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This is a follow-up question to the good old puzzle Switch The Knights:

Envoys from two rival kingdoms meet face-to-face in a very narrow corridor of a neutral castle. The corridor is a grid exactly 4 squares long and 3 squares wide.

Turning back is dishonorable for both sides. To pass each other peacefully, they must completely swap their starting positions.

The Setup:

Top Row (North Gate): Black Knight, Black Rook, Black Knight

Bottom Row (South Gate): White Knight, White Rook, White Knight

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The Rules:

  • Pieces move exactly as they do in normal chess.
  • No capturing is allowed. A piece can only move to an empty square.
  • Any piece can be moved on any turn.

Questions:

  • What is the minimum number of moves needed to completely swap the Black and White pieces?
  • If the middle pieces were Queens instead of Rooks, would the swap require fewer moves? Why or why not?
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My answer:

21 moves

21 moves ordered left-right, then top-bottom

Explanation:

To exchange the two rooks we need at least 4 moves. We ended up using 5 moves since we had to move one of the rooks by one step at the beginning. Each knight requires at least 3 moves to go to the diagonally opposite corner, but at least 5 moves to go to the other corner on the opposite side. So to minimise the moves we choose the former option. However, the path is not empty, so one of the two diagonally opposite knights needs to make a small detour worth two moves per pair. And this detour requires one of the rooks to make a small step at the beginning.

To summarise,

3 moves per knight + 4 moves for rooks + 2 moves for each pair of diagonally opposite knights due to detour + 1 move of rook to allow for the above detour = 21 moves.

When the rooks are replaced by queens, we can

save 1 move because the last three moves of the white rook can be replaced by two moves (one NW and one NE). I believe this is minimum because exchanging the queens requires at least 3 moves, which is exactly one less than what the rooks require. The rest of the argument is similar to the above.

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  • $\begingroup$ The extra rook move isn't necessary. Just don't swap each pair of knights in consecutive moves. $\endgroup$ Commented 2 hours ago
  • $\begingroup$ @DanielMathias. Sorry, I don’t see how that helps bring the number of moves to under 21. Please post an answer if it does. $\endgroup$ Commented 1 hour ago
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The minimum number of moves is

20

Solution in twenty moves

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