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Because the chess site this was on takes its puzzles from real games and gives the move numbers, potential solvers knew that the following position was reached after exactly 10 moves by each player.

[fen "2bk2nr/p1p2ppp/2p5/2p5/2P1P3/2N5/Pr3PPP/R3KB1R w - - 0 11"]


Given that information, can one prove that 11. 0-0-0+ in the above diagram is legal?

(i.e., that neither the white King nor the Ra1 has moved so far)


It seems to me like that's not enough total moves for white to have spent two
moves "shuffling", but I can't come up with any proof or a counterexample-game.

  • This might be more appropriate on puzzles.SE, but when I searched this site I saw lots of retrograde analysis problems. That position is taken from this game. – user2668 Sep 9 '14 at 8:20
  • So the mission, if one chooses to accept it, is to provide an example game that has the above position after exactly 10 moves and 0-0-0 is still possible. Presumably it is white to move? – firtydank Sep 9 '14 at 9:02
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    No. The "mission, if one chooses to accept it, is to" either "provide an example game that has the above position after exactly 10 moves and 0-0-0 is" not "still possible" or prove that white must still be allowed to castle. (It is white to move.) – user2668 Sep 9 '14 at 9:54
  • @Phonon: No. Since your definition of "redundant moves" is global rather than local, it's not quite clear that your example game does not actually have any redundant moves. – user2668 Sep 19 '14 at 15:00
  • This game is a real game. THe last moves are 9. NxB(c5) dxc5 10. QxQ+(d8) KxQ 11. o-o-o+ – Mike Jones Mar 24 '16 at 21:22
3

Reasoning by hand about proof games is fine, but is more fun for positions which have been designed by humans to be solvable and to contain interesting features. Automatic verification is standard for all but the most complex of compositions, and is normally done by specialized engine, of which there are several available for free online.

IMHO, the best proof game engine for OP's position is Natch. Running for 49.03 seconds, it found 2493 "solutions" for how to reach the position in exactly 10.0 moves. None of them contained the strings e1, a1 (or 0-0-0!) and so there is no way that White could have lost castling rights in the prior play.

Notes:
(1) The way that Natch reports consolidates results to reduce the length of the report, so there are rather more than 2493 actual proof games, but that doesn't affect the conclusion.
(2) What about other engines? Popeye is not the most performant for this kind of position, and Euclide would have terminated as soon as it found that there wasn't a unique solution. But both are excellent engines.

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17

Ok after playing around with a couple of lines, I finally found one line that shows it's still perfectly legal to play long castles for white at move 11, here it is:

[Title "possible 11.O-O-O+"]
[StartFlipped "0"]
[fen "rnbqkbnr/pppppppp/8/8/8/8/PPPPPPPP/RNBQKBNR w KQkq - 0 1"]

1.e4 e5 2.Nf3 Nc6 3.c4 Rb8 4.d4 exd4 5.Nxd4 Bc5 6.Nxc6 dxc6 (6... bxc6 7.Be3 d6 8.Bxc5 dxc5 9.Qxd8+ Kxd8 10.Nc3 Rxb2 11.O-O-O+) 7.Qxd8+ Kxd8 8.Be3 b6 9.Bxc5 bxc5 10.Nc3 Rxb2 11.O-O-O+

Added another continuation, taking on c6 with the b-pawn, same position results and same number of moves needed!

To conclude, because I reached the given position in exactly 11 moves, and did not play any redundant moves (e.g. Nf3 then back to Ng1 again), it means that all the played moves were necessary, their order can be different, but point is there was no spare move to use and destroy white's castling (e.g. an impossible scenario would be black exchanging queen on d1, Kxd1, then black plays Kxd8, and white goes back to e1, but that took up 2 more moves than the line I showed, so impossible to reach the position you're looking for in 11 moves after such line)

Long story short, castling is perfectly possible here, and in 11 moves black could not have done anything to prevent us from castling and still manage to reach the final position we want. Interesting post by the way, +1.

Alternatively one can also just look at the final position and count the necessary number of moves, that must take place for this position, I elaborate: have the final position in mind: let's take black's point of view: Pawn moves needed to obtain the final position:

  1. e5
  2. exd4 (why black has to take on d4 and not white on e5 is explained in the diagramm below)
  3. bxc6 or dxc6
  4. b6 or d6
  5. bxc5 or dxc5,

Development moves:

  1. Nc6 (to be captured on c6, otherwise there cannot be a pawn on c6 and not have knight on b8)
  2. Bc5 (otherwise c5 pawn impossible)
  3. Rb8 (otherwise Rxb2 never possible)

Remaining necessary moves: Queen d8 capture and pawn b2 capture:

  1. Kxd8
  2. Rxb2

And we find ourselves at the 11th move again, where all we did was just consider the most straighforward ways our pieces could end up where they are in the shown puzzle.

Let's see if we reach the same position via another path, showing why exd4 is the fastest way to reach the final position:

[Title "Why black must take on d4 and not white on e5:"]
[StartFlipped "0"]
[fen "rnbqkbnr/pppppppp/8/8/8/8/PPPPPPPP/RNBQKBNR w KQkq - 0 1"]

1.e4 e5 2.Nf3 Nc6 3.c4 Rb8 4.d4 Bc5 5.dxe5 Nxe5 6.Nxe5 d6 7.Nc6 bxc6 8.Be3 Rxb2 9.Bxc5 dxc5 10.Qxd8+ Kxd8 11.Nc3 

Here we are at move 11 and we still have to play Nc3... clearly because we played 2 knight moves for black.

