I guess there'd be no other way apart from extreme luck and/or perseverance.
There is. You plot the movements as a decision tree, up-left-down-right, and do a breadth-first search on it as you go.
Random example:
Move up. Success! Mark the move as valid. Restart. (You can experimentally deduce the side button restarts the puzzle by watching the movable thingy return to its location, plotting a new tree after it returns and realizing they are identical.)
Move right. Success! Mark the move as valid. Restart.
Move down. Failure! Record the move as prohibited, put a big fat dot at the end of the line, restart (we don't know that restarting isn't necessary).
Move left instead. Success! Mark as valid, restart.
Move up again, then left. Failure! Mark as prohibited, restart.
Move up, then right. Success! Mark as valid, restart.
Move right, then up. Failure! Mark as prohibited, restart.
etc.
It hinges on the assumption that the sides are independent, but we know they are because they revert to starting position on switching.
I can't help but boast I've completed harder math puzzles: for example, in a certain unfinished game that I'm forbidden to link to, there's a hilly area with apple trees, and I had to collect apples from those trees by using a hovercar which consumed fuel according to a fairly complex formula depending on the distance and height moved (it hovers a fixed distance from the surface however high the surface might be). The task was to use up as little fuel as possible.
The movement occurred on a 15*15 grid (so, 225 points total). The game provided a tiered map (6 tiers in total I think) which I thought might lead me to the intuitive solution. But no! Turned out it was a "natural" terrain with completely arbitrary heights. So I had to collect 812 data points (yay diagonal movement!), then
learn a programming language because I didn't have access to MatLab anymore (I chose Python), import the data and run two algorithms on it.
I actually naturally did the "Ray's kind of a jerk" achievement straight away...
I have an excuse! When I first saw this sequence, I was
hacking government computers and citizens' mailboxes via USB drives and form history respectively in real life. Rule #1 of
casual hacking is
not to leave tracks! So I was totally focused on my inability to delete the password request email from the boss' inbox and on Ray doing a lookup of himself. BAD MOVE, DUDE. Depending on how magical the totally-non-PCR DNA analyzer was, he could've swiped just about anything belonging to the secretary. At least
try it before signing off "Ray Abbot was here".
There's another easy way to solve the puzzle box without the viewing aid too. There's a walkthrough on Gamezebo which features screenshot "maps" of the correct path...
