More methods than your cube has room for.
The idea behind this meta-method (and I call it that because it encompasses so many normal methods, as you'll see in a bit) is to divide the solve into a small number of steps, each of which can be done relatively efficiently/quicky and in many different ways. There are in fact only two steps, which are as follows:
- 1: Solve the centers and all 8 of the edge pieces on D (or L, if you prefer).
- 2: Solve the rest of the puzzle.
So, why are these two steps good? The key is that the first step is very simple to do, while still leaving a lot of freedom to the solver. In this step, we build a structure similar to the cross from Fridrich; there is a lot of freedom here, because until the entire step is completed you have the freedom to do most outer layer turns and a few Rw U Rw'-type triggers. It may seem constraining, but in practice it is quite easy to set things up without destroying what you have. There are many ways to do this, from blockbuilding to a more reduction-oriented approach.
After we have done the first step, what we are left with is set up quite well for speedsolving: not only have we paired up the centers (the pieces that typically require either a lot of freedom or a lot of moves), but we have also fixed the bottom edges so we can guarantee never to have to look there. We are now free to do U, Uw, and D moves, as almost all moves here will keep the orientation of the center blocks constant, and we can of course use any of the common F2L triggers on the outer layers. Even though there aren't a lot of layers that we can freely turn without breaking up the solved pieces, almost everything we can do has a useful effect on the cube and is also very suitable for speedcubing.
Step 1: The Cross
There are a lot of ways to do the cross step. Here are some of the ways I've tried, although they are by no means all of the ways to do this step:
The actual K4 method sets up some corners as well, so this is a variation on it. The first step is to set up two opposite centers, one of which should be your cross color. Next, using the M layer, you pair up 3 of the cross edges, placing them correctly on the left face (if you're right-handed). Finally, you finish the centers without disturbing the cross edges, and then pair up the last cross edge and insert it.
First, simply solve all centers like in reduction. Then pair up the four cross edges (you can do this by pairing two edges at a time, or with a more 'freestyle' approach where you freely do double layer turns on one slice) and place them.
Start by doing only the D center, and place it on the bottom. Then set up the rest of the pieces by doing eight columns - each column has two center pieces and one edge piece. You can do the columns in pretty much any order, but it is important to realize that you can insert a column with Ru'R' type moves, not just rU2r'.
First, solve two centers and place them on R/L, but make sure they are not your cross color centers. Next, pair up (using the M slice) and solve the DL and DR cross edges. Finally, use the M slice to build the remaining three centers and two edge pairs in whatever order seems most convenient, making sure not to move the DL/DR edges. Notice that there's still a lot of freedom since B, U, and F are free, so you can do any kind of rUr' type triggers, and you shouldn't have to use too many slice moves.
Step 2: The Rest
Again, there are many ways to finish off the method; here are the ones I've played around with:
First, place the corners of the first layer (note that it is a bit more efficient to do this during Step 1 if you can). Next, fill in the second and third layer by inserting the 8 edge pieces with commutators. Finally, solve the last layer with CLL and then ELL.
Place the corners of the first layer, using F2L techniques to also place one edge piece (on average) per slot. Next, fill in the rest of the second and third layer with commutators, although you should only need about 4 commutators on average instead of 8. Finally, use CLL and ELL to finish the last layer.
First, pair up the edges; 3-2-2 pairing (start with a u, place three edges and do a u', then do 2-pairing for the rest) is suggested. Next, solve the F2L and LL with whatever variant of Fridrich you are most comfortable with.
Solve the corners and then finish the rest of the edges two at a time with commutators.
Place the first layer's corners along with as many edges as you can, using F2L techniques. Then use OLL and PLL to try to place as many last layer edge pieces as you can. Finally, solve the remaining edges two at a time with commutators.
The Method(s) In Practice
The three well-known 4x4 methods built around this are K4 (with two major variants) and Yau. The standard version of K4 uses the "K4 style" cross and the "traditional K4" rest, although it has the slight optimization of doing the four first-layer corners during the cross step. There is also a variation of K4, sometimes used by Dan Cohen, which is a "K4 style" cross followed by an "F2L-style K4" rest. Finally, the Yau method uses a "K4 style" cross with a "reduction" rest.
Sub-minute averages have been achieved with all three of these methods, and Dan Cohen has even broken the 50-second barrier using Yau. Yau was also the method he used to achieve the current WR single of 36.46 seconds. It is clear that this meta-method has some serious potential; all that remains is to practice a lot