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.
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.
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.
I'm sorry if I'm missing something, but did you mean to reply to a different post here? I don't quite see the relevance of what you're saying to anything that the original post covered.
Not sure about the products/companies you're listing, but cult of personality must have a lot to do with the extra press. People will click on links to stories about Tesla/Elon/SolarCity without the journalists having to do as much.
They can[0]; however, this requires some further construction.
One major benefit of the sin/cos basis is that each basis function maps uniquely to a dirac delta in frequency space. This allows for power/energy symmetries between time and frequency domain representations of a signal and allows for more tenable frequency filtering, e.g. removing 60/50Hz noise or isolating a particular frequency band.
A haar wavelet or modified square wave basis may provide you with a simple orthonormal signal basis, but each wavelet has a frequency representation with infinite support in frequency space. This is (a) untenable and (b) eliminates the possibility of frequency-specific filtering. Wavelet analysis is more useful in specific cases.
When you use the term "frequency space", you are already thinking in terms of the frequency of sin/cos functions. If you actually think in terms of frequency in a different orthonormal basis, you'll find that the dirac delta characteristic you mention is not specific to sin/cos, thereby invalidating your argument.
24hr service would be very expensive, but it seems that they should run the transit services until a little after the majority of bars and clubs close. It was nice here in Boston when they ran the T until a little past 2AM; they've since pulled that back to 12:30 or so. It can be kind of a pain, but Uber/Lyft are well established here.
It's certainly usable for that in LA, where last call is also 2am. I believe every train runs until 2 or 3 AM (on Fridays/Saturdays), and starts back up again by 5. Not quite 24 hours(except on NYE, when they don't stop), but still very usable.
