So, while he decides to get out of line from his place in front of me and fill out some other slip he forgot to fill out on the first go-round, I polish off a quick solve before he can really even turn around and look. He then asks me "So you know all the combinations?" I said yes, and went on with the rest of my bank business, but by the time I was back in my car again I felt like I had given the laziest answer possible. I realize that not answering "Yes" would not be particularly helpful at that moment, since there was no reason I needed to lecture the poor gentleman merely trying to make conversation about how many combinations there really are and why you didn't need to know all of them. This post is merely me trying to make amends for my incredibly lazy answer to that question.
Let's start with the obvious (well, obvious to me) part. I do not, and can not know all the combinations. I say that I can not know all the combinations because there are a little over 43 quintillion combinations. If I had a second apiece to learn each one it would take more than a trillion years. As it turns out, however, it is not necessary to know all the combinations - it is only necessary to know all the combinations of smaller, well-defined groups. For example, if you had completed one face of a cube and wanted to put in the edges of the middle layer, let's look at the possibilities. At this point, there would only be eight edges unsolved, as the four on the first face would already be in their correct places. Since I'm old school, we'll call the solved layer the Up, or U face. As you look for edges to place in the middle layer, and you look at each one in turn, they would be:
- In the middle layer already, correctly placed and oriented,
- In the middle layer already, correctly placed but not correctly oriented,
- In the middle layer already but not correctly placed,
- In the Down (D) layer such that the color of the edge piece on the D face is to the left of its eventual location in the middle layer, or
- In the Down (D) layer such that the color of the edge piece on the D face is to the right of its eventual location in the middle layer.
Of course, I could have explained all this to the guy that I was talking to in line at the bank and he could have said, "Yeah, that's what I meant."