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The following numberphile video has me stumped.

Why is this Puzzle Impossible? - Numberphile

It describes the use of parity in permutations to show that the following sequence is impossible

{15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, blank}

because the parity of moves required is odd, and only even parity moves are achievable in the 15 puzzle.

However the following sequence does have even parity and therefore should be achievable.

{15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, blank, 1}

From the puzzle's perspective it is trivial to go from this sequence to the impossible solution which is a contradiction. What is the error in this logic ?

Marky0
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    permutations which fix the blank space are impossible iff odd. – Angina Seng Apr 26 '20 at 14:10
  • OK. thanks. After more thought I now get it. {15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, blank, 1} is not possible either as it would require odd parity, and it is even. doh. – Marky0 Apr 26 '20 at 15:04

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