Centers Last (old)

(Disclaimer: I wrote this guide in June 2007, on twistypuzzles.com. Here it is, with just a few edits. I haven't changed up the method much if at all since then.)

This is a method for solving arbitrarily large cubes. I decided to make a topic because I want to keep a permanent reference for this method, and because I just recently improved it a bit and I thought I'd like to share that with you.

From personal experience, it becomes faster than the reduction method for cubes of a certain size or greater, but that size is determined by a lot of factors (such as how fast you can make slice moves and how good you are at finding pieces for reduction). Suffice it to say, though, that if I ever attempt a truly huge cube (say 40x40x40) I'm not going to use reduction. This method does use about 30-50% more moves than a reduction strategy for a given size of cube, though, and although you have better recognition you will need more physical endurance to be able to complete a cube with this strategy.

The basic method is just this:
1) Solve all of the edges and corners. They must be solved relative to the centers on an odd cube.
2) Solve all of the centers without disturbing the edges and corners (much).

There are a lot of subtleties, though. Each step has many different ways to complete it, and there are a lot of small (but very useful) shortcuts. Throughout this guide, I'm going to use the r* notation; what this means is, on an NxNxN cube, to do a clockwise turn on all R-side slices from the second slice to the floor(N/2)th slice. For example, on a 6x6x6 cube r*' would mean to turn the 2nd and 3rd R slices counterclockwise. This is useful for generalized algorithms.

My current step 1 is done by matching up pieces of edge groups. I'll mostly be using 3x3x3-like moves for this part, so think of everything as corners, edge groups, and center groups. Using only the outer layers, move an edge group with a lot of similar pieces to DF (but if you can't find one in 4-5 seconds don't bother), and choose a color pair of edges that you will place in that spot. Make sure to remember which color is on top. Now, looking at the R and L layers, find edge pieces which match that color pair and move them to UF or UB using R/U/L moves, then shoot them down to DF with the appropriate slice move. Once DF is completely grouped up, you can do something like R' L F to move it into the R or L layer; keep doing this, while never disturbing the previously grouped edge groups. When you have 7 of them, you will not be able to do this any more; then, move the M layer to the E layer (just do z), and group another edge group together using horizontal slices. You can use moves like RUR' to get the fifth unfinished edge group into E. When you are done, move the finished group into U or D. At this point, U and D should each have 4 finished edge groups. You can actually finish one more edge group in E quickly (remember, you can do things like R2 because centers are irrelevant) before you have to do moves like (vertical slice) R F' U R' F (vertical slice)' to finish off the remaining edges. Note that you can only be sure of solving 11 of 12 edges like this. Don't try to use last-two-tredges or parity algorithms yet, though.

When you have paired up at least 11 edges, it's time to solve the edges and corners into a cage. Do it like the 3x3x3 step. Make sure it's solved relative to the centers if you're doing an odd cube. Here is a trick: when you get to the orient step (OLL for Fridrich, step 3 for Petrus) it's very helpful if the last edge group, the one which will almost certainly have parities in it, is one of the edges you are going to orient. If you're not color neutral, you can force this easily: for example, I solve Fridrich with a yellow last layer, so I can simply pair up all of the edge groups which don't contain yellow first, which will ensure that my parity edge is in the last layer when I get to it. So fix all of the orientation parities at the same time, then do OLL, then fix the permutation parity if you need to, then do PLL. The permutation parity is very easy, by the way: just do r*2 U2 r*2 U2 r*2.

So edges and corners are all solved; now onto step 2. In this step, we use commutators to solve all of the centers. The goal is to solve one center completely, then the opposite center, and then any other center, an adjacent center, and the last two. But again, there are a lot of subtleties and tricks here. First of all, a lot of people tend to use commutators like r U' r' d r U r' d'. There's nothing wrong with that, but I much prefer commutators in the form of r U l' U' r' U l and r U' l' U r' U' l. These commutators are very fast to execute because all of the interior slices are on the same axis, and you can see what is going on by only looking at two faces (where the centers are coming from and where they are going), leaving you free to look for the next center on an adjacent face. You'll have to experiment with this algorithm to get a feel for it, but you should know a couple things: (a) you can vary the affected centers in the 3-cycle by varying which slices you use (and the r- and l-like slices don't even have to be the same depth, i.e. r U 3r U' r' U 3r works fine); (b) x-centers can only be solved by one of these, but others can be solved by both; (c) the first move of the commutator should move a center piece that you want to move to the U layer into a space in U that does not already have a U-color center in it; (d) it's possible to bring entire blocks up to U by using more than one slice in place of one (or both) of the component slices of the commutator, but the first move has to bring the block to a place in U that is mostly or fully unoccupied, and some blocks are simply too large to do this with; and (e) it's best to only use setup moves involving U and only try to solve one piece or block at a time, so that the edges and corners don't get further than an AUF away from solved.

Now that we've gone over the small-scale details, let's concentrate on the large-scale ones. When you're solving the first center, pieces can come from all five centers, but you will notice that bringing them from the opposite center is time-consuming for a couple of reasons, notably 180-degree turns and slow recognition. There is a solution, however: the move r* F2 r*' l*' F2 l* will swap the F and D centers, completely on an even cube and mostly on an odd cube, although it messes up some edges so you will have to undo it after you solve the U center. So one way you could do the first center group (say white) is to bring every white center cubie from one face into the white face, then swap that face's center group (which contains no white cubies now) with the center group opposite white, and finally just bring all of the white centers (from the center group that used to be on bottom) onto the white face before reswapping the two center groups. Personally I like to hold the target face on U, and then bring all the target center pieces on F and R into the target face before doing y2 and doing it again. Once you have the first center, the opposite center is the same, but this time every center piece you want to bring in will be in an adjacent face, which makes it a little easier.

Now, you have four centers left to do. Pick one and hold it on U, then transfer the target pieces from F to U, swap F and D, transfer the target pieces from F to U, swap F and D, do y2, transfer the target pieces from F to U. If you want to be fast, this should be mostly mechanical, as it's a lot of the same type of move done from more or less the same angle. Now pick an adjacent face and hold it on U, with the just-finished face on B. Transfer the target pieces from F to U, swap F and D, transfer, swap. Remember that if you do one swap, you must do a second one, because otherwise the edges will get messed up. Now hold the last two centers on F and U (the order is irrelevant) and finish the cube. You're done!

By the way, if you want to use this on cubes smaller than 6x6x6, there are a couple of key differences. Depending on how good you are at reduction, it may be faster for you to just hold the edges on the E layer and pair them up normally. If that works for you, do it. Also, on the smaller cubes, you shouldn't bother with swapping centers, because there are not enough centers to swap to make it worthwhile. If you want to reduce the number of centers you have to move from one face to an opposite one, you can try to find a face that has as many centers as possible on that face and the adjacent ones, and start solving there. But unless you are very slow at the centers part of reduction, this will probably be slower for you for small cubes, so I don't recommend it for the small ones.