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3x4 Knight Swap (Posted on 2010-02-13) Difficulty: 3 of 5
Eight white chess knights are placed on a 3x4 chessboard, four along the top row and four along the bottom, which are labeled as P, Q, R, S, T, U, V and W, as shown in the figure.

Exchange the positions of P and T, Q and U, R and V, and S and W in minimum possible number of moves.

P Q R S
T U V W


Let the squares be identified from 1 through 12 being read from left to right down.

No Solution Yet Submitted by K Sengupta    
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re(2): One possible solution Comment 6 of 6 |
(In reply to re: One possible solution by brianjn)

P  Q  R  S  T  U  V  W
4  6   6  4  2  4  4   2<o:p></o:p>

I believe this is almost certainly a minimum solution. The reason is that we can easily eliminate swaps of bilateral pairs, the four corner pieces (and hence also the four centre pieces) and the two side fours PQtu and RSvw as possible subjects of translation. If we then graph possible knight’s moves starting from the corner and the centre edge squares, we find that the 4 move translations only use the side 4 squares (plus one in the middle) for a corner knight and the centre 4 squares (plus one in the  middle) for a centre edge knight which we already have ruled out as solutions. The next shortest tours are of length 6 for a centre edge knight and 8 for a corner knight. By symmetry, at least 2 of the centre edge knights are therefore going to have to take one for the team in terms of a 6-move walk!<o:p></o:p>


  Posted by broll on 2010-04-09 11:04:48
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