[_-__---]

I like the rescramble idea. You could maybe go one step further by varying the difficulty of the rescrambling - I'm not sure if there's an established distance metric between rc states (something like a levenshtein distance, "minimum-number-of-changes-needed"), but the program could shuffle the cube out to some threshold on that metric to get easy, medium, hard, etc.

[bunderbunder]

There is a metric for the # of turns needed to solve a cube from a particular state. I can't remember its name offhand, but the word for the maximum possible number of moves - 20 - is known as God's Number.

I'm not sure it's relevant to the difficulty of the scramble for a human, though. Most solving methods follow an approach that isn't even trying to find the minimum possible number of moves. The one that most of the fastest speedcubers use is, IIRC, the least move-efficient of all the standard methods. It just manages to be faster in terms of time because it's such rote method that, if you learn it well and get it burned into your muscle memory, you don't really have to ever even stop to think about what you're doing in the middle of a solve.

[nomel]

I'm assuming that this cube could not be handled as a normal cube. The gearing would give it an odd feel, like moving a servo by hand, and giving too much torque would probably wear teeth.

To make it feel "normal", you could "assist" the user with the motors, but I'm guessing speed would be severely limited compared to a normal cube.

[Retra]

Just make the faces buttons and have the button press signal a rotation.