Basic Concepts and Notations

Many people look at the cube and see 54 colored squares. This is bad. There are not 54 individual pieces to move. There are only 26 pieces (8 corners, 12 edges, and 6 centers), and only 20 of them move (the centers do not). Realizations like this are the key to understanding the Rubik's cube and how to solve it. The notes below outline some of the main insights to remember when working with any Rubik's cube puzzle. Don't just read these notes. Understand them and make them unconciously obvious.

Centers

A center is always a center. No moves can ever make a center be an edge or a corner. A center has only 1 color on it and this color never changes. The centers never move, they only rotate. This is especially key. The relative positions of the 6 center pieces will never change. This defines the relative positions of the colors for you. In most cases (though not all since color patterns are not compeltely standard), white is opposite yellow, red is opposite orange, blue is opposite green. Looking straight at red with yellow on top, green is to the right. That defines your cube. Learn this color pattern well.

Edges

An edge is always an edge. No moves can ever make an edge be a center or a corner. An edge has 2 colors on it, and which 2 colors never change. An edge is only solved when both colors are on the correct faces, not just one of them.

Corners

A corner is always a corner. No moves can ever make a corner be a center or an edge. A corner has 3 colors on it, and which 3 colors never change. A corner is solved when any two of them are on the correct face, but not just one of them.

Wait? Not all 3 need colors need to be on the correct face? But can't you have the red-blue-yellow corner in between the red and yellow faces such that they look right but have the blue side on the green face? The answer is no, though it's not immediately obvious, and brings us to the next point to understand.

Not all color patterns are possible.

Why? Because there aren't 54 pieces, there are 26. In the example above, the red-blue-yellow corner always exists in the order "red-blue-yellow" when you move clockwise around looking at it. To be in the red-yellow-green position with yellow and red correct, but blue on the green face would require it to look like "red-yellow-blue" (again clockwise) which it never ever can (if it does, then someone took the stickers off and put them back on wrong and the only way to solve it is to take them off and put them back on correctly). The only color patterns possible are those which can be made from the 24 pieces which inherently lock some color squares together in particular ways. But wait! There's more!

Not all piece combinations are possible.

Why aren't all arrangements possible? Because you can't arbitrarily move the pieces. You can only rotate entire faces at a time. Thus only those arrangements attainable by a series of such moves is possible. Thus, for example, you can't have everything right except one edge flipped with the wrong orientation (though 2 edges flipped wrong is possible). There are about 519 quintillion (519 followed by 18 zeros) different ways of putting the pieces into the cube. But only 1 in 12 (43 quintillion) of them are possible to get from a solved cube (and thus are possible to solve). If you meet one of the others, someone has taken the cube apart and put it back together wrong, and the only possible way to sovle it is to take it apart and put it back together correctly. Along the way through the solution here it'll be pointed out how to tell if this has happened. And while we've just hinted at it, let's say this explicitly:

Not all moves are possible.

The only moves you can make is to rotate a face. Further, there are only 3 such rotations possible for each face: rotate 90 degrees clockwise, rotate 90 degrees counter-clockwise, or rotate 180 degrees. Anything else is equivalent to one of those (3 90-degree clockwise is equal to a single 90-degree counter-clockwise, for example, and 13578 rotations of 90-degrees counter-clockwise is equal to a single 180 degree rotation). Everything you can ever do is just some combination of these 3 moves on the 6 faces. There is no other way to move pieces around.

Position and Orientation

When we talk about the position of a piece, we mean where it is located relative to the centers. A corner, for example, is always located between two centers, and is considered to be in the same position regardless of which of its colors are on those two faces. Similarly a corner is in the same position if it is between the same three faces, regardless of which of its colors is on each face. The two different ways an edge can be in a particular position, or three different ways a corner can be in a particular position, are different orientations of the piece. To be solved, every piece on the cube must both be in the correct position and have the correct orientation. But during the process of solving it, it will sometimes be easier to only worry about one of these two aspects at a time.

Good then. So now instead of seeing 54 colored squares that can move randomly about, you see 20 (moveable) pieces, 6 faces (defined by the center pieces, which never move), and only 18 possible moves. These facts make the cube a much more tractable puzzle, and with this understanding we can now solve it.