Try to sort the scrambled numbers.
I'm not entirely convinced the puzzle can be solved, I got this far and no further, @kisonecat.
0 1 2 3 4 7 8 5 6
Can this be solved from here with the understanding the first digit is zero and the last is eight ?
Ah! For that, you'll need to rotate around the end by moving right and left so the "cursor" straddles both ends of the sequence.
The permutation is two swaps away (swap 6 and 8, swap 7 and 5), so it is in the alternating group A_9, and A_9 is generated by adjacent transpositions. So it will work -- but it isn't instantly obvious.
I am wondering if this sort of thing could form the core of a fun game. Here the basic swap is 1234 -> 2143, and you could build higher levels with other sorts of swaps.
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