Smith uses a method called the Petrus method invented by Swedish speed cuber Lars Petrus, currently working in place to get a share of the bounty when you tolerate ads on Google and related sites like gmail, googlegroups, blogger (yes this site) to name a few, read Google. Yes, you can laugh. I learned it from the man himself from his website http://lar5.com/cube .
News is I've uploaded some videos of different steps of the process I took, since at that time, I found fewer videos available for beginners and intermediates on this method. Video's can be watched from here or on Youtube.
People depending heavily on memory be forewarned - this method is logic intensive and doesn't use algorithms till step 5 (last layer). There are several other methods available which use such. The Petrus method is known for requiring fewer steps.
One common misconception is that you can move stickers around. You can't. The Rubik's cube is not made of 9x6x6 stickers that can be moved independently. It's not made of 27 smaller cubes either.
A Rubik's cube is made of a frame with 6 centres - the centre pieces cannot be moved relative to each other, only twisted. Apart from this, the Rubik's cube has 8 corner pieces and 12 edge pieces.
Step 1 - Build 2x2x2
2x2x2 refers to a corner and the 3 adjacent edges.
Step 2 - Expand to 3x2x2
Now we add an adjacent corner and it's two adjoining edges. Ensure you don't mess up the earlier 2x2x2. Beginners are recommended to hold one hand on the solved portion.
Step 3 - Correct incorrectly twisted edges
This step will save lots of trouble later on. By now, two faces should remain. Observe the colour of the centres of the two unsolved faces. Observe the position of the colour sticker on each of the unsolved edge pieces.
4 rules to determine incorrectly twisted edges:
1. If the edge piece has the same colour adjacent to the center of one of the unsolved faces, it is good
2. If the edge piece has the same colour away from the center of one of the unsolved faces, it is bad
3. If the edge piece has the opposite colour away from the center of one of the unsolved faces, it is good
4. If the edge piece has the same colour adjacent to the center of one of the unsolved faces, it is bad
Step 4 - Solve two layers
Fixing corner positions:
Twisting corners correctly:
Fixing edge positions: