Optimized Blockbuilding LBL
also known as: OBLBL
There are a bunch of layer-by-layer (LBL) methods for big cubes out there, and the big problem is that they're all slow and commutator-heavy. In general, you quickly end up with very little freedom, and then waste a lot of moves making sure you don't break the solved parts. There are also a bunch of blockbuilding methods that people have tried, where you build a large block of solved pieces (often building up to a 4x4x5 block), and those have a big problem too: it's too hard to find the pieces, and keep track of what moves are available and how to move them around. I'd tried both types of method a long time ago, and although I sometimes used them for fun I never really thought either style had the potential to be fast.
But recently I started playing around with the idea of building a LBL method and leaving the front face free - still building up from bottom to top, but using the front face's freedom to insert pieces efficiently. It was pretty fast, and after playing around with it I realized that making the layers in a blockbuilding style would save a lot of time and moves over inserting pieces haphazardly. A few modifications later, this method was born. Like many other fast bigcube methods, this method tries to cut down on slice moves.
The first step is pretty basic - just build two centers, on opposite sides of the cube. I think it's best to do one full center and then the other, but you can try something else if you want. This is similar to the first part of Reduction, so if you need help look up a tutorial for that.
Choose one center, hold it on the left, and add on a block made of three paired edges and two corners. To pair edges, you can do any outer-layer moves to set things up, and then r and l moves to pair up the individual pieces. There are two basic ways to build the block; you can place one edge, and then place the corners while placing the next edges, or you can place all three edges first and insert the two corners later. Keep in mind that you can use any 3 of the 4 possible edges for this, but you'll want to keep track of which one you didn't use, so you can do the next steps without wasting time pairing the wrong pieces.
At this point, you're ready to start adding layers to the block. Hold the block on the left, so that the top layer is free to move. Because you can now freely turn all layers on the R-L axis (except the leftmost one) and the top layer, this whole section will use almost exclusively M-layer moves (such as M, l, r) and U turns. This part of the block goes through three centers and two edges; what this means for you is that each layer of the block will contain three rows of centers, and two individual edge pieces between them. For this layer, nothing else is built, so you have the greatest freedom. Remember that you have to make three rows of centers (each with two pieces on 4x4x4 and three pieces on 5x5x5) and then insert the correct two edge pieces (there will be two with each pair of colors, so be careful).
Here's how I personally do this step, looking at the colors on the left block. First, I'll use the top face to make a row of centers with the front color, and then do a 2L' to place it on the back. From now on, I try to avoid turning the second-to-left layer when I'm building pieces. Then, I use the top face again to make a row of centers with the bottom color, and find the front-bottom edge and insert it using outer-layer moves. Finally, I do another 2L' to place the bottom color on back, and build a row of centers with the back color on top, and insert the last edge using outer-layer moves. A final 2L' solves this layer.
This step only happens on the 5x5x5, so if you're solving the 4x4x4 you should skip past it. Here, we build the middle block layer, but this time it's different because there are now fixed centers to work around. I use a similar strategy to Step 3, using M-layer moves as rarely as possible, and moving each row to the back face when I'm done working with it. It's important to notice here that the edges are all middle edges, which means that although there's only one of them it does have an orientation. So be sure to insert it the right way.
This is the final part of the block, and after this step all the centers will be solved. The only layers you can work with here are r and U, so you will be doing a lot of trigger-type moves such as r U r'. The general strategy is the same as in Step 3, though: build rows of centers, place them in the back when you're done, and insert an edge whenever you're done with the two rows aroud it. Keep in mind that since this is the last of the layer steps you will be using every center of the front, bottom, and back colors, and you will also be using the final piece of each edge (although this means that there's only one possible edge, which makes it easier to find).
Alright, pairing edges. This is a reduction step, which means that nothing is really solved, but you pair stuff up to make the next steps easier - after this step, you will be treating the cube like a 3x3x3, and only using outer-layer moves. When I do this, I put the two unfinished faces on U and F. I use a u or u' move to make a pair, and then a trigger (like R U R', F' U F, etc) to put another edge in its place so that we make another pair when we undo the u layer move. This approach makes it difficult to use the edge on DF, so I usually try to pair one edge first and then place it there to get it out of the way. You may prefer to use a more freeform approach to edge pairing. Overall this step is similar to the pairing in Reduction (specifically 2-at-a-time 4x4x4 solving and AvG pairing on 5x5x5), but with less freedom because you don't want to break up the block.
On 5x5x5, half the time you will end up with two edges swapped, and you can't do this with the normal 3-cycle method. For this case, my preferred algorithm is (r U2 r U2) (F2 r F2) (l' U2 l) U2 r2.
This step is identical to Step 3 of Petrus, and that site explains it very well. The idea here is to flip the edges so that they can all be solved with moves of only two faces (R and U). For this step it's best to have the two remaining faces on U and F.
On 4x4x4, half the time you will end up with an odd number of edges flipped. If that's the case, you can flip only one edge by using r U2 x (r U2 r U2) (r' U2 l U2) (r' U2 r U2 r' U2) r'.
Rotate the cube so the two remaining faces are on U and R. Then, solve the rest of the F4L (really the F2L, looking from a 3x3x3 standpoint) using only moves of those two faces. This is identical to Step 4 of Petrus, so if you have trouble you should check that page out.
This is the same as the last layer step in a 3x3x3 method, although all of the edges will be oriented already. The two basic ways to do this are OCLL/PLL and COLL/EPLL, and you should use whichever one is closest to what you current feel comfortabe with. It's also possible to do this part in one step (this is known as ZBLL) - but there are about 300 different cases, so I don't recommend it.