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9, A, 4, Q, J, 7, 3, 2, 10, 5, K, 8, 6
After two "shuffles", the first postion has gone from holding an Ace to holding a 10. The 10 position has gone from holding a 10 to holding a Jack, the Jack (11th) position has gone from holding a Jack to holding a 6, etc.
That sequence of "position is replaced by" can be represented as follows:
A -> 10 -> J -> 6 -> 3 -> Q -> 2 -> 9 -> 5 -> K -> 7 -> 4 -> 8 -> A
The "square root" of this transformation can be obtained by going forward 7 or backward 6 in the cycle, so that the first position would go from Ace to 9, and the 9 to the 10, and the 10 to the 5, etc.:
A -> 9 -> 10 -> 5 -> J -> K -> 6 -> 7 -> 3 -> 4 -> Q -> 8 -> 2 -> A
So after one iteration instead of two, the Ace has been replaced by a 9, the deuce by an ace, the trey by a 4, etc:
9, A, 4, Q, J, 7, 3, 2, 10, 5, K, 8, 6
which is the solution.
To verify:
The 9, in the A position after one "shuffle" is moved to the 2 position as required.
The A, in the 2 position after one "shuffle" is moved to the 8 position as required.
The 4, in the 3 position after one "shuffle" is moved to the 7 position as required.
The Q, in the 4 position after one "shuffle" is moved to the 3 position as required.
The J, in the 5 position after one "shuffle" is moved to the 10 position as required.
The 7, in the 6 position after one "shuffle" is moved to the K position as required.
The 3, in the 7 position after one "shuffle" is moved to the 6 position as required.
The 2, in the 8 position after one "shuffle" is moved to the Q position as required.
The 10, in the 9 position after one "shuffle" is moved to the A position as required.
The 5, in the 10 position after one "shuffle" is moved to the 9 position as required.
The K, in the J position after one "shuffle" is moved to the 5 position as required.
The 8, in the Q position after one "shuffle" is moved to the 4 position as required.
The 6, in the K position after one "shuffle" is moved to the J position as required.
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