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Math Help - Transpositions

  1. #1
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    Question Transpositions

    A well known children's puzzle has 15 numberes pieces arranged inside a square as shown. A move is made by sliding a piece into the empty position. Consider the empty position as occupied by the number 16, so that every move is a transposition involving 16. Show that the order of the pieces can never be reversed. [Hint: show that if a product of transpoitions each involving the number 16 moves 16 to an even-numbered position on the 4X4 board then the number of transpostions must be even. Also consider the sign of the permutation which takes the "start" board to the "end?" board]

    1 2 3 4
    5 6 7 8
    9 10 11 12
    13 14 15

    "start"

    15 14 13 12
    11 10 9 8
    7 6 5 4
    3 2 1


    "end?"
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  2. #2
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  3. #3
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    it just tell there is no soln when 14 switches with 15. It didn't say how it works with transposition I guess.
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