# Rubik’s Revenge

Rubik’s Revenge, solved

The official 4x4x4 magic cube is sold under the Rubik’s brand as the Rubik’s Revenge or the Rubiks 4x4x4. There are heaps of other brands that sell this puzzle, and most follow the 4×4 naming convention; for example, the Eastsheen 4x4x4, Shengshou 4x4x4, QJ 4×4, LanLan 4×4, GhostHand 4×4, and so on.

The Rubik’s Revenge has all the characteristics of the magic cube series, and of face-turning cube puzzles in general. For example, the face center axes of rotation are a generating set for the isometries of the cube, so any piece can be placed in any isometric position. Note, however, that edge pieces are off-centre, so an edge piece is not isometric to it under a 180° rotation. This means it is not possible to place a piece in position but flipped, and equally it is not possible to place a piece in its neighbouring edge position without flipping it. This is not easily apparent, as neighbouring edge pieces are identical, so between the two of them you can get both orientations in both positions.

### Solving by reduction

Solving squares of face centres and matching up edge pairs reduces the Rubik’s Revenge to a Rubik’s Cube.

The most common approach to solving the Rubik’s Revenge is to solve each square of centres, and then match up each pair of edges. This reduces the Rubik’s Revenge to a Rubik’s Cube— a lovely example of reducing a group to a subgroup. This reduction isn’t without its difficulties however:

• Since this puzzle has an even number of layers, there is no fixed centre piece on each face: every centre piece can be moved from face to face. It is therefore possible to solve each square of face centres on the wrong faces! Without a fixed centre piece for guidance, care must be taken to ensure that the squares of face centres are correctly positioned.
• Rubik’s Revenge with OLL parity — a pair of edges need to be flipped.

It is impossible to flip a single Rubiks Cube edge, but one can flip a pair of Rubiks Revenge edges by placing each one, flipped, in the other’s position. Doing so means the Rubiks Revenge will be reduced to a Rubiks Cube that is in the wrong orbit. Parity errors like this one usually won’t be apparent until you’re near the end of your attempt to solve the reduced Rubiks Revenge using the usual Rubiks Cube algorithms. You will then need a couple of additional algorithms that flip or swap edge pairs without disrupting the rest of the cube.

## Number of positions

According to Wikipedia, the Rubik’s Revenge has 7,401,196,841,564,901,869,874,093,974,498,574,336,000,000,000 possible positions; on the short scale you would call this about 7.4 quattuordecillion.

## God’s Number

A Rubik’s Revenge in one of its 7.4 quattuordecillion positions

The question of God’s Number for the Rubik’s Revenge is wide open. It has been shown that some positions cannot be solved in less than 30 moves, and it has also been shown that there are no positions that cannot be solved in 68 turns or less.

## Variations and related puzzles

### Shell variants

There are several shell variants on the Rubiks Revenge:

• The Rhombus Dodecahedron is a vertex-turning rhombic dodecahedron puzzle with a Rubik’s Revenge core. Since the rhombic dodecahedron has the same rotational symmetry as the cube, this is a non-jumbling puzzle. What makes it different from the Rubik’s Revenge is that the centre pieces of the Rubik’s Revenge, which do not need to be oriented, are edge pieces in the Rhombus Dodecahedron, and therefore must been oriented correctly. On the other hand, Rubik’s Revenge edges correspond to the face centres in the Rhombus Dodecahedron, so their orientation doesn’t matter. This eliminates one of the two parity errors encountered when solving the Rubik’s Revenge by reduction to the Rubik’s Cube.
• The 4×4 Dodecahedron has uniformly coloured square faces and no edge pieces at all, making it relatively easy to solve.

The 4×4 Dodecahedron is a 4x4x4 face-turning puzzle in the shape of a somewhat irregular truncated rhombic dodecahedron; that is, a rhombic dodecahedron in which the order-4 vertices have been cut off. This shape can also be created by cutting the edges off a cube, and for this puzzle that is a much better way to visualise it. It is essentially the same as the Rubiks Revenge, except that the edges have been cut off. Note the misnomer: a dodecahedron has 12 faces, whereas a truncated rhombic dodecahedron has 18. The name is somewhat justified by the fact that the six square faces on the 4×4 Dodecahedron are all the same colour, so do not need to be solved. This, together with the complete absence of edge pieces, renders the 4×4 Dodecahedron much easier to solve than the Rubiks Revenge.
• The Scopperil is an extremely difficult shell variant.

Closely related to the two above, the Scopperil (or Gyro Top) is shaped like a rhombic dodecahedron in which the order-3 vertices are cut off. Since the term truncated rhombic dodecahedron is usually reserved for the rhombic dodecahedron with the order-4 vertices cut off, I don’t know what to call this shape. This shell variant is extremely difficult to solve, at least for those who persist in solving it by reduction to the Rubik’s Cube. The four edge pieces that correspond to the Rubik’s Revenge centres are all different colours even when solved, so it is very hard to keep track of them and figure out when they are “solved”. The face pieces that correspond to the Rubik’s Revenge edge pieces are also not the same when solved. Matching up edges in the Rubiks Revenge requires repeatedly breaking and restoring both centres and edges, and this step is mind-bogglingly difficult to keep track of on the Scopperil.
• The 4x4x4 Octahedron — equivalent to the Scopperil

The 4x4x4 Octahedron is a 4x4x4 vertex-turning octahedron puzzle. Since the octahedron is the dual of the cube, a vertex-turning octahedron puzzle is built around the same core as a face-turning cube puzzle. Considering both the core and the number of layers is the same for the 4x4x4 Octahedron as for the Rubiks Revenge, we might expect the two to be quite similar. This is precisely the case: the 4x4x4 Octahedron is equivalent to the Scopperil.

### Related puzzles

The Rubiks Revenge is the 4x4x4 in a long series of magic cube puzzles, ranging from the 2x2x2 Pocket Cube, through the 3x3x3 Rubiks Cube and the 5x5x5 Professors Cube, and beyond. There are also cuboids with a non-cubic aspect; for example the 2x3x4.