Paradigm

The essence of my solution is a new paradigm. A new way to look at the cube.The way you look at something affects how you think about it, and affects what you think you can do to it. If you see it as a cube with six faces that you can turn, you would very soon be frustrated by the wild number of interactions between the things you can do. Each of your rotation makes changes to the four adjacent faces, and soon you lose track of what is happening.

Alternatively, you may view the cube as three layers, and attempt to solve them one by one. You would have a hard time trying to move pieces around on the second and third layer without destroying the layer you have built.

In the theory below, we work on subsets of the cube in manageable bits. We look only at the subset and forget about the rest of the cube, so that the complexity will be contained in the subset. More important, we would ask what we want to do in the subset, and why we choose the particular movements to achieve our goals.