
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 evennumbered 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?"


it just tell there is no soln when 14 switches with 15. It didn't say how it works with transposition I guess.