I have a 4x4 Rubik's cube (aka Rubik's Revenge), which I can solve by reducing it to a 3x3 and then solving the 3x3. However, this can result in "parity" cases, where a single edge of the 3x3 is flipped. To resolve these, I have to remember a complicated and unintuitive algorithm, which I find unsatisfying.
Because of this, I would like to learn a different method for solving the 4x4, which doesn't rely on reducing it to a 3x3. I know that such methods exist, but I haven't had much luck finding any good explanations of them. All of the explanations I can find are for variants of the reduction method. (E.g. the cage method, Yau method, etc. all involve pairing up the edges and then solving as a 3x3.)
Note that speed solving isn't really my thing. My goal here is to be able to forget about the cube completely and still be able to solve it in several years' time. (Perhaps with a bit of work, but without looking anything up.) I can solve the 3x3 this way, just by knowing a bunch of fairly intuitive algorithms that are enough to cover all cases. I'm looking for a similar set of algorithms for the 4x4.
(I did try reducing it to a 2x2 instead of a 3x3, which presumably would avoid the parity cases if I could get it to work. However, I didn't have much luck with this, as I can't assemble the last few blocks of the 2x2 without messing up the already completed ones. If someone has a way to do this it would be a nice solution.)