...and more often than not, I respond with "How do you get to Carnegie Hall?"
It's not a question that I get all the time, but if I'm out walking around with my Rubik's cube and a bystander asks me that question, it's the question that's the most likely to get me to stop.
I think the most common misconception about the Rubik's Cube is that people think there's more going on that what there is. Visually, there are 54 stickers (or colored tiles). It usually takes me showing a person specifically what an edge and what a corner is to get them to realize what they're looking at. More or less, this is the rundown that I give them.
Ok, so the first thing that you want to see is that the center pieces don't move relative to each other. On this cube, red is always opposite orange, white is always opposite yellow, and blue is always opposite green. These six pieces are on a center spindle like a U-Joint in your car, and can't go anywhere. The next thing is that you want to notice is that each edge piece and each corner piece is unique. For example there's only one orange and white piece, and there's a one-to-one correspondence between the colors on the piece and where it has to go when the cube is solved.
At this point I would turn the cube so that the orange and white piece is in the right place.
So now, you can see that the place that the orange and white piece has to go is that spot between the orange center and the white center. That piece goes at the intersection of those two faces.
While I turn the cube to show the position of the piece, I'm looking for one of the two adjacent corner pieces.
Here's the corner piece that goes next to it. Orange, white, and green.
That piece has to go at the corner that corresponds to the orange, white, and green centers. There are twelve edge pieces with two colors each, and eight corner pieces with three colors each. So now, for every piece on the cube, you can say: Is the piece in the right place, and is it turned around the right way?
If I think that I've lost them, I pick a different piece and place it in an incorrect orientation in the correct location, and if I haven't lost them I try to extrapolate a little more.
So this now means that if you do one side, like most people try to figure out first, it's not going to just be that side. If you have one side done correctly, it's going to be an entire layer solved.
From here it's usually dependent on the person and how we're doing and what follow-up questions that they're asking.
Another nice version of this interchange is that every once in a while (and it's becoming more frequent) I get to watch someone else solve a cube, and I'm able to offer some pointers, or recommend what to work on next or what to look up. The majority of the time is still me trying to demonstrate and explain as much as is requested to people that are unfamiliar.
Sometimes it's hard to know when we're done, but sometimes it's really easy. Once in a while after a solve demonstration, especially since I'm not as fast as whatever they may have seen on the internet or TV, they say things like "Did you see that there are some kids that can do it really fast, but they're doing it mathematically?" The last time that someone did that, I just walked off.
Had you seen it yourself, you might have thought that I assumed that it was the two people talking to each other and I was no longer in the conversation, since I didn't have a polite response prepared. And then you go - Hey, wait a minute, isn't there math here? What's your problem with the question? So, maybe I should explain.
1) Practically nobody successfully solves a cube without a plan. Even if you don't start with a plan, you're going to need one by the time you finish.
2) Nobody solves a cube fast without a plan, and having memorized algorithms beforehand and putting time into executing those algorithms as efficiently as possible.
3) Nobody solves a cube really fast without a plan, memorized algorithms, and lots of practice at piece tracking - looking ahead at finding the pieces you're going to need for the next algorithm by the time you finish the one you're doing.
So the question you have to ask yourself, is that math? I would say that it's just pattern recognition and execution of specific operators that have some basis in set theory or group theory, but you're not doing any set theory while you're solving the cube (unless you're working it out slowly from scratch like case #1 above.) Considering how infrequently set theory and group theory come up in casual conversation, I don't even have a good feel for what people would say is or is not math. Perhaps that can be left as an exercise for the reader to determine what is, or is not math here. (I'm not suggesting that set or group theory isn't math, I'm suggesting that those are the sort of things about which a layperson might be inclined to say, "No, I mean like real math.")
It would also appear that I really need to watch that movie about Edward Snowden. I was disappointed to find out he doesn't have a WCA ID.