This page covers a solution algorithm for the 4x4 Rubiks Cube, a.k.a. Rubiks Revenge.

## Solution Algorithm Steps

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 Double-Edges

Next step is to align the two cubies in each double-edge so that they have the same color and orientation.

Start by aligning the four double-edges on the top of the cube, then align the next four double-edges on the bottom of the cube.

The last four double-edges can be aligned as well, but will require a slightly different algorithm to keep from undoing the double-edges 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 Double-Edges

The last layer is where things can diverge from the 3x3 cube.

#### Even vs Odd Parity on the Cube

On the 3x3 cube, and on any cube with an even number of cubies per edge, the total number of states that the cube can reach is cut in half because of the way the Rubik's cube is constructed - on an odd cube, the center faces cannot be rotated, so half of the states (those with the center faces rotated) are inaccessible without physically dismantling the cube.

On the 4x4 cube, and on any cube with an even number of cubies (greater than 2) per edge, there is no fixed center piece, so any face cubie can be swapped with any other face cubie. This means all states are accessible on the 4x4 cube.

#### Even vs Odd Parity on the Last Layer

Step 4 of the solution requires us to solve the last layer on the 4x4 cube. This solution procedure splits into two cases: an even parity last layer scenario, and an odd parity last layer scenario.

#### Odd Parity (3x3 Equivalent) Case

Let us first consider the odd parity case, which is easier because solving it is equivalent to solving the last layer of the 3x3 Rubiks Cube/Layer Method.

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 last-layer configurations for the 3x3 cube are:

• A single square (four squares) on the top is in-place
• 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: Eight-Square L Configuration: Eight-Square I Configuration: Twelve-Square Cross Configuration:    #### Even Parity (No 3x3 Equivalent) Case

There are a second set of outcomes that can occur on the last layer of the solve that have no 3x3 equivalent:

• Ten cubies on the top are in place (four center cubies and six double edge cubies), forming a T shape
• Six cubies on the top are in place (four center cubies and two double edge cubies), forming an incomplete/stumpy I shape

The algorithm that cycles through the four last layer configurations on a 3x3 will not solve either of these arrangements, but when applied to one even parity last layer case, it will result in the other even parity last layer case.

Rubiks Revenge: Parity Situations Occurring On 4x4 Only
Single Dedge Inside-Out (T configuration): Three Dedges Inside-Out (Incomplete I configuration):  Get to the T shape on the top layer before applying the algorithm to swap a single inside-out double edge to be right-side in. Each double-edge should be located on the correct edge (i.e., the two colors of the double edge cubies should match the two colors of the faces they connect) and all but one double edge on the last layer should be correctly oriented (i.e., the two color of the double edge cubies match the two colors of the faces they connect and are correctly oriented).

Once the cube has the T configuration on the top of the cube, orient the cube so the front top double edge is the inside-out double edge. Then apply the following algorithm to flip the double edge pieces.

Parity Algorithms
Fix Single Inside-Out Dedge: NOTE: Only apply this to the T configuration (above left).

Start with the cube oriented with the inside-out 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
```

### 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 re-oriented 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 re-oriented 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 Cheat Sheet

### Double Edge Alignment Algorithms

Double Edge Algorithms
Bring Together Two Opposite Double Edge Cubies This algorithm should be applied once all four top double edges and all four bottom double edges are solved, and there only remain four double edges to orient.

This algorithm should be applied when two double edge pieces are twins, but both are located across from one another on a cube face. Orient the two twin double edges so they mirror one another, then execute the algorithm:

```d R F'
U
R' F d'
```

(Note this algorithm is also applicable to the Professors Cube, which also has a solution step requiring double edges to be correctly oriented.)

### Last Layer Algorithms

Parity Algorithms
Fix Two Swapped Corners: Start with the cube oriented with the two swapped corners on the top-left and top-right 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 Inside-Out Dedge: Start with the cube oriented with the inside-out 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
```

## Patterns

It is also possible to apply algorithms to the solved Rubiks Revenge cube to create various interesting patterns.