Finally let's demonstrate why in the fastest path to the final position, white has to be taking on d8 (queen exchange) and not the other way around:

[Title "A line white cannot castle but the final position cannot be reached within the 10 required moves (so impossible for our purposes)"]
[StartFlipped "0"]
[fen "rnbqkbnr/pppppppp/8/8/8/8/PPPPPPPP/RNBQKBNR w KQkq - 0 1"]

1.e4 e5 2.Nf3 Nc6 3.c4 Rb8 4.d4 exd4 5.Nxd4 Bc5 6.Nxc6 dxc6 7.Nc3 Qxd1+ 8.Kxd1 b6 9.Be3 Kd8 10.Bxc5 bxc5 11.Ke1 Rxb2

We lost 2 king moves(capture on d1 then back to e1) for white in this line, hence the Rxb2 only happening at the 11th move.

So it was shown that the only line leading up the the final position within 10 moves, is one where white still must be able to castle.

EDIT: A summary of the discussed elements in the comments:

The proof presented in this answer is purely deduction based, in the sense that the simple fact of having reached the position in exactly 11 moves without having made any redundant (or insensitive to the position) moves, implies that 11.O-O-O+ should be legal without exception.

What does redundant mean here? "Redundant moves": here, are defined as moves that do not bring us any closer to the final position, or would even deviate us from it. For example playing Nf3, then back to Ng1, would be redundant. Playing Be2 then back to Bf1 would be redundant, and so on.

Point being, in any variation, you may come up with, that would take away white's castling rights, will imperatively entail redundant moves, that will in turn delay reaching the final position by a couple of moves. (As an exercise, try some of your ideas, it is interesting, and see how many moves it takes you.)

Looking at such problem from a combinatorics point of view, may be possible, but would be too complicated as we're looking at a depth of moves(tree lines) resulting from 11 moves. Instead, like most chess puzzles, one has to look at it from a purely heuristic point of view, and find the right ideas that would go in the direction of proving the question at hand. Finally, in chess, one usually has an easier time looking for counter examples(proof by contradiction), which is why it is encouraged to look into some of the lines on your own.

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    That it was possible to reach the position in 11 moves was obvious from the question, that was given. What's asked is proof that it can't be done any faster. – RemcoGerlich Sep 9 '14 at 14:12
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    Unfortunately, this answer just considers "the most straighforward ways our pieces could end up where they are in the shown puzzle", rather than all possible ways, so it doesn't give a proof. – user2668 Sep 9 '14 at 18:20
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    We are not given that the position was reach "without having made any redundant (or insensitive to the position) moves". How do you show that black must have played e5? How do you show that black must have played b6 or d6? (as opposed to taking on c6 twice and playing c5 in-between) How do you show that black's queen-knight must have been captured on c6? (as opposed to moving to a6 or d7; I believe that based on this answer I can prove it wasn't captured on d8) How do you show that "otherwise c5 pawn impossible"? How do you show that black's queen must have made a capture? – user2668 Sep 9 '14 at 18:35
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    Looking at such problem from a combinatorics point of view, may be possible, but would be too complicated as we're looking at a depth of moves(tree lines) resulting from 11 moves. Instead, like most chess puzzles, one has to look at it from a purely heuristic point of view, and find the right ideas that would go in the direction of proving the question at hand. Finally, in chess, one usually has an easier time looking for counter examples(proof by contradiction), which is why I encourage you look into some of the lines on your own. It would be my pleasure to further discuss it with you. – Ellie Sep 9 '14 at 18:59
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    As a physicist, I consider this sufficient proof ;) +1 – Danu Sep 10 '14 at 12:00
3

We can deduce nine moves that white absolutely had to make to get to this position.

  • At least three moves were required to put white's pawns and its queen's knight there (3 total)
  • Either the c- or d-rank pawn was captured after it moved. That's another move (4 total)
  • The pieces captured on c5 and c6, whatever they were, needed at least two moves to get there. That's four extra moves. (8 total)
  • If the piece captured in c6 isn't white's missing knight (which needs 3 move to get there), that knight had to be captured in d4 by the e pawn, and that means the queen or bishop had to capture it on their way to capture on the c-file, adding another move anyway (not to mention what it means about the missing white pawn). That's another move. (9 total)

This position without possible castling requires two more moves (moving the castle or king back and forth), and it's impossible to fit them in.

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  • Aren't you assuming that the Black king captured a piece? Technically it's possible for the b and d pawns to be captured even if they don't move. – Wes Sep 11 '14 at 15:15
  • Dear Circeus, you are just repeating the elements of the list I had posted in my answer. Please try to add something original to your argumentation, that I had not already covered. Thanks, and please by no means should you take this as offensive. – Ellie Sep 11 '14 at 15:25
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    @Phonon My solution is not the same. Yours looks at black moves and deduces "there was no spare move to use and destroy white's castling". My solution deduces white does not have the spare moves to move its castle or king back and forth. – Circeus Sep 11 '14 at 15:38
  • @Wes I had realized that and was fixing my solution while you were commenting. – Circeus Sep 11 '14 at 15:40
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    Yes but that's the same point, turned upside down, I didn't add another list for white because it would been pretty much a list of similar elements, as neither sides can allow redundancies...which I also elaborated with an example (last diagram). Your post is a valid answer, but it does not bring new content I'm afraid, maybe I'm wrong. – Ellie Sep 11 '14 at 15:42
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Sorry, no, White -cannot- castle queenside in this position. Retrograde analysis has nothing to do with it. The official rules of Chess explicitly state that, in castling (either side), neither the king -nor the moved rook- is allowed to pass over a threatened square. The Black rook on b2 ruins it.

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    A rook is allowed to pass a threatened square. Please check the official FIDE rules. – Glorfindel Mar 24 '16 at 19:34