17

From move 0, what shortest-possible sequence of moves could be played so that the pieces are arranged exactly as standard board setup (with allowed swapping of same-color knights), but it is black's turn to move? Is it even possible?

I thought about it and conjectured white would have to make an odd number of knight moves, and black would have to make an even number of knight moves. I think this is impossible when the knights need to end up where they started, or even swap with each other (because even though this is an odd number of moves for one knight, the other knight would also have to make an odd number of moves). Am I correct? How can it be proven?

I also guess that this is a famous or at least previously explored problem. Maybe it's trivial.

3
  • 1
    Would it work if white make an even number of moves and black does an odd number of moves? According to your conjecture it wouldn't work.
    – Polygorial
    May 20 at 9:10
  • 1
    Given how the initial position and all the moves are symmetric, switching black to move first would technically make no effect game-wise. It'd just swap left and right and the colors.
    – ilkkachu
    May 20 at 9:39
  • 1
    If the "first" move from Black after the swap would be a double-move with a pawn, the players could co-operate so to make it look like Black made the initial move.
    – ilkkachu
    May 20 at 10:00
31

It's impossible. Knights require an even number of moves to reach a square of the same color as where they started, or correspondingly an odd number of moves to reach the opposite color. Both sides need to make an even number of moves to reach the starting position.

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  • 18
    The conclusion is correct but the analysis is incomplete. Once a Knight moves, the Rook near it can start moving too. But that doesn't help because the Rook is limited to two squares, so when it returns to its home square the Rook has made an even number of moves too. Also, the question allowed Nb1 to switch places with Ng1, and/or Nb8 with Ng8; but that doesn't help because each of those Knights then makes an odd number of moves. May 19 at 13:31
  • 24
    @NoamD.Elkies, I don't think considering rook moves in necessary, because after rook moves the position changes, castle with that rook is no longer allowed, and therefore we would not be able to go back to the starting position.
    – Akavall
    May 19 at 14:27
  • 12
    Yes, the position would be different in that sense; but the question asked only "that the pieces are arranged exactly as standard board setup", without specifying that castling rights be retained. May 19 at 16:52
  • 8
    More simply, after each knight move, the number of knights on white squares changes by one. Initially there are an even number of knights on white squares, so after an odd number of knight moves there are an odd number of knights on white squares, and after an even number of knight moves there are an even number of knights on white squares. Therefore, in order to have knights on the same squares as in the initial position (never mind whether they are the same knights), there must have been an even number of knight moves.
    – bof
    May 20 at 10:04
  • 3
    @bof You can even extend that argument to include rooks: after each move, the total number of knights and rooks on white squares changes by one. May 20 at 11:40
6

This is impossible because.

  • In the initial position, the 4 knights cover 2 white and 2 black squares.
  • each white move will lead to the knights being on 3 white squares and 1 black square or 1 white and 3 black.
  • each other black move would lead to 2 & 2 or 4 & 0.

In order for "black to go first" the knights will need to be on the initial position (2 white, 2 black).
This is not possible after a white's move as shown above.

5

You are basically trying to make a knight lose a tempo, which is known to be impossible.

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    I have no idea what this means May 20 at 20:32
  • 3
    @theonlygusti, see: en.wikipedia.org/wiki/Tempo_(chess) Basically, you can't "waste" a single move with a knight, because of the alternation between black and white squares. That's what the existing answers already say, and much better.
    – ilkkachu
    May 20 at 21:05
  • I was about to say this! I wish this could be the accepted answer. @theonlygusti it's a rule of endgames that knights can't lose a tempo. Do you know like triangulation or losing a move or reaching the same position but with the opposite player move? (eg it's currently your move but you can maneuver pieces to make it the same position but your opponent's move) You can do triangulation with kings and bishops (and i guess rooks and queens), but you can't with knights! Remark: Your question about the opening is answered by an endgame principle. (Wait I'll make this comment a cw answer.)
    – BCLC
    Sep 6 at 11:23
0

converting to comment to answer, but just a cw answer to emphasise postoronnim's answer:

I was about to say this! I wish this could be the accepted answer. @theonlygusti it's a rule of endgames that knights can't lose a tempo. Do you know like triangulation or losing a move or reaching the same position but with the opposite player move? (eg it's currently your move but you can maneuver pieces to make it the same position but your opponent's move) You can do triangulation with kings and bishops (and i guess rooks and queens), but you can't with knights!

Remark: Your question about the opening phase is answered by a principle of the endgame phase.

Re triangulation, try checking out: 1, 2

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