I did write a program last weekend to try to find the correct path (I have played around with the Knight's Tour, so I had a base to work from). I let it run for about 30 hours before I stopped it. I was afraid there was still a bug in my algorithm, and the running application maxed out my computers' processors. I did some more testing and started it again. I stopped the second run after about 20 hours, and put it aside until last night. I incorporated Y-san's constraints/hints, and now the algorithm runs to completion in a matter of minutes. But still no solution that hits all 64 squares. I have come close - I found 3 solutions that hit 60 of the squares before landing on the exit platform, but not all the squares. More then willing to post the solutions or the program if anyone is interested.
Here is what I would try: (1,0,0) (1,0,0) (1,0,0) (0,0,1) (0,0,1) (0,0,1) (0,0,1) (0,0,0) (0,0,0)
Why? Others picked up on the sphere indicating a 3-dimentional answer. I looked at the keys as a direction, instead of a point. When you add them all up, you get a point that is (5,8,4). So the additional keys will bring the point to (8,8,8) – a perfect cube. Additionally, I added two points at the end so that the total number of keys pressed is 25 (5x5), the witch gave us the first 16 (4x4).