Rubiks Revenge
From charlesreid1
Contents
Rubiks Revenge
4x4 Rubiks Cube
Solution Procedure
I use a modified "beginner's method" to solve the 4x4. This involves reducing the 4x4 to a 3x3, solving the 3x3, then dealing with parity issues in the last step.
Step 1: Centers
Start by aligning the four cubies in the center of each face. White/yellow, blue/green, and red/orange are all on opposite faces. Line up four of the six faces in any order, but when lining up the last two faces, make sure you have things oriented correctly! Red is to the right of green, etc.
Step 2: Align DoubleEdges
Next step is to align the two cubies in each doubleedge so that they have the same color and orientation.
Start by aligning the four doubleedges on the top of the cube, then align the next four doubleedges on the bottom of the cube.
The last four doubleedges can be aligned as well, but will require a slightly different algorithm to keep from undoing the doubleedges that are already oriented.
Step 3: Solve the 3x3
At this point, things get a little easier. Now that the centers are a solid color and the double edges are all matching, the two middle cubies of the 4x4 cube can be treated as a single cubie, turning the 4x4 cube into a 3x3 cube. You can now quickly solve the bottom three layers of the 4x4 cube by simply applying 3x3 algorithms (line up the cross on the bottom, arrange the corners, then align the middle two layers.
Step 4: Last Layer DoubleEdges
The last layer is where things diverge from the 3x3 cube.
When solving the cross on the last layer of a 3x3 cube, there are 4 possible configurations. These four configurations can be cycled through by repeatedly applying a single algorithm. The four configurations are:
 A single square (four squares) on the top is inplace
 Three squares (eight squares) on the top are in place (forming an L shape)
 Three squares (eight squares) on the top are in place (forming an I shape)
 Five squares (twelve squares) on the top are in place (forming a cross)
This arrangement will occur some of the time on the 4x4 (this is the "easy case"  solving the 4x4 is exactly like solving the 3x3, no new algorithms required).
Rubiks Revenge: Parity Situations Occurring On 4x4 Or 3x3  

Center Square Configuration:  EightSquare L Configuration:  EightSquare I Configuration:  TwelveSquare Cross Configuration: 
On the 4x4 cube, there are two sets of configurations. The first set are the four configurations listed above, that occur on the 3x3. The same 3x3 algorithms can be applied to the 4x4 as to the 3x3. However, the second set of configurations is:
 Ten squares on the top are in place (forming a T shape)
 Six squares on the top are in place (forming an incomplete I shape)
The same algorithm that cycles through the four configurations on the 3x3 can be applied to these configurations on the 4x4 to switch between them. Depending on the orientation of the cube, the T shape can be permuted into another T shape, the incomplete I shape can be permuted into another incomplete I shape, or the T and incomplete I can be turned into one or the other.
You can always get to the T shape from an incomplete I shape. There is an algorithm that fixes the one side of the T shape that is not oriented correctly. This is the situation where there is a single doubleedge that is "inside out."
Rubiks Revenge: Parity Situations Occurring On 4x4 Only  

Single Dedge InsideOut (T configuration):  Three Dedges InsideOut (Incomplete I configuration):  
Step 5: Last Layer Corners
When solving the corners of the last layer of a 3x3 cube, only two situations can occur:
 All four corners have the correct matching colors, and simply need to be reoriented to solve the cube.
 One of the four corners has the correct matching colors, and three corners need to be swapped/cycled.
However, in a 4x4 cube, because there are an even number of cubies, you can end up with one additional situation:
 All four corners have the correct matching colors, and simply need to be reoriented to solve the cube.
 One of the four corners has the correct matching colors, and three corners need to be swapped/cycled.
 Two of the four corners have the correct matching colors, and two corners need to be swapped/cycled.
The case of two swapped corners requires a special algorithm.
Rubiks Revenge Corner Situations  

Two Swapped Corners and a Swapped Dedge:  Two Swapped Corners:  
Algorithms
Last Layer Algorithms
Parity Algorithms  

Fix Two Swapped Corners:  Start with the cube oriented with the two swapped corners on the topleft and topright corners of the front face. Then execute the algorithm:
R U' R B2 L' D L B2 R2 U 2R2 F2 2R2 f2 2R2 2F2  
Fix Single InsideOut Dedge:  Start with the cube oriented with the insideout dedge at the top of the front face. Then execute the algorithm (broken into pieces to make it easier to remember):
r' D' 2U' u' r 2F r' u r 2F' r' 2F' u' 2F u 2F 2U D r 
Flags
Rubiks Cube notes on solving 3x3 and 4x4 Rubik's Cube
Cubes and Solutions: 2x2 Cube: Pocket Cube 3x3 Cube: Rubiks Cube · Rubiks Cube/Layer Method 4x4 Cube: Rubiks Revenge
Combinatorics and Permutations: Counting cube permutations: Rubiks Cube/Numbers Representation of cube permutations: Rubiks Cube/Tuple Permutation Algebra: Rubiks Cube/Permutations
