3^4

Roice Method

This method is the 4D version of the "ultimate solution for the rubik's cube's.
check it out here: "The Ultimate Solution to the 3^4"

Notation

View-based

The notation is similar to that of the 3^3. There are 8 cells, six of them using the same letters as that in the 3^3: U (up), D (down), F (front), B (back), R (right), L (left). The "near" and "far" cells are named A (ana) and K (kata), respectively.
Each cell in turn uses WCA Official Notation (standard notation).
The cell to be rotated is first stated, followed by the rotation. For example:
RR means to click the right face of the right cell.
AF means to click the front face of the far cell.
To state general moves (move subsets), use -. For example: R- means the set of all possible turns of the right cell.
<RR,A-> moves mean to only use RR and A moves. This method is used by Matthew Sheerin. This type of move can replicate 3^3 algorithms on one cell only (i.e. the far cell). An example of this is RR AF RR AF' RR' AF RR' AF'. This is the <RR,A-> version of R U R' U'.

Face-Based

The letters are the same as that for view-based notation, but the second letter is determined by the face the piece to turn is connected to. For example:
RR is invalid. RA means to click the face connecting the right cell to the far one.
AF means to click the face connecting the far cell to the front cell.
Order does not matter in face-based notation.

Sheerin-Zhao Method (Hybrid)

This method is an attempt to make a 4D analogue of the Friedrich, or CFOP method. Using this method results in around 650 to 900 moves, even if no shortcut turns are used. This method needs significant improvement because of the move count required for the last layer.

Prerequisites

• Knowledge of how the cube rotates.
• The Friedrich Method (F2L/OLL/PLL). 2-look OLL/PLL is enough.

Summary of the Method

1. Cross: Make a cross by solving 8 2C pieces on the far cell.
2. F2L: Fill in F2L slots by joining F2L pairs (2C/3C) together and inserting them into their respective slots.
3. S2L: Fill in S2L slots. This is done by moving a 3C piece and its respective 4C piece onto the farthest cell (i.e. the last layer). Then, OLL algorithms are used to orient the pieces so that they can be joined using <RA, AF> moves. They are then inserted by using more <RK, AF> moves.
4. OLL: Orient the LL 2C pieces, then the 3C pieces, using OLL-C (corner OLL) algorithms, such as the sune/antisune. Try to get as many corners oriented as possible as well. Setup moves are highly recommended here.
5. Pre-PLL: Permute the 2C of the LL using U-perms.
6. PLL: Solve the LL like a 3^3 by using <RK,AF> moves.
7. Parity: If the "top face" of the LL is 180 degrees off from the rest of the puzzle, use the <RA,AF> variant of the supercube centers algorithm (R U R' U five times)

Method

Cross

1. Pick the colour that you will be using for the near cell (the cell not inside the viewpoint). In this example, cyan is chosen.
2. Rotate the cube so that the cell with that specified colour is now the far cell.
3. Twist the puzzle so that all 2C (face) pieces connecting to the near cell are oriented and permuted correctly.
   This image shows the solved cross.
   

F2L

1. Find any 2C piece that does not include a sticker with the colour of the far cell (i.e. grey).
2. Find the 3C piece with the stickers the same colour as that of the 2C piece's and of the near cell. For example, if the 2C piece's colours were red and yellow and the near cell piece was cyan, the 3C piece would be red-yellow-cyan.
3. Intuitively align the two pieces so that they lie on the same slice on the far cell.
4. Join the pair together and insert the pair into the slot using -U moves.
5. Note: It is possible to insert the slot flipped. There are many shortcuts in forming pairs too. The image below shows the flipped pair.
   
6. Here is what it should look like when you are done:
   

S2L

1. Find any 3C piece that does not include a sticker with the colour of the far cell (i.e. grey).
2. Find its respective 4C piece (see F2L for details).
   note: try finding pairs that are already on the far cell first.
3. If either piece is already inserted and there are no more pairs on the far cell, go down to the insertion step.
4. Orient the pieces so that the same colours are on the far cell. Sune and antisune is used. Setup moves can help.
5. Join the pair together using <RK,A-> moves.