There are 10 people standing in a row, shortest to tallest. A “mini-shuffle move” consists of three people leaving the line and then returning to occupy the three empty spots, with none of the three going to the spot where they had been. How many “mini-shuffle moves” are needed to fully reverse the order of people standing in the row, so they are standing tallest to shortest?
"mini-shuffle moves": reversing order in a line
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combinatorics
1 Answers
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Notice that you want to realize the permutation $(1,10)(2,9)(3,8)(4,7)(5,6)$ this is an odd permutation, and because of this cannot be written as a product of even permutations. Notice that mini-shuffle moves are $3$-cycles, in other words even permutations. So it cannot be done, no matter how many permutations mini-shuffle moves you make.
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0I'm a little confused; while your reasoning makes sense, in your sequence of shuffles you only switch two elements at a time, not three. – 2017-01-21
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0Oh yeah, my bad. Let me fix it, I had missread the definition of mini shuffle move. – 2017-01-21
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0Fixed, it can never be done, no matter how many moves you use. – 2017-01-21