Mdthbs

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.

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. 0-0


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.

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

(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, but this issue resolved itself, see next edit)

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:
  So back to the question of if it was a Black piece blocking the check. Well, then, if it was the e-pawn that moved last, then the White light-squared Bishop would be stuck. In short, there wouldn't be enough time for Black to capture this Bishop and run back to its starting square, because White can't move anything else at this point without ruining ability to castle or ruining the Black Rook-White Knight duo on a1 and b1.)


• 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.