A Knight's move is as in chess, an L shaped move, 2 squares in one direction and 1 square in the other direction.)

A Knight's Tour is one where the knight passes through each square exactly once.

You may start on any square you wish.

* For extra credit, come up with a re-entrant tour: at the end, the knight is exactly one knight's move away from the starting square.

* For EXTRA extra credit, make sure that the path is, in some way, symmetrical.
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Since "Knight's Tour" is a term used outside the scope of this problem, I'm sure you can find an answer on the internet. Please find an independent solution.

This does not require a computer program.

Adding to the comments Eric made, there are 64 squares on the board. For a symmetrical pattern in four section, which would seem natural, each 'section' of moves would need to include 16 squares, distributed equally with some sort of radial symmetry.

What I did, then, is to start in each corner, and make the same move simultaneously in all four corners (to make sure that none of the patterns crosses another), and trying to get the patterns to end up in a position to move into the next corner. I'm not sure if that will work; it's late and I haven't really thought about it, but it seems good enough..