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## Question

Six (chess) knights are placed on a 4x3 chessboard, three along the top row and three along the bottom, which are labeled as P, Q, R, X, Y, Z, as shown in the figure.

Exchange the positions of P and X, Q and Y and, R and Z, in minimum possible number of moves.

4X3 CHESSBOARD.bmp

Edited by K Sengupta

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The minimum number of moves is 26

It takes 5 moves to move each corner to the opposite corner

It takes 3 moves to move each middle to the opposite middle

4 corners * 5 moves = 20

2 middles * 3 moves = 6

20 + 6 = 26

My methods for exchanging corners/middles

Corner 1:

In order to move X from a1 to a4 perform these moves

a1 to c2

c2 to a3

a3 to c4

c4 to b2

b2 to a4

Corner 2: (needed so that pieces don't end up on the same square while swapping)

In order to move X from a1 to a4 perform these moves

a1 to b3

b3 to c1

c1 to a2

a2 to c3

c3 to a4

Middle 1:

In order to move Y from b1 to b4 perform these moves

b1 to c3

c3 to a2

a2 to b4

Middle 2: (this is the mirror of the first way to prevent the two middles from ending up in the same square while swapping)

In order to move Y from b1 to b4 perform these moves

b1 to a3

a3 to c2

c2 to b4

Using these 4 methods you can swap all pieces without any piece ever being in the same square.

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There will be a total of 26 moves taken. P, R, X, and Z will each require 5 moves to swap positions on the board. Q and Y will each need 3 moves to swap positions.

Here are the moves that each piece will need to make:

P: 4a to 3c, 3c to 2a, 2a to 1c, 1c to 3b, 3b to 1a (5 moves)

Q: 4b to 2a, 2a to 3c, 3c to 1b (3 moves)

R: 4c to 3a, 3a to 2c, 2c to 1a, 1a to 3b, 3b to 1c (5 moves)

X: 1a to 2c, 2c to 3a, 3a to 4c, 4c to 2b, 2b to 4a (5 moves)

Y: 1b to 3a, 3a to 2c, 2c to 4b (3 moves)

Z: 1c to 2a, 2a to 3c, 3c to 4a, 4a to 2b, 2b to 4c (5 moves)

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I believe that it takes 26 total steps...5 for each knight on the corner, and 3 for each knight in the middle. There are a lot of move orders possible...just make sure that your pieces don't step on each others' toes. For each corner, there are three 5-step paths to the opposite side. For example, X could go a1-c2-a3-c4-b2-a4, a1-c2-b4-a2-c3-a4, or a1-b3-c1-a2-c3-a4. Each non-corner knight has only two 3-step paths to the opposite side. For example, Y could go from b1-a3-c2-b4, or b1-c3-a2-b4. Again, we simply need to coordinate these paths so that the pieces don't get in each other's way. One possible solution:Za2, Xb3, Xc1, Zc3, Xa2, Pb2, Za4, Ra3, Pc4, Xc3, Zb2, Xa4, Yc3, Rb1, Ya2, Qc2, Qa3, Yb4, Rc3, Qb1, Pa3, Zc4, Ra2, Rc1, Pc2, Pa1. Done.

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26 moves

Six (chess) knights are placed on a 4x3 chessboard, three along the top row and three along the bottom, which are labeled as P, Q, R, X, Y, Z, as shown in the figure.

Exchange the positions of P and X, Q and Y and, R and Z, in minimum possible number of moves.

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