-pawn. We conclude that the White h-pawn captured the Black Rook and then promoted on g8 to a capturable piece, say, a second Queen.

Ok, so the Black g-pawn captures all those 5 pieces. Here's where we're basically at:

Notice that I'm keeping the White light-squared Bishop alive. This is very important.

See, here's the problem now: White has only two movable pieces left: The Knight and the light-squared Bishop. Somehow, Black was able to 1) Promote on b1 to a Rook, 2) Move it to a1, and then 3) White was able to move the White Knight to b1. If the light-squared Bishop was dead, this would be impossible, because the White King would be in check while we were doing all the maneuvering! It couldn't have just been the White Knight blocking the Rook, because then it would be pinned!

The solution? The White Bishop covers the King by moving to d1. This fact is undeniable, and we will use it very soon.

(Edit: Could it have instead been a Black piece protecting the White King, say, the Bishop on f8? Actually yes, and that's the problem with this answer!)

Now we're all set for the main argument. The main problem that is not immediately obvious is that of the last move.

Looking back at the original board, we first ask: Who moved last? If White moved last, what piece was moved?

• Clearly it wasn't the b-pawn, because the Black pawn had to get in somehow.
• It wasn't the e-pawn for sure, because the light-squared Bishop had to get out at some point in order to protect the King from the promoted Rook's check. (Edit: Yeah so apparently Black actually has just enough time to get his Bishop back to f8 since White has the extra move of e3. Whoopsie.)
• It couldn't have been the Knight, because then it would be moving to block the check, i.e. when the Rook got to a1 it checked the king. That means on the move before it was NOT checking the King, so it must have promoted on a1, which we know isn't true.
• King and Rook are out of the question.

Thus, Black made the last move. Which piece moved last? Clearly it wasn't that Black rook. And if it was some other piece, it still begs the question: What was White's move before that? This is problematic because if Black's last move was, say, x.. Nb8, then we can apply the same above argument to conclude that White couldn't have made the last move.

This can only mean that Black captured a White piece on the last move. At this point, it's pretty clear that this can only be the White light-squared Bishop. It's evident that the only square it could have been captured on is g8. But neither the Black f nor h pawns have moved, so this Bishop couldn't possibly have gotten there. Contradiction. Thus, White cannot castle.

Answer 3

My goal is to describe my solution strategy in detail, even though my final solution isn't that different from that of @Bass. At multiple points, I will ask a question, which I encourage you to answer on your own before reading further.

So, let's start. The black rook on a1 is the most interesting part of the position. The black rook on h8 is gone and it seems to have done a reverse Houdini to get to a1. So, the first question is

Could the rook from h8 have gotten to a1?

No. Note that a pawn can only be on the 2nd rank if it hasn't been moved. This means that the only way for a rook to pass the second rank is through the opening on h2. But then it would have to pass both the king and rook, neither of which could have moved if white wants to castle. So, then, the next question is

Is there another way for black to get a rook on a1?

Yes. You know the movie, 'The Prestige'? It's a bit like that. You see, the rook on a1 is not the rook from h8, but a body double: the pawn from g7, promoted to a rook. The pawn can pass the 2nd rank by moving across the diagonal. Such a magic trick requires sacrifice, however. Not just from black, who has to give up the rook on h8, but also white, who has to feed the ravenous pawn so that it can move diagonally. But, can white feed enough pieces to the black pawn? First, count how many pieces white can give the black pawn.

How much pieces can white sacrifice?

5. Note that the g7 pawn has to stay on the black diagonal to get it past the pawn chain. This means that we cannot feed it the f1 bishop, as it can't reach the black fields. So, white can easily offer 4 pieces, Queen, Rook, (c1) Bishop and Knight. The final piece is the pawn on h2. This pawn can take the black rook to reach the g line and can promote to Queen on g8 after the g7 pawn has moved to f6, at which point it is easily fed to the pawn.

Is this sacrifice enough?

Yes. While the pawn needs 6 diagonal to promote on a1, the pawn is not promoting to rook on a1, but on b1! This means white can sacrifice the rook on field b2. However, this must be done without checking the king. First, this is the current board:

Now, as white still has its spare bishop, it can be used to shield the king and we can reach the following position:

Almost there, right? Not quite. The problem here is that now the bishop has served its purpose, white will have to get rid of it. To do that, it has to be taken by a black piece. Then, black needs at least one more move to return that piece to its starting position. The problem is that black doesn't have this move, because white's turn is first, and white can only move the rook or king, preventing castle! Is there another way to shield the king from harm?

Can we avoid checking the king?

Yes. The trick is not to use the white f1 bishop, but the black f8 bishop! It takes some maneuvering to get the bishop there, but it works. Before the next part, the white bishop should be cleaned up. We can get to the following position

We are close, but not there yet. The black bishop is one move short of a full retreat, and again white has no moves left. Almost miraculously, there is a way. There is exactly one piece in this position that white could have chosen not to move earlier.

Before reading the final question, I'd like to note that this question is not only the essence of this puzzle (it could have replaced the actual question, in fact), but also one of the key questions of the retrograde-analysis genre. The final question, of course, is:

If white can castle, then what was white's last move?

For the answer I must thank @Bass, because this is where I got stuck. Since I think this is hard, let me give another hint before giving it all away

White has made a move to be able to use a piece it doesn't need to use. It is this move that is white's last move.

And now, the final piece of the puzzle

White's last move is e3!!, allowing the bishop to retreat to f8 and reach the final position, where white may castle. White can afford delaying this move because it doesn't need to use the white bishop, this bishop can be taken on its home square by a black knight.

All in all, an excellent puzzle, not only on a technical level, but also on a psychological one. Why?

One part of the puzzle that struck me was that all pieces sacrificed have been nessecary: reaching this position but with any of them on their home square is impossible. That is, for all but one piece: the f1 bishop. While this bishop is 'dead weight' to reach the end, it is of vital importance for the puzzle. First of all, it makes the solution itself more involved. But more importantly, it creates the deception that moving this bishop is nessecary. This makes finding the solution truly challenging, or as @greenturtle3141 said "The White Bishop covers the King by moving to d1. This fact is undeniable."