Here is the solution to the 4x4x4 (Rubik's Revenge) puzzle. Please note that Tony Forbes provided a portion of the material seen here, and I am simply expanding on what I learned from him.
Let's begin with notation. The inner left column will be henceforth named "l" and the inner right will be named "r". the outer left and right will be named "L" and "R" respectively. The trend of lowercase letters determining the inner slices continues throughout. You will also hear me use terms like "center block" and "working face". A center block is the area that is bordered by the edges and corners of the cube (in this case the center block is a 2x2 area). The working face is the face that you're building. The "desired" piece is the piece that you want to bring onto the working face. I will also refer to a carrier piece. This is simply a piece that, in the centers solution, is brough up onto another face and brings a piece back down with it.
The first step is to solve one center. This is fairly intuitive, but if you have trouble you can position the center that you're working on as the F face. If there is already a piece of the color that you're working on in the central area of the front face and another piece of that same color on an adjacent face, you can rotate the cube so that the face that you're working on is in an F position and the adjacent face with the desired piece is in a U position. Then turn the F face to bring the carrier piece to a DL position (within the F center block), turn the U face to bring the desired piece to an FR position (close to you and to your right) and execute l'/U/l. This will bring the desired piece in with the other one(s). Repeat as necessary. If there is a piece that you want that is not on an adjacent face, simply turn the cube so that the face with desired piece is in an F position. From there, execute that same l'/U/l formula to bring it to a face that is adjacent to the working face. It is then possible to execute that same l'/U/l formula. Please note that if necessary it is possible to mirror or rotate the formula. To mirror, bring the carrier piece to a DR position in the center block, with the working face and adjacent face in the same position as described above, then execute r/U/r'. This will, once again, bring in the desired piece. You can also have the carrier piece in a UR/UL position and the desired piece in a BL/BR position (on the U face) and execute the formula and it will still work.
As you may have been able to tell, the center blocks of the 4x4 can change in relationship to each other. To determine which face goes where, solve the four corners of the face that you just solved. Then you can continue on to solve the rest of the face centers as described above.
The next step is to solve the edge pairs (the set of two pieces that are in between two corners). This can be achieved by finding two pieces of the same color and bringing them so that they are opposite from each other (you should see the same two colors on the same face in UL and DL positions). This part of the edge pair solving is intuitive. Once you have them in that position, execute the seven move formula l/F/U'/R/U/F'/l'. This will bring the two pieces together. Do this until all of the edge pairs are together. Once you have accomplished this, it is possible to solve like a normal rubik's cube.
You may encounter a parity situation at the end of the solve. Either there will be two edges that need to be switched or one that needs to be flipped. If there are two that need to be switched, invert the cube and have one of the switched edges facing you. Execute l2/U2/l2/Uu2/l2/u2. The Uu2 means to turn both the inner and outer U slices. If one edge needs to be flipped, invert the cube and bring the offending edge so that it is facing you. Execute l2/B2/l'/U2/r/B2/D2/l/D2/r'/B2/l/U2/B2/l2. This should solve the parity, though no guarantees are given on any formula seen here.
Let's begin with notation. The inner left column will be henceforth named "l" and the inner right will be named "r". the outer left and right will be named "L" and "R" respectively. The trend of lowercase letters determining the inner slices continues throughout. You will also hear me use terms like "center block" and "working face". A center block is the area that is bordered by the edges and corners of the cube (in this case the center block is a 2x2 area). The working face is the face that you're building. The "desired" piece is the piece that you want to bring onto the working face. I will also refer to a carrier piece. This is simply a piece that, in the centers solution, is brough up onto another face and brings a piece back down with it.
The first step is to solve one center. This is fairly intuitive, but if you have trouble you can position the center that you're working on as the F face. If there is already a piece of the color that you're working on in the central area of the front face and another piece of that same color on an adjacent face, you can rotate the cube so that the face that you're working on is in an F position and the adjacent face with the desired piece is in a U position. Then turn the F face to bring the carrier piece to a DL position (within the F center block), turn the U face to bring the desired piece to an FR position (close to you and to your right) and execute l'/U/l. This will bring the desired piece in with the other one(s). Repeat as necessary. If there is a piece that you want that is not on an adjacent face, simply turn the cube so that the face with desired piece is in an F position. From there, execute that same l'/U/l formula to bring it to a face that is adjacent to the working face. It is then possible to execute that same l'/U/l formula. Please note that if necessary it is possible to mirror or rotate the formula. To mirror, bring the carrier piece to a DR position in the center block, with the working face and adjacent face in the same position as described above, then execute r/U/r'. This will, once again, bring in the desired piece. You can also have the carrier piece in a UR/UL position and the desired piece in a BL/BR position (on the U face) and execute the formula and it will still work.
As you may have been able to tell, the center blocks of the 4x4 can change in relationship to each other. To determine which face goes where, solve the four corners of the face that you just solved. Then you can continue on to solve the rest of the face centers as described above.
The next step is to solve the edge pairs (the set of two pieces that are in between two corners). This can be achieved by finding two pieces of the same color and bringing them so that they are opposite from each other (you should see the same two colors on the same face in UL and DL positions). This part of the edge pair solving is intuitive. Once you have them in that position, execute the seven move formula l/F/U'/R/U/F'/l'. This will bring the two pieces together. Do this until all of the edge pairs are together. Once you have accomplished this, it is possible to solve like a normal rubik's cube.
You may encounter a parity situation at the end of the solve. Either there will be two edges that need to be switched or one that needs to be flipped. If there are two that need to be switched, invert the cube and have one of the switched edges facing you. Execute l2/U2/l2/Uu2/l2/u2. The Uu2 means to turn both the inner and outer U slices. If one edge needs to be flipped, invert the cube and bring the offending edge so that it is facing you. Execute l2/B2/l'/U2/r/B2/D2/l/D2/r'/B2/l/U2/B2/l2. This should solve the parity, though no guarantees are given on any formula seen here.
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