Clock Face Puzzle

In Final Fantasy XIII-2, you need to solve a puzzle based off of a clock face.
Below, I have a brute-force solver to find the order to solve the puzzle.

Input the numbers clockwise from the top (The positions):
0
1
2
3
4
5
6
7
8
9
10
11
12

Result:

Other information

For those interested, this problem is finding a directed Hamiltonian path. Nathaniel Johnston did a nice write-up on about the problem.