A clock game

Part of solving “paradoxes” in FFXIII-2 involves solving puzzles: this is one such puzzle appearing in the game. A circle of numbers is laid out, and you need to “visit” every number.

You may start anywhere. After visiting a number n, the clock hands will swing around to the two faces which are n steps away from the visited number. You may not visit the same number twice.

This appears to be “hard” to solve even on a computer, in that the most efficient algorithm is a brute force of all possible moves. The program which generates the puzzles on this page does this every time you request a new puzzle.