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For every reachable position, is the equivalent position for black also reachable?

If so, can you prove it?

If not, which reachable positions for white are not reachable for black and why?

Definitions:

  • position: includes castling and en passant possibilities, excludes 50-move considerations
  • equivalent position for black: swap the colours of all pieces, flip the pieces vertically (swapping rank 1 pieces with rank 8 pieces etc), swap whose turn it is to move, castling rights and e.p. ability. (e.g.: the equivalent position for black after 1.e4 would be the starting position, but with black's E pawn on e5 and with white to move, both players able to castle both sides).
  • legal: it's possible to get to this position by some sequence of legal moves (disregarding 50 move rule)
Laska
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Steve Bennett
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    Do you really mean rotate the board 180, or do you mean flip the board across the center line? If you rotate the board 180, the equivalent position after 1.e4 has a black pawn on d5, not e5, and kings and queens not in their starting positions. – Alexander Woo Apr 04 '22 at 20:08
  • Sorry, good catch. – Steve Bennett Apr 04 '22 at 22:46
  • What you describe is the CPT theorem :-) for chess, in an article in SCHWALBE I asserted (without proof) it is always true. Since Rosie describes a parity problem, I must have overlooked a detail in your "equivalence" definition... – Hauke Reddmann Apr 05 '22 at 07:31
  • Already asked sort of: https://chess.stackexchange.com/questions/29073/is-the-number-of-legal-chess-positions-odd-or-even – Laska Apr 12 '22 at 11:21
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    Thank you! That question is definitely closely related. The upvoted answer there seems to be wrong (or to have misunderstand their definition of "mirroring") though. I like the terminology of "vampire position" for ("unreachable equivalent position for black") – Steve Bennett Apr 13 '22 at 02:54
  • @SteveBennett there is no upvoted answer in the earlier question – Laska Apr 17 '22 at 09:57
  • "reachable position" is exactly "legal position" – Laska Apr 17 '22 at 11:23
  • I think "reachable" is much clearer. It is obvious that a pawn on A1 is not a legal position, but it is less obvious whether or not one would consider a position "legal" simply because it could not actually be reached in a game. (I don't think "legal position" is a concept in the rules, so I'm not sure what is gained by changing the term here.) – Steve Bennett Apr 18 '22 at 10:36

2 Answers2

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It's perhaps easier to understand the set of those positions whose flipped versions are not reachable. Suppose all the following are true:

  • all pawns, rooks, bishops and knights are on the board
  • the a and h pawns have not moved more than one square; the other pawns are at home
  • the bishops are at home

Then no bishop or queen has ever moved, and every move has been from a black square to a white square or vice versa. White starts with 8 units on black squares (four pawns, Ra1, Bc1, Ke1 and Ng1). Later, if this count is even, White has made an even number of moves; if odd, odd. The same can be said of Black units on white squares. The sum of these two numbers is the total number of White and Black moves made so far. This total's parity (i.e. whether it's even or odd) determines whose turn it is to move: even, White's turn; odd, Black's turn.

Now flip the position. The total is the same. Therefore the same player must have the move. Thus if, as you require, we must also swap whose turn it is to move, then the resulting position is unreachable.

By contrast, if anything else has happened, one or other player can "lose a move" and enable the flipped position to occur. For example, given that we may play into a Sicilian with White to move, by 1 e4 c5, we may also play into a reversed Sicilian with Black to move, by 1 c3 e5 2 c4. Another example: given that we may play into a Indian with White to move, by 1 d4 Nf6, we may play into a reversed Indian with Black to move, by 1 Nf3 d6 2 Ng1 d5 3 Nf3.

Thanks to Steve Bennett for the idea which leads to the following additional positions. Suppose that at least one of the players retains at least one castling-right, their f-pawn has advanced one square, and (except for the f-pawn) the above conditions all hold. Now flip the position. The only way to reach the flipped board position with the other player to move is to change the parity of the number of moves, by moving this other player's king out via the f-file. But moving their king destroys both their castling-rights, so the position is different.

SecretAgentMan
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Rosie F
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    I think your set of unreachable flipped positions is slightly incomplete. – Steve Bennett Apr 04 '22 at 06:22
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    To be specific: the equivalent position for black after 1.f3 would be unreachable. (Because although it's possible for the black king to make an odd number of moves while white makes an even number of knight moves, it would forfeit its castling rights in the process.) – Steve Bennett Apr 04 '22 at 12:29
  • Interestingly, the reason that the position after 1.f4 *is* reachable for black is that white could take two moves to get there (f3, f4) while black can do it in one (f4). – Steve Bennett Apr 04 '22 at 23:22
  • With respect to the author, who is one of my favourite posters, I don't know why this has been ticked as a correct answer – Laska Apr 17 '22 at 10:02
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Alternate set of unflippable positions (inspired by Lord Dunsany's chess problem):

One side has the b-g pawns, both bishops, the king and queen all on their original squares.

The rotated version of this position is not possible because the pawns and bishop prevent the king and queen from moving, yet they're on the wrong squares.

ralphmerridew
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