Monday, April 3, 2017

"What's the secret?" they ask...

...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.
When I was using older cubes, I often took one of the edge pieces out at this point to show that you can't really change the pieces and that it really was a matter of getting the selected piece 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.


Tuesday, January 17, 2017

A brief disruption, and a small explanation.


The other morning I went to the beach to walk for a few miles, as per usual, with the minimum accompaniment of my wife, our dog, and my Rubik’s brand cube that only goes to the beach. It’s a newer cube (2013 or later) with plastic tiles and the newer mechanism. This particular cube is rather loose as a fair amount of sand has gotten in it over the past couple of years and worn away at the plastic. Periodically I rinse the cube out, and then re-lubricate it, and it’s OK again.

We meet a lot of people at the beach. Most of them are tourists, but I suppose that there are a fair number of locals. Every once in a while I get to talk to people about the cube, but usually I have to already be stopped by dog socialization or people who need to ask my wife about some dog-related thing. This particular morning I got an unusual request – a kid, I can only assume that he's 10 or so - asked me if he could attempt to solve it. Since I run into other solvers so infrequently, I immediately hand the kid the cube, waiting to see what his ability level is. Disappointingly, he immediately grabbed a corner piece and twisted it in place, which resulted in me asking for the cube back a second later. I untwisted the corner, and ran through a rather clumsy corners-first solve while he watched. When I was down to three pieces left to solve, he asked me if I was still attempting to solve it, and seemed genuinely surprised four moves later (R2 E R2 E’) when I was done.

Our little interchange over, I felt bad as I walked off. Certainly, I could have let him keep going. It’s not like twisting one corner was going to make a big difference in a solution being attempted by a person that decides that twisting a corner is a viable move. Unfortunately, it struck me as a rather familiar scenario – when some people are presented with something complex they may not understand, often disruptive behavior can help them make breakthroughs when a more conventional strategy seems hopeless and time-consuming or analysis seems unnecessary or unwanted. Many a Street Fighter match or a football game has been won by a hare-brained strategy that a reasonable opponent wouldn’t even think to defend against. Our very history as a country in America has been defined, and is still being defined in new ways, by disruptive behavior.

On a small closed group, however, every unconventional disruption has to be undone for the group to return to its natural order. Had the kid managed to get closer to solving the cube, that single corner twist that he did would have to have been reversed at some point in the process. I don't know if he felt like I forced convention on him, or if he even understood the implication of what he did at all. At that moment, I felt like I had to show him that there was a right way, and it didn't take shortcuts. Unfortunately (whether for me or for him will be left as an exercise for the reader) I only could show him by solving the cube, instead of offering up some sort of explanation.
 
For a moment, let's imagine a rather simple cube. Since we're only concerned about the corner pieces for the purpose of this thought experiment, let's imagine a 2x2x2 cube that only has stickers on the U and D layers. When we talk about orientation, we will say that if it's correct - we'll assign a numerical value of 0 to that. If it's counterclockwise from the correct position, we'll call that -1/3, and if it's clockwise from the correct position, we'll call that 1/3.  If you turn the U or the D layer, no change in orientation has occurred. If you turn F, B, L, or R 90 degrees, what happens is that you get two of the pieces changed by -1/3, and two of the pieces changed by 1/3. Add the numerical values of each piece up and you get 0.

That's not to say that you always get 0 - if you started from the solved state and did the move R' D R you will have three pieces with an orientation of -1/3 and all of the rest of them correct. Add those up and you get -1. If you had done R' D' R you get three pieces with orientation 1/3 which add up to 1. As it turns out, no matter how many moves you do, it will always add up to an integer. 

This lends itself to a basic of cube behavior. Normally when I explain it, I would tell people that a single corner cannot be out of orientation by itself. They can be in opposite pairs (1/3 and -1/3) or all three in the same direction, like in our R' D R example.

If you study the edge pieces on a 3x3x3 in the same way, you discover that edges cannot be individually out of orientation for a similar reason, and you will only find an even number of edges can be out of orientation. If you try the thought experiment for edges out fully, remember an incorrectly oriented edge would have a numerical value of 1/2, since an edge only has two possible positions.

Luckily it is rather difficult to disrupt the orientation of an edge cube by hand due to the way it sits in the mechanism, but it is a concern if someone were to reassemble a cube at random. There is also a possible problem with the parity of the pieces if a 3x3x3 cube is reassembled at random, where you could possibly get to the end of the cube and only have two pieces out of place, which is also not normally possible. You can have a minimum of three pieces out of place, like three corners or three edges, or you can have two corners and two edges for a total of four pieces out of place, but no less.

At a certain point, the only solution for disruption is disassembly and careful reassembly. 

(Don't even get me started on what to do with an older cube whose stickers have been moved around.)