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How many moves are needed to exchange the positions of black and white pawns? That is, all black pawns should be located at the below line and vice versa. But, all black pawns are identical and so are all white pawns.

Each pawn can be moved to an adjacent blank square.

enter image description here

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4
  • 1
    $\begingroup$ Normal chess rules don't apply? $\endgroup$
    – ACB
    Mar 8, 2023 at 2:09
  • 1
    $\begingroup$ @ACB No. Each pawn moves just four direction: up, down, left, right. $\endgroup$
    – P.-S. Park
    Mar 8, 2023 at 3:35
  • $\begingroup$ If the two blank spaces are together and a pawn moves two steps at once, does that count as one move or two? $\endgroup$ Mar 8, 2023 at 6:15
  • $\begingroup$ @JaapScherphuis I would count it as two moves because it is natural and convenient for computer programming. Of course, someone may prefer one move. $\endgroup$
    – P.-S. Park
    Mar 8, 2023 at 13:42

2 Answers 2

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I used a computer to find an optimal solution.

52 moves.

 52 BBBBB    39 BBB.W    26 BBW.W    13 B.WWW
    X.X.X       XBX.X       XBXWX       X.XWX
    WWWWW       WBWWW       B.WBW       BBBWB

 51 B.BBB    38 BBB.W    25 BBWWW    12 .BWWW
    XBX.X       XBXWX       XBX.X       X.XWX
    WWWWW       WBW.W       B.WBW       BBBWB

 50 BB.BB    37 BBBWW    24 BBWWW    11 ..WWW
    XBX.X       XBX.X       XBX.X       XBXWX
    WWWWW       WBW.W       BW.BW       BBBWB

 49 BBB.B    36 BBBWW    23 BBWWW    10 .W.WW
    XBX.X       XBX.X       XBX.X       XBXWX
    WWWWW       WB.WW       BWB.W       BBBWB

 48 BBBB.    35 BBBWW    22 BBWWW    9  W..WW
    XBX.X       XBX.X       XBX.X       XBXWX
    WWWWW       W.BWW       BWBW.       BBBWB

 47 BBBB.    34 BBBWW    21 BBWWW    8  W.W.W
    XBXWX       XBX.X       XBXWX       XBXWX
    WWW.W       .WBWW       BWB..       BBBWB

 46 BBBB.    33 BBBWW    20 BBWWW    7  W.WWW
    XBXWX       XBXWX       XBXWX       XBX.X
    WW.WW       .WB.W       BW.B.       BBBWB

 45 BBBB.    32 BBBWW    19 BBWWW    6  WW.WW
    XBXWX       XBXWX       XBXWX       XBX.X
    W.WWW       .W.BW       BW..B       BBBWB

 44 BBBB.    31 BBBWW    18 BBWWW    5  WWW.W
    X.XWX       XBXWX       XBXWX       XBX.X
    WBWWW       ..WBW       B.W.B       BBBWB

 43 B.BB.    30 BBBWW    17 BBWWW    4  WWW.W
    XBXWX       X.XWX       X.XWX       XBXWX
    WBWWW       .BWBW       BBW.B       BBB.B

 42 BB.B.    29 B.BWW    16 BBWWW    3  WWW.W
    XBXWX       XBXWX       X.XWX       XBXWX
    WBWWW       .BWBW       BB.WB       BB.BB

 41 BBB..    28 BB.WW    15 B.WWW    2  WWW.W
    XBXWX       XBXWX       XBXWX       XBXWX
    WBWWW       .BWBW       BB.WB       B.BBB

 40 BBBW.    27 BBW.W    14 B.WWW    1  WWWWW
    XBX.X       XBXWX       XBXWX       XBX.X
    WBWWW       .BWBW       B.BWB       B.BBB

                                     0  WWWWW
                                        X.X.X
                                        BBBBB

A rearrangement of a few moves allows the solution to be broken up into clear stages, making it much easier to see what is going on.

 52 BBBBB
    X.X.X
    WWWWW

 Rotate the ring anti-clockwise until spaces are at top right

 43 BB..B
    XBXWX
    WBWWW

 Swap top right corner

 39 BBB.W
    XBX.X
    WBWWW

 Rotate the ring anti-clockwise until spaces are at bottom left

 33 BBBWW
    XBXWX
    W..BW

 Swap bottom left corner

 29 BBBWW
    X.XWX
    B.WBW

 Shift the pieces in the ring left to bring spaces to the right bottom

 23 BBWWW
    XBX.X
    BWB.W

 Swap bottom right corner

 19 BBWWW
    XBXWX
    BW..B

 Rotate the ring anti-clockwise until spaces are at left top

 13 B.WWW
    X.XWX
    BBBWB

 Swap top left corner

 9  W..WW
    XBXWX
    BBBWB

 Rotate the ring anti-clockwise until all pieces are correct

 0  WWWWW
    X.X.X
    BBBBB

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  • $\begingroup$ Great work! Would love to see an animation of this. $\endgroup$ Mar 9, 2023 at 2:57
  • $\begingroup$ @DmitryKamenetsky Me, too! $\endgroup$
    – P.-S. Park
    Mar 9, 2023 at 12:38
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Not the optimal solution of 52 moves as the other answer presents, I managed to get 62 moves by trying to place correct pawns at the four corners first. Once each corner is solved and the remaining black and white pawns are all adjacent, then shifting them is trivial.

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